A348505 -id:A348505 - cp's OEIS frontend

A348503 a(n) = gcd(sigma(n), usigma(n)), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

1, 3, 4, 1, 6, 12, 8, 3, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 12, 1, 42, 4, 8, 30, 72, 32, 3, 48, 54, 48, 1, 38, 60, 56, 18, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 12, 72, 24, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72, 15, 74, 114, 4, 20, 96, 168, 80, 6, 1, 126, 84, 32

Offset: 1

Antti Karttunen, Oct 29 2021

nonn

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 72 = 8*9, where a(72) = 15 != 3*1 = a(8)*a(9).

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A034448, A048146, A348504, A348505.
Differs from A344695 for the first time at n=72, where a(72) = 15, while A344695(72) = 3.
Differs from A348047 for the first time at n=27, where a(27) = 4, while A348047(27) = 8.

f1[p_, e_] := p^e + 1; f2[p_, e_] := (p^(e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := GCD[Times @@ f1 @@@ (fct = FactorInteger[n]), Times @@ f2 @@@ fct]; Array[a, 100] (* Amiram Eldar, Oct 29 2021 *)

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A348503(n) = gcd(sigma(n), A034448(n));

a(n) = gcd(A000203(n), A034448(n)).
a(n) = gcd(A000203(n), A048146(n)) = gcd(A034448(n), A048146(n)).
a(n) = A000203(n) / A348504(n) = A034448(n) / A348505(n).

A348504 a(n) = sigma(n) / gcd(sigma(n), usigma(n)), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 1, 5, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 5, 31, 1, 10, 7, 1, 1, 1, 21, 1, 1, 1, 91, 1, 1, 1, 5, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 10, 1, 5, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 13, 1, 1, 31, 7, 1, 1, 1, 31, 121, 1, 1, 7, 1, 1, 1, 5, 1, 13, 1, 7, 1, 1, 1, 21, 1, 57

Offset: 1

Antti Karttunen, Oct 29 2021

nonn

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 72 = 8*9, where a(72) = 13 != 5*13 = a(8) * a(9).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A005117 (positions of ones), A034448, A048146, A348503, A348505.

Differs from A344696 for the first time at n=72, where a(72) = 13, while A344696(72) = 65. Cf. also A348048.

f1[p_, e_] := p^e + 1; f2[p_, e_] := (p^(e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := (sigma = Times @@ f2 @@@ (fct = FactorInteger[n])) / GCD[sigma, Times @@ f1 @@@ fct]; Array[a, 100] (* Amiram Eldar, Oct 29 2021 *)

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A348504(n) = { my(u=sigma(n)); (u/gcd(u, A034448(n))); };

a(n) = A000203(n) / A348503(n) = A000203(n) / gcd(A000203(n), A034448(n)).

A348506 Numbers k such that sigma(k) is a multiple of usigma(k), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

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, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 113, 114, 115, 118, 119

Offset: 1

Antti Karttunen, Oct 29 2021

nonn

Conjectured to be the union of A005117 and A063880.

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

Positions of ones in A348505.
Cf. A005117 and A063880.
Cf. A000203, A034448.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
isA348506(n) = !(sigma(n)%A034448(n));

cp's OEIS Frontend

A348503 a(n) = gcd(sigma(n), usigma(n)), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A348504 a(n) = sigma(n) / gcd(sigma(n), usigma(n)), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A348506 Numbers k such that sigma(k) is a multiple of usigma(k), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI