A371421 -id:A371421 - cp's OEIS frontend

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

Amiram Eldar, Mar 23 2024

Carmichael (1921) defined this arithmetic function for the purpose of studying periodic chains that are formed by repeatedly applying the mapping x -> a(x) starting 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?

			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.

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.

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

r[n_] := n/FactorInteger[n][[1, 1]]; a[n_] := r[DivisorSigma[1, n]]; Array[a, 100]

a(n) = {my(s = sigma(n)); if(s == 1, 1, s/factor(s)[1, 1]);}

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.

A371422 Numbers whose aliquot-like sequence based on the largest aliquot divisor of the sum of divisors of n (A371418) terminates in a cycle of length 2.

Original entry on oeis.org

12, 14, 15, 23, 29, 42, 44, 48, 54, 56, 60, 62, 65, 66, 69, 70, 72, 75, 76, 77, 78, 83, 84, 85, 86, 87, 88, 90, 91, 92, 94, 95, 99, 102, 107, 108, 110, 111, 112, 114, 115, 117, 118, 119, 120, 123, 124, 125, 128, 129, 131, 132, 134, 135, 136, 137, 139, 140, 142

Offset: 1

Amiram Eldar, Mar 23 2024

nonn

It is unknown whether 222 is a term of this sequence or not (see A371423).

			12 is a term because when we start with 12 and repeatedly apply the mapping x -> A371418(x), we get the sequence 12, 14, 12, 14, ...
76 is a term because when we start with 76 and repeatedly apply the mapping x -> A371418(x), we get the sequence 76, 70, 72, 65, 42, 48, 62, 48, 62, ...

Amiram Eldar, Table of n, a(n) for n = 1..108
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.

Cf. A371418, A371421, A371423.

Similar sequences: A127655, A127660, A127665.

r[n_] := n/FactorInteger[n][[1, 1]]; f[n_] := r[DivisorSigma[1, n]];
q[n_] := Module[{m = NestWhileList[f, n, UnsameQ, All][[-1]], k}, k = f[m]; k != m && f[k] == m]; Select[Range[221], q]

A371423 Aliquot-like sequence based on the largest aliquot divisor of the sum of divisors of n (A371418) that starts with 222.

Original entry on oeis.org

222, 228, 280, 360, 585, 546, 672, 1008, 1612, 1568, 1197, 1040, 1302, 1536, 2046, 2304, 949, 518, 456, 600, 930, 1152, 1105, 756, 1120, 1512, 2400, 3906, 4992, 7140, 12096, 20320, 24192, 40800, 70308, 108416, 135660, 241920, 490560, 902208, 1235456, 1309440, 2354688

Offset: 1

Amiram Eldar, Mar 23 2024

nonn

222 is the least number k for which the repeated iterations of the mapping k -> A371418(k) seem to generate an unbounded sequence.

			a(1) = 222 by definition.
a(2) = A371418(a(1)) = A371418(222) = 228.
a(3) = A371418(a(2)) = A371418(228) = 280.

Amiram Eldar, Table of n, a(n) for n = 1..1389
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.

Cf. A371418, A371421, A371422.

Similar sequences: A008892, A323328, A361421.

r[n_] := n/FactorInteger[n][[1, 1]]; f[n_] := r[DivisorSigma[1, n]]; NestList[f, 222, 60]

f(n) = {my(s = sigma(n)); if(s == 1, 1, s/factor(s)[1, 1]);}
lista(nmax) = {my(m = 222); for(n = 1, nmax, print1(m, ", "); m = f(m));}

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

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A371422 Numbers whose aliquot-like sequence based on the largest aliquot divisor of the sum of divisors of n (A371418) terminates in a cycle of length 2.

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A371423 Aliquot-like sequence based on the largest aliquot divisor of the sum of divisors of n (A371418) that starts with 222.

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