A057292 -id:A057292 - cp's OEIS frontend

A057291 Numbers k such that k | 12^k + 11^k + 10^k + 9^k + 8^k + 7^k + 6^k + 5^k + 4^k + 3^k + 2^k + 1^k.

1, 2, 3, 9, 10, 13, 26, 27, 39, 50, 81, 110, 117, 130, 169, 243, 250, 279, 310, 338, 351, 470, 507, 550, 650, 663, 729, 1053, 1209, 1250, 1430, 1521, 1550, 1690, 2187, 2197, 2750, 3159, 3250, 3410, 4030, 4043, 4069, 4394, 4509, 4563, 6250, 6561, 6591, 7150

Offset: 1

Robert G. Wilson v, Sep 22 2000

nonn

The only primes in the sequence are 2, 3 and 13. - Robert Israel, Jun 25 2025

Robert Israel, Table of n, a(n) for n = 1..720

Cf. A056739, A057292, A103438.

filter:= n -> 12&^n + 11&^n + 10&^n + 9&^n + 8&^n + 7&^n + 6&^n + 5&^n + 4&^n + 3&^n + 2&^n + 1 mod n = 0:
select(filter, [$1..10^4]); # Robert Israel, Jun 25 2025

Select[ Range[ 10^5 ], Mod[ PowerMod[ 12, #, # ] + PowerMod[ 11, #, # ] + PowerMod[ 10, #, # ] + PowerMod[ 9, #, # ] + PowerMod[ 8, #, # ] + PowerMod[ 7, #, # ] + PowerMod[ 6, #, # ] + PowerMod[ 5, #, # ] + PowerMod[ 4, #, # ] + PowerMod[ 3, #, # ] + PowerMod[ 2, #, # ] + 1, # ] == 0 & ]
Select[Range[7200],Divisible[Total[Range[12]^#],#]&] (* Harvey P. Dale, Aug 05 2017 *)

A056739 Numbers k such that k | 10^k + 9^k + 8^k + 7^k + 6^k + 5^k + 4^k + 3^k + 2^k + 1^k.

Original entry on oeis.org

1, 5, 11, 25, 55, 121, 125, 275, 365, 605, 625, 925, 1331, 1375, 2365, 3025, 3125, 6655, 6875, 14641, 15125, 15625, 22625, 27565, 32125, 33275, 34375, 73205, 75625, 78125, 123365, 161051, 166375, 171875, 366025, 378125, 390625, 541717, 660605

Offset: 1

Robert G. Wilson v, Aug 25 2000

nonn

Contains A003598. In general n=p^i * q^j => n | Sum_{k=1..2*p} k^n, where p and q=2*p+1 are prime (see Meyer ref).

All terms == 1 or 5 (mod 6). The only prime terms are 5 and 11. - Robert Israel, Jun 25 2025

Robert Israel, Table of n, a(n) for n = 1..140
Christian Meyer, On conjecture no. 22 arising from the OEIS

Cf. A001557, A003598, A057291, A057292.

filter:= n ->   10 &^n + 9 &^ n + 8 &^ n + 7 &^ n + 6&^ n + 5&^n + 4&^n + 3&^n + 2&^n + 1 mod n = 0:
select(filter, [seq(seq(6*i + j, j=[1,5]),i=0..10^6)]); # Robert Israel, Jun 25 2025

Do[ If[ Mod[ PowerMod[ 10, n, n ] + PowerMod[ 9, n, n ] + PowerMod[ 8, n, n ] + PowerMod[ 7, n, n ] + PowerMod[ 6, n, n ] + PowerMod[ 5, n, n ] + PowerMod[ 4, n, n ] + PowerMod[ 3, n, n ] + PowerMod[ 2, n, n ] + 1, n ] == 0, Print[ n ] ], {n, 1, 10^6} ]
Select[Range[700000],Divisible[Total[Range[10]^#],#]&] (* Harvey P. Dale, Nov 24 2014 *)
Select[Range[700000],Mod[Total[PowerMod[Range[10],#,#]],#]==0&] (* Harvey P. Dale, Feb 23 2023 *)

A056741 Numbers k such that k | 5^k + 4^k + 3^k + 2^k + 1^k.

