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A347102
Totally additive with a(prime(k)) = A001223(k), where A001223 gives the distance from the k-th prime to the next larger prime.
5
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
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); };
KEYWORD
nonn
AUTHOR
Antti Karttunen, Aug 19 2021
STATUS
approved