A304276 Numbers equal to the sum of their aliquot parts, each of them increased by 2.

3, 15, 68, 232, 543232, 2139136, 8796271280128, 35184745381888

Offset

1,1

Comments

Searched up to n = 10^12.

a(9) > 10^18. - Hiroaki Yamanouchi, Aug 28 2018

From Giovanni Resta, May 11 2018: (Start)

If p = 2^(1+t) + (1+2*t)*k - 1 is a prime, for some t > 0 and k even, then x = 2^t*p is in the sequence where k is the value by which the sum of aliquot parts is increased.

In this sequence k = 2; for t = 21 we get 8796271280128 which is a term greater than 2139136, but this does not exclude the existence of other intermediate terms following a different solution pattern.

(End)

Terms using odd values of k seem very hard to find. Up to n = 10^12, only three such terms are known: 2, 98, and 8450, for k = 1, 5, and -7, respectively.

Links

Table of n, a(n) for n=1..8.

Example

Aliquot part of 3 is 1 and 1+2 = 3.
Aliquot parts of 15 are 1, 3, 5 and (1+2) + (3+2) + (5+2) = 15.

Maple

with(numtheory): P:=proc(q, k) local n;
for n from 1 to q do if 2*n=sigma(n)+k*(tau(n)-1) then print(n);
fi; od; end: P(10^12, 2);

Mathematica

Select[Range[10^6], DivisorSum[#, # + 2 &] - (# + 2) == # &] (* Michael De Vlieger, May 14 2018 *)

Crossrefs

Cf. A000005, A000203, A000396, A304277, A304278, A304279, A304280, A304281, A304282, A304283, A304284.

Sequence in context: A052981 A086200 A325586 * A217451 A213451 A224749

Adjacent sequences: A304273 A304274 A304275 * A304277 A304278 A304279

Keyword

nonn,hard,more

Author

Paolo P. Lava, Giovanni Resta, May 11 2018

Extensions

a(7)-a(8) from Hiroaki Yamanouchi, Aug 28 2018

Status

approved