A031971 -id:A031971 - cp's OEIS frontend

A229309 Primitive numbers in A229305.

10, 26, 43, 55, 57, 58, 136, 155, 222, 253, 355, 381, 737, 876, 904, 1027, 1055, 1081, 1552, 1711, 1751, 1962, 2696, 2758, 3197, 3403, 3411, 3775, 3916, 4063, 4401, 5093, 5671, 6176, 6567, 7111, 8251, 8515, 8702, 9316, 9465, 10768, 11026, 12195, 12742, 13301

Offset: 1

José María Grau Ribas, Sep 24 2013

nonn

Jose María Grau, A. M. Oller-Marcen, and J. Sondow, On the congruence 1^n + 2^n +... + n^n = d (mod n), where d divides n

Cf. A014117 (numbers n such that A031971(n)==1 (mod n)).
Cf. A229300 (numbers n such that A031971(1806*n)== n (mod n*1806)).
Cf. A229301 (numbers n such that A031971(42*n) == n (mod 42*n)).
Cf. A229302 (numbers n such that A031971(6*n) == n (mod 6*n)).
Cf. A229303 (numbers n such that A031971(2*n) == n (mod 2*n)).
Cf. A229304 (numbers n such that A031971(1806*n) <> n (mod n*1806)).
Cf. A229305 (numbers n such that A031971(42*n) <> n (mod 42*n)).
Cf. A229306 (numbers n such that A031971(6*n) <> n (mod 6*n)).
Cf. A229307 (numbers n such that A031971(2*n) <> n (mod 2*n)).
Cf. A229308 (primitive numbers in A229304).
Cf. A229309 (primitive numbers in A229305).
Cf. A229310 (primitive numbers in A229306).
Cf. A229311 (primitive numbers in A229307).

g[n_] := Mod[Sum[PowerMod[i, n, n], {i, 1, n}], n]; tachar[lis_, num_] := Select[lis, ! IntegerQ[#1/num] &]; primi[{}]={}; primi[lis_] := Join[{lis[[1]]}, primi[tachar[lis, lis[[1]]]]]; primi@Select[Range[80], !g[42*#] == # &]

A229310 Primitive numbers in A229306.

Original entry on oeis.org

7, 10, 26, 55, 57, 136, 155, 222, 253, 737, 876, 1027, 1081, 1552, 1711, 1751, 1962, 3197, 3403, 3775, 3916, 4401, 5671, 6176, 6567, 8251, 8515, 8702, 9316, 11026, 12195, 12742, 13301, 13861, 14878, 15657, 15931, 18145, 20242, 22387, 23126, 25651, 26202

Offset: 1

José María Grau Ribas, Sep 24 2013

nonn

Jose María Grau, A. M. Oller-Marcen, and J. Sondow, On the congruence 1^n + 2^n +... + n^n = d (mod n), where d divides n

Cf. A014117 (numbers n such that A031971(n)==1 (mod n)).
Cf. A229300 (numbers n such that A031971(1806*n)== n (mod n*1806)).
Cf. A229301 (numbers n such that A031971(42*n) == n (mod 42*n)).
Cf. A229302 (numbers n such that A031971(6*n) == n (mod 6*n)).
Cf. A229303 (numbers n such that A031971(2*n) == n (mod 2*n)).
Cf. A229304 (numbers n such that A031971(1806*n) <> n (mod n*1806)).
Cf. A229305 (numbers n such that A031971(42*n) <> n (mod 42*n)).
Cf. A229306 (numbers n such that A031971(6*n) <> n (mod 6*n)).
Cf. A229307 (numbers n such that A031971(2*n) <> n (mod 2*n)).
Cf. A229308 (primitive numbers in A229304).
Cf. A229309 (primitive numbers in A229305).
Cf. A229310 (primitive numbers in A229306).
Cf. A229311 (primitive numbers in A229307).

g[n_] := Mod[Sum[PowerMod[i, n, n], {i, 1, n}], n]; tachar[lis_, num_] := Select[lis, ! IntegerQ[#1/num] &];primi[{}]={}; primi[lis_] := Join[{lis[[1]]}, primi[tachar[lis, lis[[1]]]]]; primi@Select[Range[500], !g[6*#] == # &]

A332517 a(n) = Sum_{k=1..n} gcd(n,k)^n.

