A353577 -id:A353577 - cp's OEIS frontend

A351954 Arithmetic derivative without its inherited divisor applied to the prime shadow of the factorial base exp-function: a(n) = A342001(A181819(A276076(n))).

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, 2, 2, 3, 5, 8, 2, 3, 3, 4, 8, 11, 5, 8, 8, 11, 7, 10, 7, 12, 12, 17, 31, 46, 1, 5, 5, 8, 2, 7, 5, 8, 8, 11, 7, 10, 2, 7, 7, 10, 3, 9, 8, 31, 31, 46, 13, 41, 1, 7, 7, 12, 8, 31, 7, 12, 12, 17, 31, 46, 8, 31, 31, 46, 13, 41, 2, 9, 9, 14, 11

Offset: 0

Antti Karttunen, Apr 02 2022

Compare the scatter plot to those of A275735, A353575 and of A353577. - Antti Karttunen, Apr 30 2022

Antti Karttunen, Table of n, a(n) for n = 0..20160
Index entries for sequences related to factorial base representation

Cf. A003415, A003557, A181819, A275735, A276076, A342001, A351576, A351945, A351952, A353575, A353577.

Cf. A051683 (positions of 1's).

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])));
A276076(n) = { my(i=0,m=1,f=1,nextf); while((n>0),i=i+1; nextf = (i+1)*f; if((n%nextf),m*=(prime(i)^((n%nextf)/f));n-=(n%nextf));f=nextf); m; };
A342001(n) = (A003415(n) / A003557(n));
A351945(n) = A342001(A181819(n));
A351954(n) = A351945(A276076(n));

a(n) = A342001(A275735(n)) = A351945(A276076(n)).

a(n) = A353577(A351576(n)). - Antti Karttunen, Apr 30 2022

Verbal description added to the definition by Antti Karttunen, Apr 30 2022

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).

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

A351954 Arithmetic derivative without its inherited divisor applied to the prime shadow of the factorial base exp-function: a(n) = A342001(A181819(A276076(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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A353524 A003557 applied to the prime shadow of primorial base exp-function: a(n) = A003557(A181819(A276086(n))).

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