A318841 -id:A318841 - cp's OEIS frontend

A318833 a(n) = n + A023900(n).

2, 1, 1, 3, 1, 8, 1, 7, 7, 14, 1, 14, 1, 20, 23, 15, 1, 20, 1, 24, 33, 32, 1, 26, 21, 38, 25, 34, 1, 22, 1, 31, 53, 50, 59, 38, 1, 56, 63, 44, 1, 30, 1, 54, 53, 68, 1, 50, 43, 54, 83, 64, 1, 56, 95, 62, 93, 86, 1, 52, 1, 92, 75, 63, 113, 46, 1, 84, 113, 46, 1, 74, 1, 110, 83, 94, 137, 54, 1, 84, 79, 122, 1, 72, 149, 128, 143, 98, 1, 82

Offset: 1

Antti Karttunen, Sep 16 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A023900, A318841.

A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A318833(n) = (n+A023900(n));

a(n) = n + A023900(n).

A126864 a(n) = gcd(n, Product_{p|n} (p-1)), where the product is over the distinct primes, p, that divide n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 4, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 3, 4, 1, 6, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 5, 2, 3, 2, 1, 4, 1, 2, 3, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 4, 1, 2, 1, 12, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 3

Offset: 1

Leroy Quet, Mar 15 2007

nonn

Product_{p|n} (p-1) is the absolute value of A023900(n) (that is, A173557(n)).

First occurrence of k: 1, 6, 21, 20, 55, 42, 203, 120, 171, 110, 253, 84, 689, 406, 465, 272, 1751, 342, 3629, 220, ..., . - Robert G. Wilson v

			The distinct primes that divide 20 are 2 and 5. So a(20) = gcd(20, (2-1)(5-1)) = gcd(20,4) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A023900, A173557, A318841, A319341.

with(numtheory): a:=n->gcd(n,product(factorset(n)[i]-1,i=1..nops(factorset(n)))); # Emeric Deutsch, Apr 11 2007

f[n_] := GCD[n, Times @@ (First /@ FactorInteger[n] - 1)]; Array[f, 105] (* Robert G. Wilson v, Sep 08 2007 *)

A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A126864(n) = gcd(n, A173557(n)); \\ Antti Karttunen, Sep 17 2018

a(n) = gcd(n, A173557(n)) = gcd(n, A318841(n)). - Antti Karttunen, Sep 17 2018

More terms from Emeric Deutsch, Apr 11 2007

A319341 a(n) = A000010(n) - A173557(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 3, 4, 0, 0, 2, 0, 0, 0, 7, 0, 4, 0, 4, 0, 0, 0, 6, 16, 0, 16, 6, 0, 0, 0, 15, 0, 0, 0, 10, 0, 0, 0, 12, 0, 0, 0, 10, 16, 0, 0, 14, 36, 16, 0, 12, 0, 16, 0, 18, 0, 0, 0, 8, 0, 0, 24, 31, 0, 0, 0, 16, 0, 0, 0, 22, 0, 0, 32, 18, 0, 0, 0, 28, 52, 0, 0, 12, 0, 0, 0, 30, 0, 16, 0, 22, 0, 0, 0, 30, 0, 36, 40, 36, 0, 0, 0, 36, 0

Offset: 1

Antti Karttunen, Sep 17 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A051953, A173557, A318841, A319340, A126864.

Cf. A013661, A307868.

a[n_] := EulerPhi[n] - Times @@ (FactorInteger[n][[;;, 1]] - 1); a[1] = 0; Array[a, 100] (* Amiram Eldar, Dec 21 2023 *)

A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A319341(n) = (eulerphi(n)-A173557(n));

a(n) = A000010(n) - A173557(n).
a(n) = A318841(n) - A051953(n).
a(A005117(n)) = 0. - Ivan N. Ianakiev, Sep 18 2018
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A059956 - A307868 = 0.136246... . - Amiram Eldar, Dec 21 2023

cp's OEIS Frontend

A318833 a(n) = n + A023900(n).

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A126864 a(n) = gcd(n, Product_{p|n} (p-1)), where the product is over the distinct primes, p, that divide n.

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A319341 a(n) = A000010(n) - A173557(n).

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