A347958 -id:A347958 - cp's OEIS frontend

A345000 a(n) = gcd(A003415(n), A003415(A276086(n))), where A003415(n) is the arithmetic derivative of n, and A276086(n) gives the prime product form of primorial base expansion of n.

0, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 16, 1, 3, 1, 2, 5, 1, 1, 4, 5, 5, 1, 2, 1, 1, 1, 10, 1, 1, 3, 12, 1, 1, 1, 2, 1, 1, 1, 4, 1, 5, 1, 2, 1, 5, 5, 4, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 12, 3, 1, 1, 2, 1, 1, 1, 12, 1, 1, 55, 10, 3, 1, 1, 16, 1, 1, 1, 2, 1, 5, 1, 140, 1, 3, 1, 16, 1, 49, 3, 2, 1, 7, 1, 28, 1, 7, 1, 2, 1

Offset: 0

Antti Karttunen, Jul 21 2021

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to primorial base

Cf. A003415, A276086, A327860, A347958 (inverse Möbius transform), A347959, A351083, A351085, A351086, A351235, A351236.
Cf. A166486 (a(n) mod 2, parity of terms, see comment in A327860).
Cf. also A324198, A327858.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A345000(n) = gcd(A003415(n), A003415(A276086(n)));

a(n) = gcd(A003415(n), A327860(n)) = gcd(A003415(n), A003415(A276086(n))).

A347959 Dirichlet convolution of A342001 with A345000.

Original entry on oeis.org

0, 1, 1, 3, 1, 7, 1, 7, 3, 9, 1, 18, 1, 11, 10, 15, 1, 16, 1, 24, 12, 15, 1, 40, 3, 17, 6, 30, 1, 54, 1, 39, 16, 21, 14, 41, 1, 23, 18, 54, 1, 72, 1, 42, 25, 27, 1, 92, 3, 24, 22, 52, 1, 30, 18, 68, 24, 33, 1, 132, 1, 35, 35, 73, 20, 96, 1, 60, 28, 92, 1, 99, 1, 41, 27, 66, 20, 114, 1, 120, 10, 45, 1, 176, 24, 47

Offset: 1

Antti Karttunen, Sep 21 2021

Antti Karttunen, Table of n, a(n) for n = 1..11550
Index entries for sequences related to primorial base

Cf. A003415, A327860, A342001, A345000.

Cf. also A347234, A347954, A347958.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A342001(n) = (A003415(n) / A003557(n));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A345000(n) = gcd(A003415(n), A003415(A276086(n)));
A347959(n) = sumdiv(n,d,A342001(n/d)*A345000(d));

a(n) = Sum_{d|n} A342001(n/d) * A345000(d).

A348027 Dirichlet convolution of Euler phi with A324198.

Original entry on oeis.org

1, 2, 5, 4, 5, 8, 7, 8, 15, 14, 11, 16, 13, 14, 37, 16, 17, 24, 19, 28, 35, 22, 23, 32, 49, 26, 45, 28, 29, 60, 31, 32, 55, 34, 41, 48, 37, 38, 65, 56, 41, 62, 43, 44, 111, 46, 47, 64, 55, 114, 85, 52, 53, 72, 59, 62, 95, 58, 59, 120, 61, 62, 123, 64, 65, 88, 67, 68, 115, 134, 71, 96, 73, 74, 293, 76, 83, 104, 79

Offset: 1

Antti Karttunen, Sep 25 2021

Antti Karttunen, Table of n, a(n) for n = 1..11550
Index entries for sequences related to primorial base

Cf. A000010, A324198.

Cf. also A346242, A347131, A347235, A347958.

s[n_] := Module[{k = n, m = 1, p = 2}, While[k > 0, m *= (p^Min[Mod[k, p], IntegerExponent[n, p]]); k = Quotient[k, p]; p = NextPrime[p]]; m]; a[n_] := DivisorSum[n, s[#] * EulerPhi[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 27 2021 *)

A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };
A348027(n) = sumdiv(n,d,eulerphi(d)*A324198(n/d));

a(n) = Sum_{d|n} phi(n/d) * A324198(d).

a(n) = Sum_{k=1..n} A324198(gcd(n,k)).

cp's OEIS Frontend

A345000 a(n) = gcd(A003415(n), A003415(A276086(n))), where A003415(n) is the arithmetic derivative of n, and A276086(n) gives the prime product form of primorial base expansion of n.

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A347959 Dirichlet convolution of A342001 with A345000.

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A348027 Dirichlet convolution of Euler phi with A324198.

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