A307409 a(n) = (A001222(n) - 1)*A001221(n).

0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 4, 0, 2, 2, 3, 0, 4, 0, 4, 2, 2, 0, 6, 1, 2, 2, 4, 0, 6, 0, 4, 2, 2, 2, 6, 0, 2, 2, 6, 0, 6, 0, 4, 4, 2, 0, 8, 1, 4, 2, 4, 0, 6, 2, 6, 2, 2, 0, 9, 0, 2, 4, 5, 2, 6, 0, 4, 2, 6, 0, 8, 0, 2, 4, 4, 2, 6, 0, 8, 3, 2, 0, 9, 2, 2, 2, 6, 0, 9, 2, 4, 2, 2, 2, 10, 0, 4, 4, 6, 0, 6, 0, 6

Offset: 1

Author

Mats Granvik, Apr 07 2019

Keywords

• nonn

Comments

a(n) + 2 appears to differ from A000005 at n=1 and when n is a term of A320632. Verified up to n=3000.

If A320632 contains the numbers such that A001222(n) - A051903(n) > 1, then this sequence contains precisely the numbers p^k and p^k*q for distinct primes p and q. The comment follows, since d(p^k) = k+1 = (k-1)*1 + 2 and d(p^k*q) = 2k+2 = ((k+1)-1)*2 + 2. - Charlie Neder, May 14 2019

Positions of first appearances are A328965. - Gus Wiseman, Nov 05 2019

Regarding Neder's comment above, see also my comments in A322437. - Antti Karttunen, Feb 17 2021

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Two less than A307408.

A113901(n) is bigomega(n) * omega(n).

A328958(n) is sigma_0(n) - bigomega(n) * omega(n).

Cf. A000005, A001221, A001222, A060687, A070175, A071625, A113901, A124010, A303555, A320632, A323023, A328956, A328957, A328964, A328965, A322437.

Programs

Mathematica

a[n_] := (PrimeOmega[n] - 1)*PrimeNu[n];
aa = Table[a[n], {n, 1, 104}];

PARI

a(n) = (bigomega(n) - 1)*omega(n); \\ Michel Marcus, May 15 2019

Formula

a(n) = (A001222(n) - 1)*A001221(n).

a(n) = binomial(A001222(n) - 1, 1)*binomial(A001221(n), 1).

a(n) = A307408(n) - 2.