A343226 -id:A343226 - cp's OEIS frontend

A343227 a(n) = sigma(n) / gcd(sigma(n), n+A003415(n)), where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

1, 1, 1, 7, 1, 12, 1, 3, 13, 18, 1, 1, 1, 24, 24, 31, 1, 1, 1, 21, 32, 36, 1, 15, 31, 42, 20, 14, 1, 72, 1, 9, 48, 54, 48, 91, 1, 60, 56, 5, 1, 96, 1, 21, 13, 72, 1, 31, 19, 93, 72, 49, 1, 8, 72, 30, 80, 90, 1, 21, 1, 96, 52, 127, 84, 144, 1, 9, 96, 48, 1, 65, 1, 114, 62, 35, 96, 168, 1, 93, 121, 126, 1, 14, 108

Offset: 1

Antti Karttunen, Apr 15 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A003415, A129283, A343226.

a(n) = A000203(n) / A343226(n) = A000203(n) / gcd(A000203(n), n+A003415(n))

A348972 a(n) = gcd(A003959(n), A129283(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

Original entry on oeis.org

1, 3, 4, 1, 6, 1, 8, 1, 1, 1, 12, 4, 14, 1, 1, 3, 18, 3, 20, 2, 1, 1, 24, 4, 1, 1, 2, 12, 30, 1, 32, 1, 1, 1, 1, 48, 38, 1, 1, 54, 42, 1, 44, 4, 12, 1, 48, 4, 1, 1, 1, 18, 54, 3, 1, 4, 1, 1, 60, 8, 62, 1, 2, 1, 1, 1, 68, 2, 1, 3, 72, 12, 74, 1, 2, 12, 1, 1, 80, 2, 1, 1, 84, 16, 1, 1, 1, 12, 90, 3, 1, 4, 1, 1, 1, 4, 98

Offset: 1

Antti Karttunen, Nov 06 2021

nonn

Cf. A003415, A003959, A129283, A348970, A348973, A348974.

Cf. also A343226.

f1[p_, e_] := e/p; f2[p_, e_] := (p + 1)^e; a[1] = 1; a[n_] := GCD[n*(1 + Plus @@ f1 @@@ (f = FactorInteger[n])), Times @@ f2 @@@ f]; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348972(n) = gcd(A003959(n),(n+A003415(n)));

a(n) = gcd(A003959(n), A129283(n)) = gcd(A003959(n), n+A003415(n)).
a(n) = gcd(A003959(n), A348970(n)) = gcd(A129283(n), A348970(n)).
a(n) = A129283(n) / A348973(n) = A003959(n) / A348974(n).

A345059 a(n) = A129283(n) / gcd(sigma(n), A129283(n)), where A129283(n) is the sum of n and its arithmetic derivative.

Original entry on oeis.org

1, 1, 1, 8, 1, 11, 1, 4, 15, 17, 1, 1, 1, 23, 23, 48, 1, 1, 1, 22, 31, 35, 1, 17, 35, 41, 27, 15, 1, 61, 1, 16, 47, 53, 47, 96, 1, 59, 55, 6, 1, 83, 1, 23, 14, 71, 1, 40, 21, 95, 71, 54, 1, 9, 71, 37, 79, 89, 1, 19, 1, 95, 57, 256, 83, 127, 1, 10, 95, 43, 1, 76, 1, 113, 65, 39, 95, 149, 1, 128, 189, 125, 1, 13, 107

Offset: 1

Antti Karttunen, Jun 07 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12800
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A003415, A129283, A211991, A343226, A343227.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A129283(n) = (n+A003415(n));
A345059(n) = { my(u=A129283(n)); (u/gcd(u,sigma(n))); };

a(n) = A129283(n) / A343226(n) = A129283(n) / gcd(A000203, A129283(n)).

A348028 Greatest common divisor of A003415 (arithmetic derivative) and sigma, the sum of divisors function.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 3, 8, 1, 1, 3, 1, 6, 2, 1, 1, 4, 1, 3, 1, 8, 1, 1, 1, 1, 2, 1, 12, 1, 1, 3, 8, 2, 1, 1, 1, 12, 39, 1, 1, 4, 1, 3, 4, 14, 1, 3, 8, 4, 2, 1, 1, 4, 1, 3, 1, 1, 6, 1, 1, 18, 2, 1, 1, 39, 1, 3, 1, 20, 6, 1, 1, 2, 1, 1, 1, 4, 2, 3, 8, 20, 1, 3, 4, 24, 2, 1, 24, 4, 1, 1, 3, 7, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Sep 25 2021

nonn

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

Cf. A000203, A003415.

Cf. also A342413, A342458, A343226.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A348028(n) = gcd(sigma(n), A003415(n));

a(n) = gcd(A000203(n), A003415(n)).

cp's OEIS Frontend

A343227 a(n) = sigma(n) / gcd(sigma(n), n+A003415(n)), where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

Views

Author

Keywords

Links

Crossrefs

Formula

A348972 a(n) = gcd(A003959(n), A129283(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A345059 a(n) = A129283(n) / gcd(sigma(n), A129283(n)), where A129283(n) is the sum of n and its arithmetic derivative.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A348028 Greatest common divisor of A003415 (arithmetic derivative) and sigma, the sum of divisors function.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula