A339905 Fully multiplicative with a(prime(k)) = prime(k+1) - 1.

1, 2, 4, 4, 6, 8, 10, 8, 16, 12, 12, 16, 16, 20, 24, 16, 18, 32, 22, 24, 40, 24, 28, 32, 36, 32, 64, 40, 30, 48, 36, 32, 48, 36, 60, 64, 40, 44, 64, 48, 42, 80, 46, 48, 96, 56, 52, 64, 100, 72, 72, 64, 58, 128, 72, 80, 88, 60, 60, 96, 66, 72, 160, 64, 96, 96, 70, 72, 112, 120, 72, 128, 78, 80, 144, 88, 120, 128, 82

Offset: 1

Author

Antti Karttunen, Dec 30 2020

Keywords

• nonn
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10205
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A003958, A003961, A151800, A339903 (the odd part).

Coincides with A003972 on A005117.

Programs

PARI

A339905(n) = if(1==n,n,my(f=factor(n)); for(i=1,#f~,f[i,1] = nextprime(1+f[i,1])-1); factorback(f));

Formula

a(n) = A003958(A003961(n)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-nextprime(p)+1)) = 0.73732173..., where nextprime is A151800. - Amiram Eldar, Dec 08 2022