Original entry on oeis.org

1, 5, 29, 274, 3129, 47515, 823549, 16843268, 387459861, 10009769725, 285311670621, 8918311856102, 302875106592265, 11112685048729175, 437893951473411261, 18447025557276459016, 827240261886336764193, 39346558373052524325225, 1978419655660313589123997

Offset: 1

Ilya Gutkovskiy, Feb 14 2020

nonn

If n is prime, a(n) = n-1 + n^n. - Robert Israel, Feb 16 2020

Robert Israel, Table of n, a(n) for n = 1..386

Cf. A000010, A008683, A018804, A031971, A069097, A226561, A228640, A321294.

[&+[Gcd(n,k)^n:k in [1..n]]: n in [1..20]]; // Marius A. Burtea, Feb 15 2020

f:= n -> add(igcd(n,k)^n,k=1..n):
map(f, [$1..30]); # Robert Israel, Feb 16 2020

Table[Sum[GCD[n, k]^n, {k, 1, n}], {n, 1, 19}]
Table[Sum[EulerPhi[n/d] d^n, {d, Divisors[n]}], {n, 1, 19}]
Table[Sum[MoebiusMu[n/d] d DivisorSigma[n - 1, d], {d, Divisors[n]}], {n, 1, 19}]

a(n) = sum(k=1, n, gcd(n, k)^n); \\ Michel Marcus, Feb 14 2020

from sympy import totient, divisors
def A332517(n):
    return sum(totient(d)*(n//d)**n for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

a(n) = Sum_{d|n} phi(n/d) * d^n.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_(n-1)(d).
a(n) ~ n^n.
From Richard L. Ollerton, May 09 2021: (Start)
a(n) = Sum_{k=1..n} (n/gcd(n,k))^n*phi(gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))*gcd(n,k)*sigma_(n-1)(gcd(n,k))/phi(n/gcd(n,k)). (End)

A249459 a(n) = Sum_{k=0..n} k^(2*n).

Original entry on oeis.org

1, 1, 17, 794, 72354, 10874275, 2438235715, 762963987380, 317685943157892, 169842891165484965, 113394131858832552133, 92465351109879998121806, 90431265068257318469676710, 104479466717230437574945525959, 140782828210237288756752539959687

Offset: 0

Vaclav Kotesovec, Oct 29 2014

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..214

Cf. A031971, A224899, A249489.

Table[Sum[k^(2*n),{k,1,n}],{n,1,20}]
Table[n!*SeriesCoefficient[Sum[Exp[k^2*x], {k, 1, n}],{x,0,n}], {n,1,20}]

a(n)=n!*polcoeff(sum(k=0, n, exp(k*x+x*O(x^n))^k), n);
for(n=1, 20, print1(a(n), ", "))

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^2*x)^k/(1-k^2*x))) \\ Seiichi Manyama, Dec 03 2021

E.g.f.: Sum_{n>=0} exp(n^2*x).
a(n) ~ exp(2)/(exp(2)-1) * n^(2*n).
G.f.: Sum_{k>=0} (k^2 * x)^k/(1 - k^2 * x). - Seiichi Manyama, Dec 03 2021

a(0)=1 prepended by Seiichi Manyama, Dec 03 2021

A120485 a(n) = n^n - (n-1)^n + (n-2)^n - ... + (-1)^(k+n)k^n + ... + (-1)^(2+n)2^n + (-1)^(1+n)1^n = Sum_{k=1..n} (-1)^(k+n)k^n.

Original entry on oeis.org

1, 1, 3, 20, 190, 2313, 34461, 607408, 12360636, 285188825, 7356173275, 209762134236, 6552069616170, 222481706868337, 8159714626124985, 321456928026650816, 13538204870285608696, 606979028986115413329

Offset: 0

Alexander Adamchuk, Jul 22 2006

nonn

p divides a(p-1) for prime p>2. p^k divides a(p^k-1) for all prime p and integer k>1. p^2 divides a(2p) and a(2p-1) for prime p>2. (p^k)^2 divides a(2p^k) for prime p>2 and integer k>0. (p^k)^2 divides a(2p^k-1) for all prime p and integer k>1.

It seems that a(n) ~ k*n^n with k = e/(e+1). - Charles R Greathouse IV, May 26 2015

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A031971, A089072.

Main diagonal of A091884.

