A304280 Numbers equal to the sum of their aliquot parts, each of them increased by 10.

11, 51, 675, 1450, 1755, 1995, 5536, 16448, 621056, 35186255331328, 144115340815630336

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 = 10; for t = 22 we get 35186255331328, which is a term greater than 621056, but this does not exclude the existence of other intermediate terms following a different solution pattern. (End)

With the same searching range, no value has been found for the similar sequence with aliquot parts decreased by 10. For n > 10^12, solutions of the kind x = 2^t*p can be found for t = 20, 33, 57, 76, 134, 422, 496, ...: e.g. 2^20*2096741 = 2198592290816.

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 11 is 1 and 1+10 = 11.
Aliquot parts of 51 are 1, 3, 17 and (1+10) + (3+10) + (17+10) = 51.

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, 10);

Mathematica

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

Prog

(PARI) isok(n) = sumdiv(n, d, if (d < n, d+10)) == n; \\ Michel Marcus, May 14 2018

Crossrefs

Cf. A000005, A000203, A000396, A304276, A304277, A304278, A304279, A304281, A304282, A304283, A304284.

Sequence in context: A321421 A317021 A199895 * A231386 A191099 A325727

Adjacent sequences: A304277 A304278 A304279 * A304281 A304282 A304283

Keyword

nonn,hard,more

Author

Paolo P. Lava, Giovanni Resta, May 11 2018

Extensions

a(10)-a(11) from Hiroaki Yamanouchi, Aug 28 2018

Status

approved