A318528 a(n) = least number > 1 that equals the sum of the n-th powers of its first k divisors for some k.

6, 130, 36, 41860, 276, 1015690, 2316, 921951940, 20196, 10009766650, 179196, 2387003305930334914, 1602516, 100006103532010, 14381676, 1880100018939820249188604888836, 129271236, 1000003814697527770, 1162785756, 19105043663614041367780, 10462450356, 10000002384185795209930, 94151567436, 226500219158007133816826003223992308820431641700

Offset

1,1

Comments

a(48) > 10^90. - Max Alekseyev, Jan 17 2025

Links

Max Alekseyev, Table of n, a(n) for n = 1..47

Formula

a(n) = 1 + 2^n + 3^n for n = p^k with prime p > 2. - Giovanni Resta, Aug 28 2018

From Charlie Neder, Jan 24 2019: (Start)

a(n) = 1 + 2^n + 3^n for n odd,

a(n) = 1 + 2^n + 5^n + 10^n for n congruent to 2 modulo 4,

a(n) = 1 + 2^n + 4^n + 5^n + 7^n + 10^n + 13^n for n congruent to 4 or 8 modulo 12 and not 16 modulo 20.

All other a(n) contain a term at least 24^n. (End)

Example

a(2) = 130 since 130 has the divisors 1, 2, 5, 10, ... and 1^2 + 2^2 + 5^2 + 10^2 = 130.

Mathematica

a[k_] := Module[{n = 2}, While[! MemberQ[Accumulate[Divisors[n]^k], n], n++]; n]; Do[Print[a[n]], {n, 1, 10}]

Prog

(PARI) a(n) = for(x=2, oo, my(div=divisors(x), s=0); for(k=1, #div, s=sum(i=1, k, div[i]^n); if(s==x, return(x)))) \\ Felix Fröhlich, Aug 28 2018

Crossrefs

Cf. A027750, A185584, A185960, A190940, A307137, A378313, A378314.

Sequence in context: A012842 A012638 A338709 * A095695 A156475 A000907

Adjacent sequences: A318525 A318526 A318527 * A318529 A318530 A318531

Keyword

nonn

Author

Amiram Eldar, Aug 28 2018

Extensions

a(12)-a(24) from Giovanni Resta confirmed by Max Alekseyev, Jan 04 2025

Status

approved