A031971 -id:A031971 - cp's OEIS frontend

A226964 Numbers n such that 1^n + 2^n + 3^n + ... + n^n == 6 (mod n).

1, 3, 4, 20, 36, 252, 10836

Offset: 1

José María Grau Ribas, Jun 24 2013

Also, numbers n such that B(n)*n == 6 (mod n), where B(n) is the n-th Bernoulli number. Equivalently, SUM[prime p, (p-1) divides n] n/p == -6 (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), A226962 (m=4), A226963 (m=5), this sequence (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, #}], #] == 6 &]

is(n)=Mod(sumdiv(n, d, if(isprime(d+1), n/(d+1))), n)==-6 \\ Charles R Greathouse IV, Nov 13 2013

1,3,4 prepended by Max Alekseyev, Aug 26 2013

A226965 Numbers n such that 1^n + 2^n + 3^n +...+ n^n == 7 (mod n).

Original entry on oeis.org

1, 2, 6, 7, 14, 294, 12642

Offset: 1

José María Grau Ribas, Jun 24 2013

Also, integers n such that B(n)*n == 7 (mod n), where B(n) is the n-th Bernoulli number, or SUM[prime p, (p-1) divides n] n/p == -7 (mod n). It is easy to see that for n>1, every prime divisor p of n, except p=7, must appear in first power, while p=7 may appear in first or second power. Moreover, the multiset P of prime divisors of all such n satisfies the property: if p is in P, then p-1 is the product of distinct elements of P. This multiset is P = {2, 3, 7, 7, 43}, implying that the sequence is finite and complete. - 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), A226961 (m=3), A226962(m=4), A226963 (m=5), A226964 (m=6), this sequence (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, #}], #] == 7&]

is(n)=Mod(sumdiv(n, d, if(isprime(d+1), n/(d+1))), n)==-7 \\ Charles R Greathouse IV, Nov 13 2013

Corrected and keywords full,fini added by Max Alekseyev, Aug 25 2013

A226966 Numbers n such that 1^n + 2^n + 3^n + ... + n^n == 8 (mod n).

Original entry on oeis.org

1, 16, 48, 336, 14448

Offset: 1

José María Grau Ribas, Jun 24 2013

Also, numbers n such that B(n)*n == 8 (mod n), where B(n) is the n-th Bernoulli number. Equivalently, SUM[prime p, (p-1) divides n] n/p == -8 (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), A226962(m=4), A226963 (m=5), A226964 (m=6), A226965 (m=7), this sequence (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, #}], #] == 8 &]

is(n)=Mod(sumdiv(n, d, if(isprime(d+1), n/(d+1))), n)==-8 \\ Charles R Greathouse IV, Nov 13 2013

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

A231409 Least k with 1^(km) + 2^(km) + ... + (km)^(km) == k (mod k*m) for m in A230311.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 5, 39607528021345872635

Offset: 1

Jonathan Sondow, Nov 30 2013

Least k with A031971(k*m) == k (mod k*m) for m in A230311.

See A031971 and A230311 for more comments and crossrefs.

			1^m + 2^m + ... + m^m == 1 (mod m) for the first 5 terms m = 1, 2, 6, 42, 1806 of A230311, so a(n) = 1 for n <= 5.

J. M. Grau, A. M. Oller-Marcen, and J. Sondow, On the congruence 1^m + 2^m + ... + m^m == n (mod m) with n|m, arXiv:1309.7941 [math.NT].

Cf. A031971, A229300, A229301, A229302, A229303, A230311.

a(2) = A229303(1), a(3) = A229302(1), a(4) = A229301(1), a(5) = A229300, a(6) = A229312(1).

A121706 a(n) = Sum_{k=1..n-1} k^n.

