A304281 Numbers equal to the sum of their aliquot parts, each of them decreased by 2.

208, 945, 1312, 6464, 8292352

Offset

1,1

Comments

Searched up to n = 10^12.

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 8795910569984, which is a term greater than 8292352, 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..5.

Example

Aliquot parts of 208 are 1 and 1, 2, 4, 8, 13, 16, 26, 52, 104 and (1-2) + (2-2) + (4-2) + (8-2) + (13-2) + (16-2) + (26-2) + (52-2) + (104-2) = 208.

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

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

Prog

(PARI) isok(n) = sumdiv(n, d, if (d < n, d-2)) == n; \\ Michel Marcus, May 14 2018

Crossrefs

Cf. A000005, A000203, A000396, A304276, A304277, A304278, A304279, A304280, A304282, A304283, A304284.

Sequence in context: A235273 A255082 A255075 * A234549 A234543 A233665

Adjacent sequences: A304278 A304279 A304280 * A304282 A304283 A304284

Keyword

nonn,hard,more

Author

Paolo P. Lava, Giovanni Resta, May 11 2018

Status

approved