A378912 -id:A378912 - cp's OEIS frontend

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

Michel Marcus, Sep 09 2018

nonn

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.

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.

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

Right border of A378912.

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

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

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

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

A074754 Number of integers k such that sigma(k) divides n.

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 2, 3, 2, 1, 1, 6, 2, 3, 3, 3, 1, 5, 1, 3, 3, 1, 1, 10, 1, 2, 2, 5, 1, 5, 3, 5, 2, 1, 2, 9, 1, 2, 4, 5, 1, 8, 1, 3, 3, 1, 1, 13, 2, 1, 2, 3, 1, 7, 1, 8, 3, 1, 1, 12, 1, 4, 4, 5, 2, 3, 1, 3, 2, 3, 1, 18, 1, 2, 3, 3, 2, 6, 1, 7, 2, 1, 1, 15, 1, 1, 2, 4, 1, 10, 4, 2, 5, 1, 1, 19, 1, 5, 2, 3, 1

Offset: 1

Benoit Cloitre, Sep 28 2002

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Amiram Eldar, Plot of Sum_{k=1..n} a(k)/(n*log(n)) for n = 10^(1..7).
Index entries for sequences related to sigma(n).

Cf. A000203, A070610.

Row lengths of A378912.

Table[Length[Select[Range[n], Divisible[n, DivisorSigma[1,#]]&]], {n, 1, 100}] (* Vaclav Kotesovec, Feb 16 2019 *)

PARI
```
a(n)=sum(i=1,n,if(n%sigma(i),0,1))
```

a(n)=if(n<1,0,polcoeff(sum(k=1,n,1/(1-x^sigma(k)),x*O(x^sigma(n))),n))

a(n) = {my(s = []); fordiv(n, d, s = setunion(s, invsigma(d))); #s;} \\ Amiram Eldar, Apr 18 2025, using Max Alekseyev's invphi.gp (see links).

Sum_{k=1..n} a(k) seems to be asymptotic to c*n*log(n) with c = 0.7...
G.f.: sum(k>=1, 1/(1-x^sigma(k))).
a(n) = Sum_{k=1..n} (1 - ceiling(n/sigma(k)) + floor(n/sigma(k))). - Wesley Ivan Hurt, Apr 21 2023

A378913 a(n) = sum of 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, 3, 4, 1, 8, 5, 11, 3, 1, 1, 28, 10, 18, 11, 11, 1, 35, 1, 23, 7, 1, 1, 87, 1, 10, 3, 33, 1, 45, 42, 63, 3, 1, 5, 77, 1, 38, 30, 57, 1, 112, 1, 47, 11, 1, 1, 202, 5, 1, 3, 13, 1, 122, 1, 107, 52, 1, 1, 205, 1, 103, 39, 63, 10, 8, 1, 71, 3, 18, 1, 389, 1, 74, 11

Offset: 1

Paolo Xausa, Dec 11 2024

nonn

Paolo Xausa, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)

Row sums of A378912.
Cf. A378915 (positions of ones).
Cf. A000203.

With[{nmax = 100}, Table[Total[PositionIndex[Divisible[n, #[[;; n]]]][True]], {n, nmax}] & [DivisorSigma[1, Range[nmax]]]]

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

A378914 Smallest positive m such that sigma(m) does not divide n, where sigma is the sum-of-divisors function (A000203).

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 7, 2, 2, 3, 2, 2, 3

Offset: 1

Paolo Xausa, Dec 11 2024

Paolo Xausa, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)

Cf. A000203, A051549, A378912.

A378914[n_] := Module[{m = 1}, While[Divisible[n, DivisorSigma[1,++m]]]; m];
Array[A378914, 100]

a(n) = my(m=1); while (!(n % sigma(m)), m++); m; \\ Michel Marcus, Dec 11 2024

cp's OEIS Frontend

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

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A074754 Number of integers k such that sigma(k) divides n.

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A378913 a(n) = sum of all positive m such that sigma(m) divides n, where sigma is the sum-of-divisors function (A000203).

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Mathematica

PARI

A378914 Smallest positive m such that sigma(m) does not divide n, where sigma is the sum-of-divisors function (A000203).

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI