Saturday, January 18

Anagrammatic sums

Éric Angelini asked about anagrammatic sums on MathFun on January 12: "Let a + b = c and a < b < c and a, b, c = anagrams of each other." Halfway down his sausage article, he lists the Gilles Esposito-Farèse calculation for 3- to 5-digit results: 1 @ 3-digit, 25 @ 4-digit, and 648 @ 5-digit. In that spirit, here are 17338 @ 6-digit results. I count 495014 @ 7-digit and 17565942 @ 8-digit.

The idea for these has been around a few years. Claudio Meller's A160851 appears to be a (currently) somewhat misguided attempt at enumeration, while Rajesh Bhowmick's A203024, fleshed out by Charles Greathouse, provides a seemingly complete listing of sums, including 9449 6-digit terms. My 17338 6-digit results yield only 9443 distinct sums. Why six fewer?

Apparently 6-digit sums are the first that allow the sums to be twice one of the addends (i.e., a = b). In A023086 we see that there are twelve such. It turns out that six of these are the six that are not in my 17338 sums (because I did not allow a = b):

251748 = 2 * 125874
257148 = 2 * 128574
285174 = 2 * 142587
285714 = 2 * 142857
517482 = 2 * 258741
825174 = 2 * 412587

The other six are included because they each had an alternate solution:

517428 = 2 * 258714 = 241587 + 275841
571428 = 2 * 285714 = 142857 + 428571
571482 = 2 * 285741 = 158724 + 412758
825714 = 2 * 412857 = 241587 + 584127
851742 = 2 * 425871 = 127584 + 724158
857142 = 2 * 428571 = 142857 + 714285 = 275418 + 581724 = 285714 + 571428

Addendum: Éric and Gilles had created A331468 for their triples.

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