A238922 Numbers n such that Sum_{i=1..j} 1/d(i) - Sum_{i=1..k} 1/p(i) is an integer, where p are the prime factors of n, counted with multiplicity, and d its divisors.

1, 12, 18, 220, 396, 17296, 24016, 287532, 4661056, 64288512, 334144656, 358585488, 555192576, 568719616, 2172649216, 2451538112, 2645953344, 2955423888, 6704333824, 26996772032, 88734733632, 147861504000, 311063879024, 371226582848, 429391876096

Offset: 1

Author

Paolo P. Lava, Mar 07 2014

Keywords

• nonn
• hard

Comments

A212128 and A230164 are subsets of this sequence.

a(26) > 10^12. - Giovanni Resta, Mar 11 2014

Examples

			Divisors of 12 are 1, 2, 3, 4, 6, 12 and 1/1 + 1/2 + 1/3 +1/4 + 1/6 + 1/12 = 7/3. Prime factors of 12 are 2^2, 3 and 1/2 + 1/2 + 1/3 = 4/3. Finally 7/3 - 4/3 = 1 that is an integer.
		

Crossrefs

Cf. A003415, A212128, A224346, A230164.

Programs

Maple

with(numtheory); P:=proc(q) local a,b,c,k,n;
for n from 1 to q do if not isprime(n) then b:=sigma(n)/n;
a:=ifactors(n)[2]; c:=add(a[k][2]/a[k][1],k=1..nops(a));
if type(b-c,integer) then lprint(n,b-c); fi; fi; od; end: P(10^6);

Extensions

a(9)-a(10), a(13)-a(17), a(19)-a(25) from Giovanni Resta, Mar 11 2014