Saturday, July 17, 2021

What goes up

My previous post managed to ensnare Ray Chandler into helping extend OEIS sequence A293355, which had reached term #384. Ray has access to "yafu" which allows him to factor large integers quickly. Sadly, I don't know how to get that program to run on my Mac. At any rate, thanks to Ray's persistent effort, he was able to extend the sequence past a local maximum at #398 and then down the other side to a point where the numbers were finally small enough that I could once again automate the procedure in Mathematica. The downslope reaches a local minimum at #923, then rises again, slowly and erratically:

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The ratios graph (that I had previously posted to term #384) extends now past term #3280:
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It clearly shows the differing "resonances" about which I had speculated.

Sunday, July 11, 2021

The unknown soldier

I have been cleaning up my primary iMac's virtual "desktop" for a week or so. One of the files thereon was a tabular listing of OEIS sequence A293355 up to term #291. When I submitted the sequence back in 2017, I included a listing of terms only up to #250, so clearly it was something I was extending at the time and hoping to extend even further. But other projects got in the way.

I had a look at term #291 and found that factordb had it fully factored in its database. If we have the full factorization of a sequence term, one may find the subsequent term by applying an algorithm to that factorization. If it's an easy factorization I can even automate the procedure in Mathematica and have it determine several terms at once. Unfortunately when one is trying to fully factorize a 90-plus-digit number, chances are that the second-largest factor is itself large (say, > 30 digits) and this makes it difficult for Mathematica to find in a reasonably short amount of time.

So I went back and forth (between Mathematica and factordb) yesterday trying to extend A293355 beyond term #291. Incredibly, every time I put a difficult-to-factor term into factordb it came back with the number fully factored! This was, clearly, no coincidence. Someone, three-or-so years ago, was also extending the sequence. Except that when they entered a term it did not come back fully factored. Instead they had the time-consuming task of factoring the number by other means. Whoever it was soldiered on, extending the sequence to term #384 which contained a 115-digit composite that my hero was apparently unable to resolve (unless, tragically, they succumbed in the battle). I submitted the updated, much larger "b-file" (an indexed tabular listing) to OEIS. The point of A293355 was to present an infinitary aliquot sequence (the one here starting with 6216) that seemed to evolve into monotonically increasing (never decreasing, by magnitude) terms. The actual term-by-term increases are much more interesting than the conjecture might suggest:

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Perhaps the sequence is just in some sort of "resonance" (for lack of a better word) that will eventually be disturbed by an unkind factorization — one that portends a much more chaotic future.

An abundance of additional terms (July 17): What goes up

Thursday, July 01, 2021

O Canada!

The Globe, Toronto, Saturday, 13 May 1911
Historic Home of a Vanished Canadian People
by Janet Grant Needham (1866-1939)

Canada is making history every day through her wealth of natural resources, agriculture, mines, and her tide of immigration. "Forward" is the keynote of our land from sea to sea. Nevertheless there is a spot in the very heart of this great Dominion where history has been made already; the thousands of her inhabitants have come and gone, their records sealed and closed. We, too, have our Herculaneum and Pompeii offering a rich treasure to the excavator. The savage Huron people are gone, but they have left traces of their race.

It seems a far cry to the summer of 1615, when Samuel de Champlain, in response to a promise he had made to a band of fur-trading Hurons at their rendezvous, "Falls of St. Louis", set out to assist them in their wars against the fierce Iroquois ("Voyage de Champlain", page 276). Besides giving aid from a sense of duty against these raiders, who terrorized every other tribe, he thought it would open a way for their conversion, for to Champlain "the saving of a soul was more than the winning of an empire". Notwithstanding, this promise was the fatal mistake of his Indian policy, producing the lasting enmity of the Iroquois and the ultimate extermination of the Huron race. He brought out from France the same year four Recollect friars, one of the three branches of the Franciscan Brotherhood. Father Joseph Le Caron elected to come to the Hurons. Arrayed in the habit of his order, a coarse grey garment girdled at the waist by a stout cord, a peaked hood, sandals of wood an inch in thickness, he gathered without delay his church equipment and left the spot where now stands Montreal, accompanied by twelve Frenchmen and a few savages. Eight days later, early in July, Champlain, with two Frenchmen and ten savages, followed. Both parties journeyed up the Ottawa and Mattawa Rivers, over a short portage to Lake Nipissing, near the spot where North Bay is built, down the French River, past Byng Inlet and the rocky east coast, and through the island enchantment of the Georgian Bay to its extremity, landing on the first day of August a few miles west of Penetanguishene. Passing through a country abundant with raspberries, grapes, plums, squashes, Indian corn and wild rice, through village after village of unpronounable names where now are Sturgeon Bay, Fesserton, Victoria Harbor, Coldwater and other towns, they reached a large village, Carhagouha, with triple palisades thirty-five feet high, situated near the locality where Waubaushene now stands. There they overtook Le Caron. In these wilds on August 12 Father Joseph said the first Mass in the Huron country, holding the Host high above his kneeling countrymen and the awe-stricken savages. A cross was planted at this spot. The explorer proceeded thirty-five miles to Cahaigue, the capital, with 200 cabins. The warriors assembled gladly welcomed him.

On September 1 all was in readiness, and 2200 warriors, decorated with war paint and with small supplies of smoked fish and wild rice, passed by Lake Couchiching through the Narrows connecting it with Lake Simcoe, out by the Trent waters to Lake Ontario. They crossed at the eastern end and soon stood before the camp of the astonished Iroquois, who defeated and put to rout the panic-stricken Hurons. The Iroquois never forgave this assault. Champlain, twice wounded, wintered with a friendly chief on the shore of Lake Simcoe. Towards spring he visited the Tobacco Nation, or Petuns, in the Collingwood district, then finally left these parts. This is our first knowledge of the Huron Nation.

