Thursday, February 2

With multiplicity

It may be noted in Tuesday's "what's so special about 10928094208 in base 77" A281335 curio that the factorization contained an exponent. Arithmeticians have another way of expressing factors that have greater-than-one exponents and that way is called "with multiplicity". For example, in base 43 the smallest solution to A281335 is 10969263:

(3,8,41,23,6) = 3^6 * 41 * (8,23)

But if we insist that the factors be expressed with multiplicity:

(3,8,41,23,6) = 3 * 3 * 3 * 3 * 3 * 3 * 41 * (8,23)

Which is no longer a solution. So I created another OEIS sequence, A281336, to deal with that situation. In base 43 the smallest solution to A281336 is 12505821873:

(1,42,2,41,3,29,35) = 3 * 29 * (42,1,41,2,35)

Yes, there is no multiplicity in this particular example but that is not a necessity. As with my other sequence, I have an even bigger number as a solution for a smaller base (37) but have yet to discover it.

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