A371418 The largest aliquot divisor of the sum of divisors of n; a(1) = 1.

1, 1, 2, 1, 3, 6, 4, 5, 1, 9, 6, 14, 7, 12, 12, 1, 9, 13, 10, 21, 16, 18, 12, 30, 1, 21, 20, 28, 15, 36, 16, 21, 24, 27, 24, 13, 19, 30, 28, 45, 21, 48, 22, 42, 39, 36, 24, 62, 19, 31, 36, 49, 27, 60, 36, 60, 40, 45, 30, 84, 31, 48, 52, 1, 42, 72, 34, 63, 48, 72

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.

Periodic chains of cycle 2 are amicable pairs (A371419 and A371420). Are there any longer cycles?

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

a(n) = A032742(A000203(n)).

a(n) = A000203(n)/A071189(n).

a(n) = A000203(n)/2 if n is in A028983 (i.e., n is not in A028982).

a(k) = 1 if and only if k = 1 or k is in A023194.

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

Cf. A000203, A000396, A005820, A023194, A028982, A028983, A032742, A071189, A371419, A371420, A371421, A371422, A371423.

Sequence in context: A133904 A245182 A209160 * A340406 A257881 A268182

Adjacent sequences: A371415 A371416 A371417 * A371419 A371420 A371421

Keyword

nonn,easy

Author

Amiram Eldar, Mar 23 2024

Status

approved