OFFSET
1,1
COMMENTS
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.
From Giovanni Resta, May 11 2018; updated by Max Alekseyev, Jun 18 2025: (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 = 20 we get a(11) = 2198850240512.
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 = 2^31*4294967357*297528134446815421.
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)
Some large terms not of the form 2^t*p: 2310371603427730330910457856, 2412353826707071170936175618, 5418674794944518417979629633536, 2744212714409470551042856619088019456, 10874213664581953026714846132140941646319545382600704. - Max Alekseyev, Feb 18 2026
EXAMPLE
Aliquot parts of 350 are 1, 2, 5, 7, 10, 14, 25, 35, 50, 70, 175 and (1-4) + (2-4) + (5-4) + (7-4) + (10-4) + (14-4) + (25-4) + (35-4) + (50-4) + (70-4) + (175-4) = 350.
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 *)
CROSSREFS
KEYWORD
nonn,hard
AUTHOR
Paolo P. Lava and Giovanni Resta, May 11 2018
EXTENSIONS
a(11) from Giovanni Resta confirmed and a(12)-a(16) added by Max Alekseyev, Jul 24 2025
a(17)-a(19) from Max Alekseyev, Feb 18 2026
STATUS
approved