A199767 Numbers m for which Sum_{i=1..k} (1+1/p_i) + Product_{i=1..k} (1+1/p_i) is an integer, where p_i are the k prime factors of m (with multiplicity).

21, 45, 432, 740, 1728, 3456, 3888, 5616, 12096, 23760, 46656, 52164, 131328, 152064, 186624, 195656, 233280, 311472, 606528, 618192, 746496, 926208, 933120, 979776, 1273536, 1403136, 2985984, 3221456, 3732480, 5178816, 5412096, 5971968, 9704448, 13651200

Offset: 1

Author

Paolo P. Lava, Nov 22 2011

Keywords

• nonn
• easy

Comments

The numbers of the sequence are the solution of the differential equation m’=(a-k)*m-b, which can also be written as A003415(m)=(a-k)*m-A003958(m), where k is the number of prime factors of m, and a is the integer Sum_{i=1..k} (1+1/p_i) + Product_{1=1..k} (1+1/p_i).

The numbers of the sequence satisfy also Sum_{i=1..k} (1-1/p_i) - Product_{i=1..k} (1+1/p_i) = some integer.

Examples

			740 has prime factors 2, 2, 5, 37. 1 + 1/2 + 1 + 1/2 + 1 + 1/5 + 1 + 1/37 = 967/185 is the sum over 1+1/p_i. (1+1/2) * (1+1/2) * (1+1/5) * (1+1/37) = 513/185 is the product over 1+1/p_i. 967/185 + 513/185 = 8 is an integer.
		

Links

• J. M. Borwein and E. Wong, A survey of results relating to Giuga’s conjecture on primality, May 8, 1995.
• Romeo Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.

Crossrefs

Cf. A003415, A003958, A007850, A198391.

Programs

Maple

isA199767 := proc(n)
    p := ifactors(n)[2] ;
    add(op(2,d)+op(2,d)/op(1,d),d=p) + mul((1+1/op(1,d))^op(2,d),d=p) ;
    type(%,'integer') ;
end proc:
for n from 20 do
    if isA199767(n) then
           printf("%d,\n",n);
    end if;
end do: # R. J. Mathar, Nov 23 2011