A199767 Numbers n 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 n (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

1,1

Comments

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

Links

Table of n, a(n) for n=1..34.

J. M. Borwein and E. Wong, A survey of results relating to Giuga’s conjecture on primality, May 8, 1995.

R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.

Example

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.

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

Crossrefs

Cf. A003415, A003958, A007850, A198391.

Sequence in context: A372290 A102603 A147222 * A044098 A044479 A139581

Adjacent sequences: A199764 A199765 A199766 * A199768 A199769 A199770

Keyword

nonn,easy

Author

Paolo P. Lava, Nov 22 2011

Status

approved