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

%I A376417 #18 Nov 04 2024 01:37:03
%S A376417 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,
%T A376417 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,
%U A376417 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
%N A376417 a(n) = n - A276076(A276075(n)), where A276075 and A276076 are factorial base log and exp-functions.
%C A376417 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.
%C A376417   This implies also that the least k for which A276075(k) = n is k = A276076(n).
%C A376417 Differs from similar A376418 for the first time at n=625, 1250, 1875, 2500, 3125, 3375, 3750, 4375, 4500, 5000, 5625, ...
%H A376417 Antti Karttunen, <a href="/A376417/b376417.txt">Table of n, a(n) for n = 1..65537</a>
%H A376417 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%e A376417 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.
%e A376417 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.
%e A376417 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.
%o A376417 (PARI)
%o A376417 A276075(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*(primepi(f[k, 1])!)); };
%o A376417 A276076(n) = { my(m=1, p=2, i=2); while(n, m *= (p^(n%i)); n = n\i; p = nextprime(1+p); i++); (m); };
%o A376417 A376417(n) = (n - A276076(A276075(n)));
%Y A376417 Cf. A007623, A276075, A276076, A276078 (indices of 0's), A276079 (of terms > 0), A376418.
%K A376417 nonn
%O A376417 1,8
%A A376417 _Antti Karttunen_, Nov 03 2024