Original entry on oeis.org

0, 1, 9, 98, 1300, 20515, 376761, 7907396, 186884496, 4914341925, 142364319625, 4505856912854, 154718778284148, 5729082486784839, 227584583172284625, 9654782997596059912, 435659030617933827136, 20836030169620907691465

Offset: 1

Alexander Adamchuk, Aug 16 2006

nonn

n^3 divides a(n) for n in A121707.
It appears that p^(3k-1) divides a(p^k) for all integer k > 1 and prime p > 2:
  for prime p > 2, p^2 divides a(p), p^5 divides a(p^2) and p^8 divides a(p^3).
Additionally, p^3 divides a(3p) for prime p > 2.
For prime p > 3, p divides a(p+1) and p^3 divides a(2p+1);
  for prime p > 5, p divides a(3p+1) and p^3 divides a(4p+1);
  for prime p > 7, p divides a(5p+1) and p^3 divides a(6p+1):
It appears that p divides a((2k+1)p+1) for integer k >= 0 and prime p > 2k+3, and p^3 divides a(2kp+1) for integer k > 0 and prime p > 2k+2.
p divides a((p+1)/2) for primes in A002145: primes of the form 4n+3, n >= 1.
p^2 divides a((p+1)/2) for primes in A007522: primes of the form 8n+7, n >= 0.
n*(2*n+1) divides a(2*n+1) for n >= 1. - Franz Vrabec, Dec 20 2020

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A121707, A031971, A000312, A002145, A007522.

A121706 := proc(n)
    (bernoulli(n+1,n)-bernoulli(n+1))/(n+1) ;
end proc: # R. J. Mathar, May 10 2013

Mathematica
```
Table[Sum[k^n,{k,1,n-1}],{n,1,35}]
```

a(n)=sum(k=1,n-1,k^n) \\ Charles R Greathouse IV, May 09 2013

a(n)=subst(sumformal('x^n),'x,n-1) \\ Charles R Greathouse IV, May 09 2013

a(n) = Sum(k^n, k=1..n) - n^n = A031971(n) - A000312(n) for n > 1.

a(n) = zeta(-n) - zeta(-n, n).

Edited by M. F. Hasler, Jul 22 2019

A344110 Triangle read by rows: T(n,k) = 2^(n*k), n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 2, 1, 4, 16, 1, 8, 64, 512, 1, 16, 256, 4096, 65536, 1, 32, 1024, 32768, 1048576, 33554432, 1, 64, 4096, 262144, 16777216, 1073741824, 68719476736, 1, 128, 16384, 2097152, 268435456, 34359738368, 4398046511104, 562949953421312

Offset: 0

Mohammad K. Azarian, May 10 2021

T(n, k) is the number of relations from an n-element set into a k-element set, n >= 0, 0 <= k <= n.

T(n,k) is the size of the right principal ideal generated by A where A is an n X n matrix over GF(2) having rank k. The right principal ideal of A contains precisely the matrices whose image is contained in the image of A. - Geoffrey Critzer, Sep 25 2022

			T(3,3) = number of relations from a 3-element set into a 3-element set=2^(3*3)=512.
Triangle begins:
   1
   1   2
   1   4      16
   1   8      64      512
   1  16     256     4096      65536
   1  32    1024    32768    1048576    33554432
   ...

Michael De Vlieger, Table of n, a(n) for n = 0..1325 (rows n = 0..50, flattened)
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141.

Cf. A000079, A089072, A008292, A031971, A117401, A344260 (row sums).

Cf. A022166, A288853.

Mathematica
```
Table[2^(n*k), {n, 0, 10}, {k, 0, n}]
```

T(n,k) = 2^(n*k).

T(n,k) = Sum_{j=0..k} A288853(n,j)*A022166(n,j). - Geoffrey Critzer, Jan 02 2023

A349886 a(n) = Sum_{k=0..n} k^(k*n).

Original entry on oeis.org

1, 2, 18, 19749, 4295498995, 298024323402930834, 10314425729813391637014599924, 256923578002288684397369021397408936103993, 6277101735598268377660667072561845282166297358613176925573

Offset: 0

Seiichi Manyama, Dec 03 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..26

Cf. A031971, A062970, A249459.

