A226365 Composite numbers such that 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).

152, 432, 1620, 1728, 3456, 4752, 22464, 46656, 80892, 139968, 186624, 237168, 326592, 746496, 1651968, 2052864, 2426112, 2985984, 5971968, 10257408, 12177216, 12690432, 14048240, 14183424, 20155392, 20901888, 26127360, 38817792

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) - 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.

Links

Table of n, a(n) for n=1..28.

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

237168 has prime factors 2, 2, 2, 2, 3, 3, 3, 3, 3, 61. 4*(1 + 1/2) + 5*(1 + 1/3) + (1 + 1/61) = 2504/183 is the sum over the 1 + 1/p_i. (1 + 1/2)^4 * (1 + 1/3)^5 * (1 + 1/61) = 3968/183 is the product of the 1 + 1/p_i. The difference over sum and product is 2504/183 - 3968/183 = -8, an integer.

Maple

with(numtheory); ListA226365:=proc(q) local a, d, n, p;
for n from 1 to q do if not isprime(n) then p:=ifactors(n)[2];
a:=add(op(2, d)+op(2, d)/op(1, d), d=p)-mul((1+1/op(1, d))^op(2, d), d=p);
if type(a, integer) then print(n); fi; fi;
od; end: ListA226365(10^9);

Crossrefs

Cf. A003415, A003958, A198391, A199767, A224912.

Sequence in context: A137632 A259740 A253359 * A085775 A208492 A208485

Adjacent sequences: A226362 A226363 A226364 * A226366 A226367 A226368

Keyword

nonn

Author

Paolo P. Lava, Jun 12 2013

Status

approved