"Quelles hures!" (What hyenas!) exclaimed an astonished Frenchman, attracted by the fantastical modes of dressing their hair, chief of which was the head shaved, leaving a ridge of bristles from the forehead over the crown to the neck, hence the word "Hurons". The original name was Ouendat. Tourists or the curious may see to-day at the Narrows (two and a half miles from Orillia) the "fishing weirs" used in Champlain's time and to which he refers. They are of heavy tamarac driven into the mud a great depth, exposing only six inches, and in an excellent state of preservation considering the lapse of nearly 300 years. The population at this time was 10000 huddled in eighteen villages varying in size from three or four hamlets to twelve acres, one even fifteen.

Let us look into a Huron cabin or long house. It is 100 feet long, built of young sapling supports placed at intervals, curving towards the centre, and over which are laid sheets of bark. A foot space is left the entire length at the top to admit light and for the escape of smoke more or less always present. The place is thronged with beings, for it is the home and common property of ten families, each averaging six. On the ground, at intervals down the centre, are five fires fed with fat pine logs. Round about are hung weapons, ornaments and clothing on long arms of wood. A broad shelf of bark, on supports, is built along both sides three feet from the ground, under which is firewood. Around the fitful gleams of the fires are grouped on a winter's night battle-scarred warriors and women, feasting, cooking, gambling, dancing or idly jesting; young men eager for the fray, girls decorated with ornaments, unruly children, aged, feeble squaws, the "burden-bearers", besides numerous half-starved dogs. During a snow-storm the smoke is so dense the inmates breathe with nostrils to the ground.

The brain of the Huron-Iroquois has been proved to be larger than that of any other tribes of American aborigines, with doubtful exceptions. They spoke the same dialect, used the same system of warfare, customs of marriage and ceremonies in burial of their dead; besides, their long houses were after the same style — all indicating that they were descendants of one common stock, though deadly enemies. The Huron Nation was divided into four great tribes or clans, located in townships as follows: — Bear clan in Tiny, Wolf in Tay, Heron or Cord people in Medonte, and Falcon or Rock people in Oro — overlapping, naturally others. (Tiny, Tay and Flos townships are named after Lady Colborne's three lapdogs.) The 35th Regiment, Simcoe Foresters, carry on their uniforms to this day the Bear totem. The Hurons were an agricultural people, clever and ingenious, but much behind the Algonquins in the work of ornamentation and bone needle work. Established custom with them was law, and, though proud, vindictive, selfish, great thieves and gamblers, they were witty and social. Much generosity and harmony prevailed amongst them, thus enabling thousands of these untamed creatures to live in peace. Sagard ("Le Grand Voyage aux Pays des Hurons, 1632") distinguishes the Hurons, Algonquins and Montagnais as the nobles, burghers and paupers of the forest. The Iroquois, more than the Hurons, reached the highest civilization possible under benighted conditions; in short, they were the aristocracy, yet this raised them very little above the animal. Mission work among the aborigines has many dangers and much to shock the finer feelings of the cultivated missionary, and that at our own doors proved no exception. The first settlers came in 1819, a band of fugitives from Lord Selkirk's Red River expedition. This was added to greatly in 1832 by other settlers. With the agriculturist at work, comes definite information regarding the haunts and habits of these people. Soon were brought to the surface great quantities of pottery, cooking utensils, strings of wampum, arrowheads, stone and French tomahawks. After the troubles of 1837 a great flood of immigration took place. So thickly settled now is the county that one is safe in saying the average farmer has given or thrown away bushels of curios and relics, apart from collections made by interested persons.

The public is greatly indebted to Mr. Andrew Hunter, M.A., of Barrie, for constant archaeological research in securing accurate information in Huron locations and "finds". The late David Boyle, and Mr. J.H. Hammond, Orillia, have also given valuable services. The latter has most of his collection of 3000 pieces in Toronto Museum. Perfect relics are now becoming rare. The Hurons occupied the higher lands and ridges. Over 400 sites have been located, though not all inhabited at the same time. These can easily be detected by the very black soil and ash pits three or more feet thick, though covered by forest. Farmers frequently come upon mounds or depressions in the ground which prove to be ossuaries or burial pits containing human skulls averaging 200, face downwards and to the east, justifying the belief held by many, from this and their ashpits, that the Hurons were sun worshippers. Their custom was to place bodies on scaffolds. At intervals of ten years the remains were collected by the clans and deposited in pits ten feet square, with wampum, axes, pipes, beaver skins, colored beads, native beads, French articles, parcels of hair, brass kettles (tomahawks in the bottom), and whatever was held most precious by them. The whole was accompanied by solemn yet frenzied ceremonies; says one "like demons from the lower world let loose". Strange to say, numerous Mexican shells were often found in these pits. An ossuary on the farm known as the Michael Braden property in Medonte was fifteen feet in diameter, containing from 700 to 1000 remains, also seventeen copper kettles two feet wide at the top, in good condition, and later used by the early settlers. An ossuary equally large, containing 1000, was found on the Oliver farm near Barrie. A great tree had grown over it. To date 140 of these ossuaries have been carefully catalogued, indicating a population of at least 25000.

The Algonquins, inhabiting more the southern district in small numbers, used single burial in circular pits, sufficient only for a crouching position. Only once is this found among the Hurons, and that in Medonte. There are evidences, too, of hasty burial conforming to neither form — where battles have been fought, arrows and tomahawks found.