[(-1)^n*(&+[(-1)^k*k^n: k in [0..n]]): n in [0..40]]; // G. C. Greubel, Nov 01 2022

Table[Sum[(-1)^(k+n)*k^n,{k,1,n}],{n,1,25}]

a(n)=abs(sum(i=1,n,i^n*(-1)^i)) \\ Charles R Greathouse IV, May 26 2015

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1+k*x))) \\ Seiichi Manyama, Dec 03 2021

[(-1)^n*sum((-1)^k*k^n for k in range(n+1)) for n in range(41)] # G. C. Greubel, Nov 01 2022

a(n) = Sum_{k=1..n} (-1)^(k+n)*k^n.
a(n) = (-1)^n*((-1+2^(n+1))*Zeta[ -n] + (-2)^n*((Zeta[ -n,(n+1)/2] - Zeta[ -n,(n+2)/2]))).
a(n) = n! * [x^n] exp(x)*(exp(n*x) + 1)/(exp(x) + 1). - Ilya Gutkovskiy, Apr 07 2018
G.f.: Sum_{k>=0} (k * x)^k/(1 + k * x). - Seiichi Manyama, Dec 03 2021

A226960 Numbers n such that 1^n + 2^n + 3^n +...+ n^n == 2 (mod n).

Original entry on oeis.org

1, 4, 12, 84, 3612

Offset: 1

José María Grau Ribas, Jun 24 2013

Also, numbers n such that B(n)*n == 2 (mod n), where B(n) is the n-th Bernoulli number. Equivalently, SUM[prime p, (p-1) divides n] n/p == -2 (mod n). - Max Alekseyev, Aug 25 2013

M. A. Alekseyev, J. M. Grau, A. M. Oller-Marcen. Computing solutions to the congruence 1^n + 2^n + ... + n^n == p (mod n). Discrete Applied Mathematics, 2018. doi:10.1016/j.dam.2018.05.022 arXiv:1602.02407 [math.NT]

Cf. A031971, A014117.
Subsequence of A124240.
Solutions to 1^n+2^n+...+n^n == m (mod n): A005408 (m=0), A014117 (m=1), this sequence (m=2), A226961 (m=3), A226962 (m=4), A226963 (m=5), A226964 (m=6), A226965 (m=7), A226966 (m=8), A226967 (m=9), A280041 (m=19), A280043 (m=43), A302343 (m=79), A302344 (m=193).

Select[Range[10000], Mod[Sum[PowerMod[i, #, #], {i, #}], #] == 2 &]

is(n)=if(n%2,return(n==1)); Mod(sumdiv(n/2,d, if(isprime(2*d+1), n/(2*d+1)))+n/2,n)==-2 \\ Charles R Greathouse IV, Nov 13 2013

a(1)=1 prepended by Max Alekseyev, Aug 25 2013

A226961 Numbers n such that 1^n + 2^n + 3^n + ... + n^n == 3 (mod n).

Original entry on oeis.org

1, 2, 3, 18, 126, 5418

Offset: 1

José María Grau Ribas, Jun 24 2013

Equivalently, numbers n such that B(n)*n == 3 (mod n), where B(n) is the n-th Bernoulli number. Equivalently, SUM[prime p, (p-1) divides n] n/p == -3 (mod n). - Max Alekseyev, Aug 25 2013

M. A. Alekseyev, J. M. Grau, A. M. Oller-Marcen. Computing solutions to the congruence 1^n + 2^n + ... + n^n == p (mod n). Discrete Applied Mathematics, 2018. doi:10.1016/j.dam.2018.05.022 arXiv:1602.02407 [math.NT]

Cf. A031971.

Solutions to 1^n+2^n+...+n^n == m (mod n): A005408 (m=0), A014117 (m=1), A226960 (m=2), this sequence (m=3), A226962 (m=4), A226963 (m=5), A226964 (m=6), A226965 (m=7), A226966 (m=8), A226967 (m=9), A280041 (m=19), A280043 (m=43), A302343 (m=79), A302344 (m=193).

Select[Range[10000], Mod[Sum[PowerMod[i, #, #], {i, #}], #] == 3 &]

is(n)=Mod(sumdiv(n, d, if(isprime(d+1), n/(d+1))), n)==-3 \\ Charles R Greathouse IV, Nov 13 2013

1, 2, 3 prepended by Max Alekseyev, Aug 25 2013

A226963 Numbers n such that 1^n + 2^n + 3^n + ... + n^n == 5 (mod n).

Original entry on oeis.org

1, 2, 5, 10, 30, 210, 9030, 235290, 11072512110

Offset: 1

José María Grau Ribas, Jun 24 2013

Also, numbers n such that B(n)*n == 5 (mod n), where B(n) is the n-th Bernoulli number. Equivalently, SUM[prime p, (p-1) divides n] n/p == -5 (mod n). - Max Alekseyev, Aug 26 2013

There are no other terms below 10^31. - Max Alekseyev, Apr 04 2018

M. A. Alekseyev, J. M. Grau, A. M. Oller-Marcen. Computing solutions to the congruence 1^n + 2^n + ... + n^n == p (mod n). Discrete Applied Mathematics, 2018. doi:10.1016/j.dam.2018.05.022 arXiv:1602.02407 [math.NT]

Cf. A031971.

