A319068 -id:A319068 - cp's OEIS frontend

A319048 a(n) is the greatest k such that A000010(k) divides n where A000010 is the Euler totient function.

2, 6, 2, 12, 2, 18, 2, 30, 2, 22, 2, 42, 2, 6, 2, 60, 2, 54, 2, 66, 2, 46, 2, 90, 2, 6, 2, 58, 2, 62, 2, 120, 2, 6, 2, 126, 2, 6, 2, 150, 2, 98, 2, 138, 2, 94, 2, 210, 2, 22, 2, 106, 2, 162, 2, 174, 2, 118, 2, 198, 2, 6, 2, 240, 2, 134, 2, 12, 2, 142, 2, 270, 2, 6, 2, 12, 2, 158, 2, 330

Offset: 1

Michel Marcus, Sep 09 2018

nonn

Sándor calls this function the totient maximum function and remarks that this function is well-defined, since a(n) can be at least 2, and cannot be greater than n^2 (when n > 6).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
József Sándor, On the Euler minimum and maximum functions, Notes on Number Theory and Discrete Mathematics, Volume 15, 2009, Number 3, Pages 1—8.

Cf. A000010 (Euler totient), A061026 (the totient minimum function).
Cf. A319068 (the analog for the sum of divisors).
Cf. A070633, also A057635, A057826, A069932, A071181.
Right border of A378638.

a[n_] := Module[{kmax = If[n <= 6, 10 n, n^2]}, For[k = kmax, True, k--, If[Divisible[n, EulerPhi[k]], Return[k]]]];
Array[a, 80] (* Jean-François Alcover, Sep 17 2018, from PARI *)

a(n) = {my(kmax = if (n<=6, 10*n, n^2)); forstep (k=kmax, 1, -1, if ((n % eulerphi(k)) == 0, return (k)););}

\\ (The first two functions could probably be combined in a smarter way):
A014197(n, m=1) = { n==1 && return(1+(m<2)); my(p, q); sumdiv(n, d, if( d>=m && isprime(d+1), sum( i=0, valuation(q=n\d, p=d+1), A014197(q\p^i, p))))} \\ From A014197 by M. F. Hasler
A057635(n) = if(1==n,2,if((n%2),0,my(k=A014197(n),i=n); if(!k, 0, while(k, i++; if(eulerphi(i)==n, k--)); (i))));
A319048(n) = { my(m=0); fordiv(n,d, m = max(m,A057635(d))); (m); }; \\ Antti Karttunen, Sep 09 2018

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

a(n) = Max_{d|n} A057635(d). - Antti Karttunen, Sep 09 2018

A378912 Irregular triangle read by rows: row n lists all positive m such that sigma(m) divides n, where sigma is the sum-of-divisors function (A000203).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 1, 2, 5, 1, 4, 1, 3, 7, 1, 2, 1, 1, 1, 2, 3, 5, 6, 11, 1, 9, 1, 4, 13, 1, 2, 8, 1, 3, 7, 1, 1, 2, 5, 10, 17, 1, 1, 3, 19, 1, 2, 4, 1, 1, 1, 2, 3, 5, 6, 7, 11, 14, 15, 23, 1, 1, 9, 1, 2, 1, 3, 4, 12, 13, 1, 1, 2, 5, 8, 29, 1, 16, 25, 1, 3, 7, 21, 31

Offset: 1

Paolo Xausa, Dec 10 2024

			Triangle begins:
  n\k|  1   2   3   4   5   6 ...
  -------------------------------
   1 |  1;
   2 |  1;
   3 |  1,  2;
   4 |  1,  3;
   5 |  1;
   6 |  1,  2,  5;
   7 |  1,  4;
   8 |  1,  3,  7;
   9 |  1,  2;
  10 |  1;
  11 |  1;
  12 |  1,  2,  3,  5,  6, 11;
  13 |  1,  9;
  14 |  1,  4, 13;
  15 |  1,  2,  8;
  16 |  1,  3,  7;
  17 |  1;
  18 |  1,  2,  5, 10, 17;
  19 |  1;
  20 |  1,  3, 19;
  ...

Paolo Xausa, Table of n, a(n) for n = 1..10634 (rows 1..2000 of triangle, flattened).
Index entries for sequences related to sigma(n)

Cf. A074754 (row lengths), A319068 (right border), A378913 (row sums).

Cf. A000203.

With[{nmax = 50}, Table[PositionIndex[Divisible[n, #[[;; n]]]][True], {n, nmax}] & [DivisorSigma[1, Range[nmax]]]]

row(n) = select(x->(!(n % sigma(x))), [1..n]); \\ Michel Marcus, Dec 11 2024

T(n,k) <= n (see A319068).

A375228 a(n) is the largest number k such that usigma(k) divides n where usigma(k) is the sum of unitary divisors of k (A034448).

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 1, 7, 8, 9, 1, 11, 1, 13, 4, 7, 16, 17, 1, 19, 2, 1, 1, 23, 4, 25, 8, 27, 1, 29, 1, 31, 32, 16, 4, 24, 1, 37, 2, 28, 1, 41, 1, 43, 8, 1, 1, 47, 1, 49, 16, 25, 1, 53, 4, 39, 2, 1, 1, 59, 1, 61, 8, 31, 64, 32, 1, 67, 2, 52, 1, 71, 1, 73, 4, 37

Offset: 1

Amiram Eldar, Aug 06 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Bhabesh Das and Helen K. Saikia, On the Sum of Unitary Divisors Maximum Function, AIMS Mathematics, Vol. 2, No. 1 (2017), pp. 96-101.

The unitary analog of A319068.

Cf. A034448.

usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; a[n_] := Module[{k = n}, While[!Divisible[n, usigma[k]], k--]; k]; Array[a, 100]

usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}
a(n) = {my(k = n); while((n % usigma(k)), k--); k;}

cp's OEIS Frontend

A319048 a(n) is the greatest k such that A000010(k) divides n where A000010 is the Euler totient function.

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A378912 Irregular triangle read by rows: row n lists all positive m such that sigma(m) divides n, where sigma is the sum-of-divisors function (A000203).

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A375228 a(n) is the largest number k such that usigma(k) divides n where usigma(k) is the sum of unitary divisors of k (A034448).

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PARI