A126864 -id:A126864 - cp's OEIS frontend

A348928 a(n) = gcd(n, A003958(n)), where A003958 is multiplicative with a(p^e) = (p-1)^e.

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, 4, 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, 4, 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

Antti Karttunen, Nov 07 2021

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

Cf. A003958, A085731, A126865, A322582, A348929, A348998 [= a(A276086(n))].

Differs from similar A126864 for the first time at n=36, where a(36) = 4, while A126864(36) = 2.

f[p_, e_] := (p - 1)^e; a[n_] := GCD[n, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Nov 07 2021 *)

A003958(n) = if(1==n,n,my(f=factor(n)); for(i=1,#f~,f[i,1]--); factorback(f));
A348928(n) = gcd(n, A003958(n));

a(n) = gcd(n, A003958(n)) = gcd(n, A322582(n)) = gcd(A003958(n), A322582(n)).

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

A173751 a(n) = gcd(n, lcm_{p is a prime divisor of n} (p-1)) = gcd(n, A173614(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, 6, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 3

Offset: 1

José María Grau Ribas, Feb 23 2010

nonn

a(n) is divisor of A126864(n).

			84 = 2^2*3*7; lcm{p-1|p is prime and divisor of 84} = lcm{1,2,6} = 6; gcd(84,6) = 6 ==> a(84)=6.

Andrew Howroyd, Table of n, a(n) for n = 1..10000
A. M. Oller-Marcen, On arithmetic numbers, arXiv preprint arXiv:1206.1823 [math.NT], 2012. From _N. J. A. Sloane_, Nov 25 2012

Cf. A173614.

fa=FactorInteger; lcm[n_] := Module[{aux = 1, lon = Length[fa[n]]}, Do[aux = LCM[aux, (fa[n][[i]][[1]] - 1)], {i, lon}]; aux] a[n_] := GCD[lcm[n], n]; Table[a[n], {n, 1, 300}]

a(n)=gcd(n, lcm(apply(p->p-1, factor(n)[,1]))) \\ Andrew Howroyd, Aug 06 2018

cp's OEIS Frontend

A348928 a(n) = gcd(n, A003958(n)), where A003958 is multiplicative with a(p^e) = (p-1)^e.

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

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A173751 a(n) = gcd(n, lcm_{p is a prime divisor of n} (p-1)) = gcd(n, A173614(n)).

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