Journal of Integer Sequences, Vol. 22 (2019), Article 19.3.4

Integers That Are Sums of Uniform Powers of All Their Prime Factors: The Sequence A068916

Spencer P. Hurd
Professor Emeritus of Mathematics
The Citadel
Charleston, SC
and P.O. Box 197
Meigs, GA 31765

Judson S. McCranie
105 Mackqueen Dr.
Brunswick, GA 31525


For integers s ≥ 1 and n ≥ 2, we define the function Ts as follows: Ts(n) = Ts(pa qb ... rc) = aps + bqs + ... + crs. Thus Ts(n) is the sum of the sth powers of the prime factors of n, counted according to multiplicity of the prime factors. The set T*(s) is defined as { n : Ts(n) = n }, and we let a(s) be the smallest element in T*(s). We consider several natural questions. Is the set T*(s) empty, finite or infinite for some particular values of s? Suppose y is a prime power, say y = pm. Is it possible that y = Ts(y) for some s? What is the smallest element a(s) in the set T*(s)? The answer for the last question is documented, but only for certain small values of s in the title sequence, a(s), for s = 1, 2, ..., namely sequence A068916 in the Online Encyclopedia of Integer Sequences. It begins 2, 16, 1096744, 3125, ... . Some sets T*(s) are known to have one or two elements, and T*(1) is infinite. Some sets have prime powers. In fact, infinitely often T*(s) contains py for some power y and prime p. For example T*(1) contains 327, which may be the value of a(24). The set T*(3) contains six known elements, and none of these are prime powers. We prove T*(3) does not contain any prime powers at all. Curiously, every known member of T*(s) for any value of s, except s = 3, is in fact a prime power. We also briefly discuss algorithms and functions related to Ts(n).

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(Concerned with sequences A067688 A068916 A268036 A268594 A318606.)

Received August 20 2018; revised versions received March 20 2019; March 21 2019; May 13 2019. Published in Journal of Integer Sequences, May 18 2019.

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