A319068 a(n) is the greatest k such that A000203(k) divides n where A000203 is the sum of divisors of n.

1, 1, 2, 3, 1, 5, 4, 7, 2, 1, 1, 11, 9, 13, 8, 7, 1, 17, 1, 19, 4, 1, 1, 23, 1, 9, 2, 13, 1, 29, 25, 31, 2, 1, 4, 22, 1, 37, 18, 27, 1, 41, 1, 43, 8, 1, 1, 47, 4, 1, 2, 9, 1, 53, 1, 39, 49, 1, 1, 59, 1, 61, 32, 31, 9, 5, 1, 67, 2, 13, 1, 71, 1, 73, 8, 37, 4, 45, 1, 79

Offset: 1

Author

Michel Marcus, Sep 09 2018

Keywords

• nonn

Comments

Sándor names this function the sum-of-divisors maximum function and remarks that this function is well-defined, since a(n) can be at least 1, and cannot be greater than n.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
• József Sándor, The sum-of-divisors minimum and maximum functions, Research Report Collection, Volume 8, Issue 1, 2005. See pp. 3-4.

Crossrefs

Cf. A000203 (sigma), A070982 (the sum of divisors minimum function).

Right border of A378912.

Programs

Mathematica

A319068[n_] := Module[{k = n}, While[!Divisible[n, DivisorSigma[1, k]], k--]; k];
Array[A319068, 100] (* Paolo Xausa, Dec 11 2024 *)

PARI

a(n) = {forstep (k=n, 1, -1, if ((n % sigma(k)) == 0, return (k)););}

PARI

a(n) = {my(d = divisors(n)); vecmax(vector(#d, i, invsigmaMax(d[i])));} \\ Amiram Eldar, Nov 29 2024, using Max Alekseyev's invphi.gp

Formula

a(p+1) = p, for p prime. See Sándor Theorem 2 p. 4.