A348030 -id:A348030 - cp's OEIS frontend

A348029 a(n) = A003959(n) - sigma(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and sigma is the sum of divisors.

0, 0, 0, 2, 0, 0, 0, 12, 3, 0, 0, 8, 0, 0, 0, 50, 0, 9, 0, 12, 0, 0, 0, 48, 5, 0, 24, 16, 0, 0, 0, 180, 0, 0, 0, 53, 0, 0, 0, 72, 0, 0, 0, 24, 18, 0, 0, 200, 7, 15, 0, 28, 0, 72, 0, 96, 0, 0, 0, 48, 0, 0, 24, 602, 0, 0, 0, 36, 0, 0, 0, 237, 0, 0, 20, 40, 0, 0, 0, 300, 135, 0, 0, 64, 0, 0, 0, 144, 0, 54, 0, 48, 0

Offset: 1

Antti Karttunen, Oct 20 2021

nonn

Inverse Möbius transform of A348030.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A003959, A005117 (positions of zeros), A013661, A065488, A348030.

f[p_, e_] := (p + 1)^e; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - DivisorSigma[1, n]; Array[a, 100] (* Amiram Eldar, Oct 20 2021 *)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348029(n) = (A003959(n)-sigma(n));

a(n) = A003959(n) - A000203(n).
a(n) = Sum_{d|n} A348030(d).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p^2-p-1)) - Pi^2/6 = A065488 - A013661 = 1.0291786... . - Amiram Eldar, May 29 2025

A348036 a(n) = gcd(n, A003968(n)), where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 36, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 2, 65, 66, 67, 34, 69, 70, 71, 72, 73, 74, 15, 38, 77, 78, 79, 10, 3, 82, 83, 42

Offset: 1

Antti Karttunen, Oct 19 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Differs from A007947 at the positions given by A347960.

Cf. A003959, A003968, A348030, A348037, A348038, A348039.

f[p_, e_] := p*(p + 1)^(e - 1); a[n_] := GCD[n, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Oct 20 2021 *)

A003968(n) = { my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
A348036(n) = gcd(n, A003968(n));

a(n) = gcd(n, A003968(n)).
a(n) = gcd(n, A348030(n)) = gcd(A003968(n), A348030(n)).
a(n) = n / A348037(n) = A003968(n) / A348038(n).
a(n) = A007947(n) * A348039(n).

A348499 Numbers k such that A003968(k) is a multiple of k, where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 72, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119

Offset: 1

Antti Karttunen, Oct 29 2021

nonn

This is a subsequence of A333634. See comments in A347892.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Union of A005117 and A347892 (terms that are not squarefree).
Subsequence of A333634.
Positions of ones in A348037.
Cf. A003968, A348030, A348036, A347960.

f[p_, e_] := p*(p + 1)^(e - 1); s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[120], Divisible[s[#], #] &] (* Amiram Eldar, Oct 29 2021 *)

A003968(n) = {my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
isA348499(n) = !(A003968(n)%n);

cp's OEIS Frontend

A348029 a(n) = A003959(n) - sigma(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and sigma is the sum of divisors.

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Formula

A348036 a(n) = gcd(n, A003968(n)), where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

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A348499 Numbers k such that A003968(k) is a multiple of k, where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

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