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A304284 Numbers equal to the sum of their aliquot parts, each of them decreased by 8. 8
120, 460, 1472, 57584, 69488, 76516, 93148, 231748, 600928, 1924096, 8009728, 8043652, 33626692, 1078034816, 2139324416, 2535523012, 8572567552, 188403300352 (list; graph; refs; listen; history; text; internal format)
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: A307933 A235239 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

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Last modified May 26 02:56 EDT 2020. Contains 334613 sequences. (Running on oeis4.)