A353576 -id:A353576 - cp's OEIS frontend

A353575 Primepi based arithmetic derivative applied to the prime shadow of the primorial base exp-function: a(n) = A258851(A181819(A276086(n))).

0, 1, 1, 4, 2, 7, 1, 4, 4, 12, 7, 20, 2, 7, 7, 20, 12, 33, 3, 11, 11, 32, 19, 53, 4, 15, 15, 44, 26, 73, 1, 4, 4, 12, 7, 20, 4, 12, 12, 32, 20, 52, 7, 20, 20, 52, 33, 84, 11, 32, 32, 84, 53, 136, 15, 44, 44, 116, 73, 188, 2, 7, 7, 20, 12, 33, 7, 20, 20, 52, 33, 84, 12, 33, 33, 84, 54, 135, 19, 53, 53, 136, 87, 219

Offset: 0

Antti Karttunen, Apr 29 2022

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

Cf. A181819, A258851, A276086, A328835, A353379.

Cf. also A342002, A353574 and A353576.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
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));
A353575(n) = A353379(A276086(n));

a(n) = A353379(A276086(n)) = A258851(A328835(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).

A353524 A003557 applied to the prime shadow of primorial base exp-function: a(n) = A003557(A181819(A276086(n))).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 4, 1, 2, 1, 1, 1, 2, 3, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 4, 1, 2, 2, 4, 4, 8, 2, 4, 1, 2, 2, 4, 3, 6, 1, 2, 2, 4, 1, 2, 1, 2, 2, 4, 1, 2, 1, 1, 1, 2, 3, 3, 1, 2, 2, 4, 3, 6, 3, 3, 3, 6, 9, 9, 1, 1, 1, 2, 3, 3, 1, 1, 1, 2, 3, 3, 1, 1, 1, 2, 1, 1, 1, 2, 2, 4, 1, 2, 1, 1, 1, 2

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. A003557, A181819, A276086, A328835, A351944, A353576, A353577.

A003557(n) = (n/factorback(factorint(n)[, 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); };
A353524(n) = A003557(A181819(A276086(n)));

a(n) = A351944(A276086(n)) = A003557(A328835(n)) = A003557(A181819(A276086(n))).

For n > 0, a(n) = A353576(n) / A353577(n).

cp's OEIS Frontend

A353575 Primepi based arithmetic derivative applied to the prime shadow of the primorial base exp-function: a(n) = A258851(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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A353524 A003557 applied to the prime shadow of primorial base exp-function: a(n) = A003557(A181819(A276086(n))).

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