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

1,3

Links

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

Index entries for primes, gaps between

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.

Example

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.

Prog

(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); };

Crossrefs

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

Sequence in context: A281965 A228074 A152803 * A187181 A211187 A241504

Adjacent sequences: A347099 A347100 A347101 * A347103 A347104 A347105

Keyword

nonn

Author

Antti Karttunen, Aug 19 2021

Status

approved