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

%I A319068 #18 Dec 11 2024 15:05:24
%S A319068 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,
%T A319068 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,
%U A319068 1,59,1,61,32,31,9,5,1,67,2,13,1,71,1,73,8,37,4,45,1,79
%N A319068 a(n) is the greatest k such that A000203(k) divides n where A000203 is the sum of divisors of n.
%C A319068 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.
%H A319068 Antti Karttunen, <a href="/A319068/b319068.txt">Table of n, a(n) for n = 1..16384</a>
%H A319068 Antti Karttunen, <a href="/A319068/a319068.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H A319068 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%H A319068 József Sándor, <a href="https://rgmia.org/papers/v8n1/art4.pdf">The sum-of-divisors minimum and maximum functions</a>, Research Report Collection, Volume 8, Issue 1, 2005. See pp. 3-4.
%F A319068 a(p+1) = p, for p prime. See Sándor Theorem 2 p. 4.
%t A319068 A319068[n_] := Module[{k = n}, While[!Divisible[n, DivisorSigma[1, k]], k--]; k];
%t A319068 Array[A319068, 100] (* _Paolo Xausa_, Dec 11 2024 *)
%o A319068 (PARI) a(n) = {forstep (k=n, 1, -1, if ((n % sigma(k)) == 0, return (k)););}
%o A319068 (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
%Y A319068 Cf. A000203 (sigma), A070982 (the sum of divisors minimum function).
%Y A319068 Right border of A378912.
%K A319068 nonn
%O A319068 1,3
%A A319068 _Michel Marcus_, Sep 09 2018