Original entry on oeis.org

1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 125, 135, 225, 243, 261, 295, 375, 405, 625, 675, 729, 925, 1125, 1215, 1875, 2025, 2187, 3125, 3375, 3645, 4077, 4833, 5139, 5625, 6075, 6345, 6561, 9375, 10125, 10935, 15625, 16875, 17895, 18125, 18225, 18495, 19683

Offset: 1

Robert G. Wilson v, Aug 25 2000

nonn

All terms are odd. The only primes in the sequence are 3 and 5. - Robert Israel, Jun 25 2025

Robert Israel, Table of n, a(n) for n = 1..300 (first 100 terms from Harvey P. Dale)

Cf. A001552, A056739, A057291, A057292.

filter:= n ->   5&^n + 4&^n + 3&^n + 2&^n + 1 mod n = 0:
select(filter, [seq(i,i=1..10^5,2)]); # Robert Israel, Jun 25 2025

Do[ If[ Mod[ PowerMod[ 5, n, n ] + PowerMod[ 4, n, n ] + PowerMod[ 3, n, n ] + PowerMod[ 2, n, n ] + 1, n ] == 0, Print[ n ] ], {n, 1, 10^6} ]
Select[Range[20000],Mod[Total[PowerMod[Range[0,5],#,#]],#]==0&] (* Harvey P. Dale, Oct 09 2021 *)

A385314 a(n) is the least positive integer m such that Sum_{k = 1 .. m} k^n is divisible by n.

Original entry on oeis.org

1, 3, 2, 7, 4, 4, 6, 15, 2, 4, 10, 8, 12, 3, 5, 31, 16, 27, 18, 24, 6, 11, 22, 31, 4, 12, 2, 7, 28, 4, 30, 63, 11, 8, 14, 40, 36, 19, 12, 31, 40, 8, 42, 16, 5, 11, 46, 31, 6, 4, 17, 32, 52, 40, 10, 31, 18, 28, 58, 31, 60, 15, 6, 127, 4, 27, 66, 8, 23, 7, 70, 80, 72, 36, 5, 47, 6, 36, 78, 31, 2

Offset: 1

Robert Israel, Jun 25 2025

nonn

If p is an odd prime, a(p) = p - 1.
If n > 1 is odd, a(n) <= n - 1.
For all n, a(n) <= n^2 - 1.

			a(3) = 2 because 1^3 + 2^3 = 9 is divisible by 3, while 1^3 is not.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A057291, A057292, A103438.

f:= proc(n) local k,t;
  t:= 0:
  for k from 1 do
    t:= t + k &^ n mod n;
    if t = 0 then return k fi;
  od:
end proc:
map(f, [$1..100]);

a[n_]:=Module[{m=1},While[!Divisible[Sum[k^n,{k,1,m}],n],m++];m];Array[a,81] (* James C. McMahon, Jun 25 2025 *)

a(n) = my(m=1); while(sum(k=1, m, k^n) % n, m++); m; \\ Michel Marcus, Jun 25 2025

cp's OEIS Frontend

A057291 Numbers k such that k | 12^k + 11^k + 10^k + 9^k + 8^k + 7^k + 6^k + 5^k + 4^k + 3^k + 2^k + 1^k.

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A056739 Numbers k such that k | 10^k + 9^k + 8^k + 7^k + 6^k + 5^k + 4^k + 3^k + 2^k + 1^k.

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A056741 Numbers k such that k | 5^k + 4^k + 3^k + 2^k + 1^k.

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A385314 a(n) is the least positive integer m such that Sum_{k = 1 .. m} k^n is divisible by n.

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