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

Professor Emeritus of Mathematics

The Citadel

Charleston, SC

USA

and P.O. Box 197

Meigs, GA 31765

USA

Judson S. McCranie

105 Mackqueen Dr.

Brunswick, GA 31525

USA

**Abstract:**

For integers *s* ≥ 1 and *n* ≥ 2, we define the function
*T*_{s} as follows:
*T*_{s}(*n*) = *T*_{s}(*p*^{a} *q*^{b} ... *r*^{c}) = *a**p*^{s} + *b**q*^{s} +
... + *c**r*^{s}. Thus *T*_{s}(*n*) is the sum of the *s*^{th} powers
of the prime factors of *n*, counted according to multiplicity of the
prime factors. The set *T*^{*}(*s*) is defined as
{ *n* : *T*_{s}(*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* = *p*^{m}.
Is it possible that *y* = *T*_{s}(*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 *p*^{y} for
some power *y* and prime *p*.
For example *T*^{*}(1) contains 3^{27},
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 *T*_{s}(*n*).

(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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