A304279 Numbers equal to the sum of their aliquot parts, each of them increased by 8.

188, 370, 568, 1612, 1648, 1892, 4832, 70384, 165632, 430508, 2394848, 76393148, 85532348, 174712264, 540655616, 2359808828, 2544998588, 2199366139904, 35236711282688, 2931045725111036, 20542625035648508, 144115310213988352, 144141640382529536

Offset

1,1

Comments

Searched up to n = 10^12.

a(24) > 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 = 8; for t = 20 we get 2199366139904, which is a term greater than 2544998588, but this does not exclude the existence of other intermediate terms following a different solution pattern.

In fact, there could also be 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 = 2^14*32771*268460149.

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..23.

Example

Aliquot parts of 188 are 1, 2, 4, 47, 94; (1+8) + (2+8) + (4+8) + (47+8) + (94+8) = 188.
Aliquot parts of 370 are 1, 2, 5, 10, 37, 74, 185; (1+8) + (2+8) + (5+8) + (10+8) + (37+8) + (74+8) + (185+8) = 370.

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);

Crossrefs

Cf. A000005, A000203, A000396, A304276, A304277, A304278, A304280, A304281, A304282, A304283, A304284.

Sequence in context: A243077 A186397 A260836 * A099945 A211814 A335592

Adjacent sequences: A304276 A304277 A304278 * A304280 A304281 A304282

Keyword

nonn,hard,more

Author

Paolo P. Lava, Giovanni Resta, May 11 2018

Extensions

a(18)-a(23) from Hiroaki Yamanouchi, Aug 28 2018

Status

approved