OFFSET
1,3
COMMENTS
Carmichael (1921) defined this arithmetic function for the purpose of studying periodic chains that are formed by repeatedly applying the mapping x -> a(x) staring at a given positive integer. This results in a sequence that is analogous to an aliquot sequence.
Periodic chains of cycle 1 are the fixed points of this sequence. 1 and the even perfect numbers (the even terms of A000396) are fixed points. Are there any other numbers k such that a(k) = k?
If a(k) = k and k is even, then a(k) is even and so is sigma(k), and therefore sigma(k) = 2*k and k is an even perfect number. If k is odd, then it is an odd multiperfect number, and no odd multiperfect number above 1 is known.
More specifically, if a(k) = k and k is odd, then k must be a square, and an m-multiperfect number (number k such that sigma(k) = m * k), with m being an odd prime number that is the least prime factor of sigma(k). For example, if there is an odd triperfect number (A005820) then it is a fixed point of this sequence.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Robert D. Carmichael, Empirical Results in the Theory of Numbers, The Mathematics Teacher, Vol. 14, No. 6 (1921), pp. 305-310; alternative link. See p. 309.
Eric Weisstein's World of Mathematics, Aliquot Sequence.
Eric Weisstein's World of Mathematics, Multiperfect Number.
Wikipedia, Aliquot sequence.
FORMULA
EXAMPLE
The sum of the divisors of 3 is 1 + 3 = 4. The divisors of 4 are 1, 2, 4. 2 is the largest aliquot divisor of 4. Therefore a(3) = 2.
MATHEMATICA
r[n_] := n/FactorInteger[n][[1, 1]]; a[n_] := r[DivisorSigma[1, n]]; Array[a, 100]
PROG
(PARI) a(n) = {my(s = sigma(n)); if(s == 1, 1, s/factor(s)[1, 1]); }
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Mar 23 2024
STATUS
approved