A304277 Numbers equal to the sum of their aliquot parts, each of them increased by 4.

5, 182, 230, 344, 1072, 3424, 11456, 12844, 321470, 2182144, 33959936, 1084153472, 8598519808, 15381952750, 36113287330004992

Offset

1,1

Comments

Searched up to n = 10^12.

a(16) > 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 = 4; for t = 64 we get 680564733841876936426822412823955505152, which is a term greater than 15381952750, 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 = 4 we have x = 2^17*500069*550959.

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..15.

Example

Aliquot part of 5 is 1 and 1+4 = 5.
Aliquot parts of 182 are 1, 2, 7, 13, 14, 26, 91 and (1+4) + (2+4) + (7+4) + (13+4) + (14+4) + (26+4) + (91+4) = 182.

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, 4);

Mathematica

With[{k = 4}, Select[Range[10^6], DivisorSum[#, # + k &] - (# + k) == # &] ] (* Michael De Vlieger, May 14 2018 *)
Select[Range[219*10^4], Total[Most[Divisors[#]]+4]==#&] (* The program generates the first 10 terms of the sequence. *) (* Harvey P. Dale, Aug 26 2020 *)

Prog

(PARI) isok(n) = sumdiv(n, d, if (d < n, d+4)) == n; \\ Michel Marcus, May 14 2018

Crossrefs

Cf. A000005, A000203, A000396, A304276, A304278, A304279, A304280, A304281, A304282, A304283, A304284.

Sequence in context: A218690 A345338 A256286 * A208403 A280797 A094081

Adjacent sequences: A304274 A304275 A304276 * A304278 A304279 A304280

Keyword

nonn,hard,more

Author

Paolo P. Lava, Giovanni Resta, May 11 2018

Extensions

a(15) from Hiroaki Yamanouchi, Aug 28 2018

Status

approved