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 = 6 and for t = 20 we get 2199280156672 that is another term greater than 137496887296 but this does not exclude the existence of other intermediate terms following a different solution pattern.

(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.

EXAMPLE

Aliquot part of 7 is 1 and 1+6 = 7.

Aliquot parts of 33 are 1, 3, 11 and (1+6) + (3+6) + (11+6) = 33.

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, 6);

MATHEMATICA

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

CROSSREFS

KEYWORD

nonn,hard,more

AUTHOR

Paolo P. Lava, Giovanni Resta, May 11 2018

EXTENSIONS

a(11) from Hiroaki Yamanouchi, Aug 28 2018

STATUS

approved