A304278 Numbers equal to the sum of their aliquot parts, each of them increased by 6.

7, 33, 148, 165, 1485, 40005, 156928, 195077085, 539705344, 137496887296, 2199280156672, 9444732995769701761024

Offset

1,1

Comments

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 = 6 and for t = 20 we get 2199280156672, which is now known as a(11). - Giovanni Resta, May 11 2018; updated by Max Alekseyev, Jan 20 2026

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.

a(13) <= 25352183624640279459399165. - Max Alekseyev, Jan 20 2026

Links

Table of n, a(n) for n=1..12.

Example

Aliquot part of 7 is 1 and 1+6 = 7.
Aliquot parts of 33 are 1, 3, 11 and (1+6) + (3+6) + (11+6) = 33.

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, 6);

Mathematica

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

Crossrefs

Cf. A000005, A000203, A000396, A304276, A304277, A304279, A304280, A304281, A304282, A304283, A304284.

Sequence in context: A089106 A211829 A227555 * A155603 A282991 A344816

Adjacent sequences: A304275 A304276 A304277 * A304279 A304280 A304281

Keyword

nonn,hard,more

Author

Paolo P. Lava, Giovanni Resta, May 11 2018

Extensions

a(11) from Hiroaki Yamanouchi, Aug 28 2018

a(12) from Max Alekseyev, Jan 20 2026

Status

approved