A304284 Numbers equal to the sum of their aliquot parts, each of them decreased by 8.

120, 460, 1472, 57584, 69488, 76516, 93148, 231748, 600928, 1924096, 8009728, 8043652, 33626692, 1078034816, 2139324416, 2535523012, 8572567552, 188403300352

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 = -8; for t = 23 we get 140734325850112, which is a term greater than 188403300352, but this does not exclude the existence of other intermediate terms following a different solution pattern.

In fact, there could be also sporadic solutions of the type x = 2^t*r*q, where r and q are prime and for which no closed form is known. E.g. for k = -8 we have x = 2^26*134442677*80216006459.

To find them, since d(n) = 4*(t+1) and sigma(n) = (2^(t+1)-1)*(1+r)*(1+q), the relation 2*n = sigma(n) + k*(d(n)-1) becomes 2^(t+1)*r*q = (2^(t+1)-1)*(1+r)*(1+q) + k*(4*t+3), which, for fixed t and k, is a quadratic Diophantine equation in r and q that could admit solutions with r and q prime.

(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..18.

Example

Aliquot parts of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60 and (1-8) + (2-8) + (3-8) + (4-8) + (5-8) + (6-8) + (8-8) + (10-8) + (12-8) + (15-8) + (20-8) + (24-8) + (30-8) + (40-8) + (60-8) = 120.

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, -8);

Mathematica

With[{k = -8}, Select[Range[10^6], DivisorSum[#, # + k &] - (# + k) == # &] ] (* Michael De Vlieger, May 14 2018 *)

Crossrefs

Cf. A000005, A000203, A000396, A304276, A304277, A304278, A304279, A304280, A304281, A304282, A304283.

Sequence in context: A235239 A337469 A235232 * A167562 A033697 A157960

Adjacent sequences: A304281 A304282 A304283 * A304285 A304286 A304287

Keyword

nonn,hard,more

Author

Paolo P. Lava, Giovanni Resta, May 11 2018

Status

approved