Solutions to 1^n+2^n+...+n^n == m (mod n): A005408 (m=0), A014117 (m=1), A226960 (m=2), A226961 (m=3), A226962 (m=4), this sequence (m=5), A226964 (m=6), A226965 (m=7), A226966 (m=8), A226967 (m=9), A280041 (m=19), A280043 (m=43), A302343 (m=79), A302344 (m=193).

Select[Range[10000], Mod[Sum[PowerMod[i, #, #], {i, #}], #] == 5 &]

is(n)=Mod(sumdiv(n, d, if(isprime(d+1), n/(d+1))), n)==-5 \\ Charles R Greathouse IV, Nov 13 2013

Terms 1,2,5 prepended and a(9) added by Max Alekseyev, Aug 26 2013

A226967 Numbers n such that 1^n + 2^n + 3^n + ... + n^n == 9 (mod n).

Original entry on oeis.org

1, 2, 3, 9, 54, 378, 16254

Offset: 1

José María Grau Ribas, Jun 24 2013

Also, numbers n such that B(n)*n == 9 (mod n), where B(n) is the n-th Bernoulli number. Equivalently, SUM[prime p, (p-1) divides n] n/p == -9 (mod n). There are no other terms below 10^30. - Max Alekseyev, Aug 26 2013

Cf. A031971.

Solutions to 1^n+2^n+...+n^n == m (mod n): A005408 (m=0), A014117 (m=1), A226960 (m=2), A226961 (m=3), A226962(m=4), A226963 (m=5), A226964 (m=6), A226965 (m=7), A226966 (m=8), this sequence (m=9), A280041 (m=19), A280043 (m=43), A302343 (m=79), A302344 (m=193).

Select[Range[10000], Mod[Sum[PowerMod[i, #, #], {i, #}], #] == Mod[9,#] &]

is(n)=Mod(sumdiv(n, d, if(isprime(d+1), n/(d+1))), n)==-9 \\ Charles R Greathouse IV, Nov 13 2013

1,2,9 prepended by Max Alekseyev, Aug 26 2013

A226962 Numbers n such that 1^n + 2^n + 3^n + ... + n^n == 4 (mod n).

Original entry on oeis.org

1, 8, 24, 168, 7224

Offset: 1

José María Grau Ribas, Jun 24 2013

Also, numbers n such that B(n)*n == 4 (mod n), where B(n) is the n-th Bernoulli number. Equivalently, SUM[prime p, (p-1) divides n] n/p == -4 (mod n). There are no other terms below 10^15. - Max Alekseyev, Aug 26 2013

Cf. A031971.

Solutions to 1^n+2^n+...+n^n == m (mod n): A005408 (m=0), A014117 (m=1), A226960 (m=2), A226961 (m=3), this sequence (m=4), A226963 (m=5), A226964 (m=6), A226965 (m=7), A226966 (m=8), A226967 (m=9), A280041 (m=19), A280043 (m=43), A302343 (m=79), A302344 (m=193).

Select[Range[10000], Mod[Sum[PowerMod[i, #, #], {i, #}], #] == 4 &]

is(n)=Mod(sumdiv(n, d, if(isprime(d+1), n/(d+1))), n)==-4 \\ Charles R Greathouse IV, Nov 13 2013

a(1)=1 prepended by Max Alekseyev, Aug 26 2013

cp's OEIS Frontend

A229309 Primitive numbers in A229305.

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A229310 Primitive numbers in A229306.

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A332517 a(n) = Sum_{k=1..n} gcd(n,k)^n.

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A249459 a(n) = Sum_{k=0..n} k^(2*n).

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A120485 a(n) = n^n - (n-1)^n + (n-2)^n - ... + (-1)^(k+n)*k^n + ... + (-1)^(2+n)*2^n + (-1)^(1+n)*1^n = Sum_{k=1..n} (-1)^(k+n)*k^n.

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A226960 Numbers n such that 1^n + 2^n + 3^n +...+ n^n == 2 (mod n).

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A226961 Numbers n such that 1^n + 2^n + 3^n + ... + n^n == 3 (mod n).

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A226963 Numbers n such that 1^n + 2^n + 3^n + ... + n^n == 5 (mod n).

A120485 a(n) = n^n - (n-1)^n + (n-2)^n - ... + (-1)^(k+n)k^n + ... + (-1)^(2+n)2^n + (-1)^(1+n)1^n = Sum_{k=1..n} (-1)^(k+n)k^n.