A376417 a(n) = n - A276076(A276075(n)), where A276075 and A276076 are factorial base log and exp-functions.

0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 7, 0, 0, 0, 5, 0, 0, 0, 6, 0, 0, 22, 7, 0, 0, 0, 14, 0, 0, 0, 31, 0, 0, 0, 10, 0, 0, 0, 11, 0, 0, 0, 43, 0, 0, 0, 13, 0, 44, 0, 14, 0, 0, 0, 15, 0, 0, 0, 59, 0, 0, 0, 17, 0, 0, 0, 62, 0, 0, 0, 19, 0, 0, 0, 35, 66, 0, 0, 21, 0, 0, 0, 22, 0, 0, 0, 23, 0, 0, 0, 86, 0, 0, 0, 25

Offset: 1

Author

Antti Karttunen, Nov 03 2024

Keywords

• nonn

Comments

All terms are nonnegative because for all n, x = A276076(A276075(n)) <= n, as any factor prime(i)^k || n (with k > i) will propagate carries (in the image of fully additive A276075) towards more significant digit positions, which A276076 will convert back to the exponents of larger primes, but for each new instance of such larger prime present in x, enough instances of smaller primes in n have been eliminated (by the carry process) so that the net change of magnitude is negative, unless there are no such factors present at all in n (i.e., when n is a term of A276078), then A276076(A276075(n)) = n, and a(n) = 0.

This implies also that the least k for which A276075(k) = n is k = A276076(n).

Differs from similar A376418 for the first time at n=625, 1250, 1875, 2500, 3125, 3375, 3750, 4375, 4500, 5000, 5625, ...

Examples

			a(625) = 618, as 625 = 5^4 = prime(3)^4, thus A276075(625) = 4 * 3! = 24, but on the other hand, A276076(24) = prime(4) = 7, and 625 - 7 = 618.
a(2500) = 2479, as 2500 = 2^2 * 5^4 = prime(1)^2 * prime(3)^4, thus A276075(2500) = 2 * 1! + 4 * 3! = 26, but on the other hand, A276076(26) = prime(2)*prime(4) = 21 (as A007623(26) = 1010), and 2500 - 21 = 2479.
a(16807) = 16796, as 16807 = prime(4)^5 = 7^5, thus A276075(16807) = 5 * 4! = 120, but on the other hand, A276076(120) = prime(5) = 11, and 16807 - 11 = 16796.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to factorial base representation

Crossrefs

Cf. A007623, A276075, A276076, A276078 (indices of 0's), A276079 (of terms > 0), A376418.

Programs

PARI

A276075(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*(primepi(f[k, 1])!)); };
A276076(n) = { my(m=1, p=2, i=2); while(n, m *= (p^(n%i)); n = n\i; p = nextprime(1+p); i++); (m); };
A376417(n) = (n - A276076(A276075(n)));