A326075 Difference between the number of prime divisors in a nonstandard factorization process based on the sieve of Eratosthenes vs. their number in the ordinary factorization of n (when counted with multiplicity): a(n) = A253557(n) - A001222(n).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 1, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 3, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 1

Offset: 1

Author

Antti Karttunen, Aug 23 2019

Keywords

• sign

Examples

			A001222(21) = 2 because A032742(21) = 7, and A032742(7) = 1, while A253557(21) = 3 because A302042(21) = 9, A302042(9) = 3, and A302042(3) = 1. Thus a(21) = 3-2 = 1.
A001222(27) = 3 because A032742(27) = 9, A032742(9) = 3 and A032742(3) = 1, while A253557(27) = 2 because A302042(27) = 7 and A302042(7) = 1. Thus a(27) = 2-3 = -1.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A001222, A032742, A250246, A253557, A302042, A323888, A326139, A326189, A326190, A326191.

Programs

PARI

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A302042(n) = if((1==n)||isprime(n),1,my(c = A078898(n), p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p));
A253557(n) = if(1==n, 0,1+A253557(A302042(n)));
A326075(n) = (A253557(n)-bigomega(n));

Formula

a(n) = A253557(n) - A001222(n) = A001222(A250246(n)) - A001222(n).

a(p) = 0 for all primes p.