A351942 -id:A351942 - cp's OEIS frontend

A351945 a(n) = A342001(A181819(n)).

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 5, 1, 2, 2, 1, 1, 5, 1, 5, 2, 2, 1, 7, 1, 2, 1, 5, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 7, 1, 3, 1, 5, 5, 2, 1, 9, 1, 5, 2, 5, 1, 7, 2, 7, 2, 2, 1, 8, 1, 2, 5, 1, 2, 3, 1, 5, 2, 3, 1, 8, 1, 2, 5, 5, 2, 3, 1, 9, 1, 2, 1, 8, 2, 2, 2, 7, 1, 8, 2, 5, 2, 2, 2, 13, 1, 5, 5, 2, 1, 3, 1, 7, 3

Offset: 1

Antti Karttunen, Apr 02 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from exponents in factorization of n

Cf. A003415, A003557, A181819, A342001, A351942, A351954 [= a(A276076(n))].

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]));
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A342001(n) = (A003415(n) / A003557(n));
A351945(n) = A342001(A181819(n));

a(n) = A342001(A181819(n)) = A351942(n) / A351944(n).

A353379 Primepi-based variant of the arithmetic derivative applied to the prime shadow of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 2, 4, 1, 7, 1, 4, 4, 4, 1, 7, 1, 7, 4, 4, 1, 11, 2, 4, 3, 7, 1, 12, 1, 5, 4, 4, 4, 12, 1, 4, 4, 11, 1, 12, 1, 7, 7, 4, 1, 15, 2, 7, 4, 7, 1, 11, 4, 11, 4, 4, 1, 20, 1, 4, 7, 6, 4, 12, 1, 7, 4, 12, 1, 19, 1, 4, 7, 7, 4, 12, 1, 15, 4, 4, 1, 20, 4, 4, 4, 11, 1, 20, 4, 7, 4, 4, 4, 21, 1, 7, 7

Offset: 1

Antti Karttunen, Apr 28 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A181819, A258851.

Cf. also A351942.

a:= n-> (m-> m*add(i[2]*numtheory[pi](i[1])/i[1], i=ifactors(m)[2]))
        (mul(ithprime(i[2]), i=ifactors(n)[2])):
seq(a(n), n=1..120);  # Alois P. Heinz, Apr 28 2022

a[n_] := If[n == 1, 0, #*Sum[i[[2]]*PrimePi[i[[1]]]/i[[1]], {i, FactorInteger[#]}]]&[Product[Prime[i[[2]]], {i, FactorInteger[n]}]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 29 2025, after Alois P. Heinz *)

A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A258851(n) = (n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i])); \\ From A258851
A353379(n) = A258851(A181819(n));

a(n) = A258851(A181819(n)).

A353576 Arithmetic derivative applied to the prime shadow of the primorial base exp-function: a(n) = A003415(A181819(A276086(n))).

Original entry on oeis.org

0, 1, 1, 4, 1, 5, 1, 4, 4, 12, 5, 16, 1, 5, 5, 16, 6, 21, 1, 7, 7, 24, 8, 31, 1, 9, 9, 32, 10, 41, 1, 4, 4, 12, 5, 16, 4, 12, 12, 32, 16, 44, 5, 16, 16, 44, 21, 60, 7, 24, 24, 68, 31, 92, 9, 32, 32, 92, 41, 124, 1, 5, 5, 16, 6, 21, 5, 16, 16, 44, 21, 60, 6, 21, 21, 60, 27, 81, 8, 31, 31, 92, 39, 123, 10, 41, 41, 124

Offset: 0

Antti Karttunen, Apr 30 2022

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A003415, A181819, A276086, A328835, A351942, A353524.
Cf. A060735 (positions of 1's).
Cf. also A353575 which has quite a similar scatter plot, while on the contrast, A353577 has a very different look, explained by the contribution of A353524.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A351942(n) = A003415(A181819(n));
A353576(n) = A351942(A276086(n));

a(n) = A351942(A276086(n)) = A003415(A328835(n)) = A003415(A181819(A276086(n))).

a(n) = A353524(n) * A353577(n).

A353577 Arithmetic derivative without its inherited divisor applied to the prime shadow of the primorial base exp-function: a(n) = A342001(A181819(A276086(n))).

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 2, 2, 3, 5, 8, 1, 5, 5, 8, 2, 7, 1, 7, 7, 12, 8, 31, 1, 9, 9, 16, 10, 41, 1, 2, 2, 3, 5, 8, 2, 3, 3, 4, 8, 11, 5, 8, 8, 11, 7, 10, 7, 12, 12, 17, 31, 46, 9, 16, 16, 23, 41, 62, 1, 5, 5, 8, 2, 7, 5, 8, 8, 11, 7, 10, 2, 7, 7, 10, 3, 9, 8, 31, 31, 46, 13, 41, 10, 41, 41, 62, 17, 55, 1, 7, 7, 12, 8

Offset: 0

Antti Karttunen, Apr 30 2022

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A003415, A003557, A181819, A276086, A328835, A342001, A351942, A351945, A353524, A353576.
Cf. A060735 (positions of 1's).
Cf. also A342002, A351954 (similar or analogous definitions).

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));
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A353577(n) = A342001(A181819(A276086(n)));

a(n) = A342001(A328835(n)) = A351945(A276086(n)) = A342001(A181819(A276086(n))).

a(n) = A353576(n) / A353524(n).

A351943 a(n) = A069359(A181819(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 5, 1, 2, 2, 1, 1, 5, 1, 5, 2, 2, 1, 7, 1, 2, 1, 5, 1, 4, 1, 1, 2, 2, 2, 3, 1, 2, 2, 7, 1, 4, 1, 5, 5, 2, 1, 9, 1, 5, 2, 5, 1, 7, 2, 7, 2, 2, 1, 10, 1, 2, 5, 1, 2, 4, 1, 5, 2, 4, 1, 8, 1, 2, 5, 5, 2, 4, 1, 9, 1, 2, 1, 10, 2, 2, 2, 7, 1, 10, 2, 5, 2, 2, 2, 13, 1, 5, 5, 3, 1, 4, 1, 7, 4

Offset: 1

Antti Karttunen, Apr 02 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from exponents in factorization of n

Cf. A069359, A181819.

Coincides with A351942 on the positions given by A130091.

A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A351943(n) = A069359(A181819(n));

cp's OEIS Frontend

A351945 a(n) = A342001(A181819(n)).

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A353379 Primepi-based variant of the arithmetic derivative applied to the prime shadow of n.

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A353576 Arithmetic derivative applied to the prime shadow of the primorial base exp-function: a(n) = A003415(A181819(A276086(n))).

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A353577 Arithmetic derivative without its inherited divisor applied to the prime shadow of the primorial base exp-function: a(n) = A342001(A181819(A276086(n))).

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PARI

Formula

A351943 a(n) = A069359(A181819(n)).

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PARI