A347102 Totally additive with a(prime(k)) = A001223(k), where A001223 gives the distance from the k-th prime to the next larger prime.

0, 1, 2, 2, 2, 3, 4, 3, 4, 3, 2, 4, 4, 5, 4, 4, 2, 5, 4, 4, 6, 3, 6, 5, 4, 5, 6, 6, 2, 5, 6, 5, 4, 3, 6, 6, 4, 5, 6, 5, 2, 7, 4, 4, 6, 7, 6, 6, 8, 5, 4, 6, 6, 7, 4, 7, 6, 3, 2, 6, 6, 7, 8, 6, 6, 5, 4, 4, 8, 7, 2, 7, 6, 5, 6, 6, 6, 7, 4, 6, 8, 3, 6, 8, 4, 5, 4, 5, 8, 7, 8, 8, 8, 7, 6, 7, 4, 9, 6, 6, 2, 5, 4, 7, 8

Offset: 1

Author

Antti Karttunen, Aug 19 2021

Keywords

• nonn

Examples

			For n = 12 = 2*2*3, the corresponding prime gaps are 1, 1 and 2, thus a(12) = 1+1+2 = 4.
For n = 42 = 2*3*7, the corresponding prime gaps are 1, 2 and 4, thus a(42) = 1+2+4 = 7.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for primes, gaps between

Crossrefs

Cf. A001223, A001414, A003961, A007814, A056239, A064989, A347101, A347123.

Programs

PARI

A347102(n) = { my(f=factor(n), s=0); for(i=1, #f~, s += f[i, 2]*(nextprime(f[i, 1]+1)-f[i,1])); (s); };

Formula

a(n) = A001414(A003961(n)) - A001414(n).

a(n) = A007814(n) + 2*A056239(A064989(A347123(n))).

For all n >= 0, a(2^n) = n.