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

Author

Antti Karttunen, Apr 02 2022

Keywords

• nonn
• base
• easy
• look

Comments

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

Links

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

Crossrefs

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

Cf. A051683 (positions of 1's).

Programs

PARI

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));

Formula

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

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

Extensions

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