A337926 Number of ways to write n as the sum of two positive integers with different numbers of distinct prime factors.

0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 4, 4, 3, 6, 4, 6, 4, 5, 6, 9, 5, 8, 7, 7, 7, 10, 6, 11, 6, 10, 6, 11, 6, 15, 10, 12, 9, 14, 11, 15, 12, 13, 11, 18, 10, 17, 11, 17, 14, 20, 11, 19, 13, 18, 15, 21, 12, 20, 16, 21, 16, 24, 13, 24, 16, 20, 18, 25, 13, 24, 19, 22, 21, 26, 17, 27, 20

Offset

1,8

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{i=1..floor(n/2)} (1 - [omega(i) = omega(n-i)]), where omega is the number of distinct prime factors (A001221) and [ ] is the Iverson bracket.

Example

a(13) = 3; 13 = 12 + 1 = 10 + 3 = 7 + 6 and omega(12) > omega(1), omega(10) > omega(3) and omega(7) < omega(6).

Maple

omega:= n -> nops(numtheory:-factorset(n)):
f:= proc(n) nops(select(t -> omega(t) <> omega(n-t), [$1..n/2])) end proc:
map(f, [$1..100]); # Robert Israel, Jan 31 2021

Mathematica

Table[Sum[1 - KroneckerDelta[PrimeNu[i], PrimeNu[n - i]], {i, Floor[n/2]}], {n, 100}]

Prog

(PARI) a(n) = sum(i=1, n\2, 1 - (omega(i) == omega(n-i))); \\ Michel Marcus, Sep 30 2020

Crossrefs

Cf. A001221, A333701.

Sequence in context: A298949 A329097 A197054 * A120502 A099480 A025783

Adjacent sequences: A337923 A337924 A337925 * A337927 A337928 A337929

Keyword

nonn,look

Author

Wesley Ivan Hurt, Sep 30 2020

Status

approved