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

Offset

1,1

Comments

Searched up to n = 10^12.

a(12) > 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 = 6 and for t = 20 we get 2199280156672 that is another term greater than 137496887296 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..11.

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

Status

approved