Table[1 + Sum[k^(k*n), {k, 1, n}], {n, 0, 10}] (* Vaclav Kotesovec, Dec 04 2021 *)
a[n_] := Sum[If[k == 0, 1, k^(k*n)], {k, 0, n}]; Array[a, 9, 0] (* Amiram Eldar, Dec 04 2021 *)

PARI
```
a(n) = sum(k=0, n, k^(k*n));
```

my(N=10, x='x+O('x^N)); Vec(sum(k=0, N, k^k^2*x^k/(1-k^k*x)))

G.f.: Sum_{k>=0} k^(k^2) * x^k/(1 - k^k * x).

a(n) ~ n^(n^2). - Vaclav Kotesovec, Dec 04 2021

A332408 a(n) = Sum_{k=0..n} binomial(n,k) * k! * k^n.

Original entry on oeis.org

1, 1, 10, 213, 8284, 513105, 46406286, 5772636373, 945492503320, 197253667623681, 51069324556151290, 16067283861476491941, 6037615013420387657844, 2670812587802323522405393, 1373842484756310928089102022, 813119045938378747809030359445

Offset: 0

Ilya Gutkovskiy, Apr 23 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..232

Cf. A000522, A006153, A031971, A063170, A072034, A256016, A277452, A332627.

Join[{1}, Table[Sum[Binomial[n, k] k! k^n, {k, 0, n}], {n, 1, 15}]]

a(n) = sum(k=0, n, binomial(n,k) * k! * k^n); \\ Michel Marcus, Apr 24 2020

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*x*exp(x))^k))) \\ Seiichi Manyama, Feb 19 2022

G.f.: Sum_{k>=0} k! * k^k * x^k / (1 - k*x)^(k+1).
a(n) = n! * Sum_{k=0..n} k^n / (n-k)!.
a(n) ~ c * n! * n^n, where c = A073229 = exp(exp(-1)). - Vaclav Kotesovec, Feb 20 2021
E.g.f.: Sum_{k>=0} (k*x*exp(x))^k. - Seiichi Manyama, Feb 19 2022

A191686 a(n) = n^(n-1) - (n-1)^(n-1) - ... - 2^(n-1) - 1^(n-1).

Original entry on oeis.org

1, 1, 4, 28, 271, 3351, 50478, 896848, 18362109, 425695015, 11023082676, 315332380452, 9876127761371, 336120888377743, 12351836713047770, 487443031053702976, 20559664804361256953, 923012267234425940655, 43944912052993796265952, 2211595951039098481159300

Offset: 1

Juri-Stepan Gerasimov, Jun 11 2011

nonn

			a(1)=1 (=1^0), a(2)=1 (=2^1-1^1), a(3)=4 (=3^2-2^2-1^2).

Seiichi Manyama, Table of n, a(n) for n = 1..387

Cf. A031971, A000169.

A191686 := proc(n) n^(n-1)-add( i^(n-1),i=1..n-1) ; end proc:
seq(A191686(n),n=1..20) ; # R. J. Mathar, Jun 11 2011

Table[n^(n-1)-Total[Range[n-1]^(n-1)],{n,20}] (* Harvey P. Dale, Sep 20 2011 *)

Corrected by R. J. Mathar, Jun 11 2011

A226872 1 together with even numbers n >= 2 such that 1^n + 2^n + 3^n + ... + n^n == n/2 (mod n).

Original entry on oeis.org

1, 2, 4, 8, 10, 14, 16, 22, 26, 28, 32, 34, 38, 44, 46, 50, 52, 56, 58, 62, 64, 68, 70, 74, 76, 82, 86, 88, 92, 94, 98, 104, 106, 112, 116, 118, 122, 124, 128, 130, 134, 136, 142, 146, 148, 152, 154, 158, 164, 166, 170, 172, 176, 178, 182, 184, 188, 190, 194, 196

Offset: 1

José María Grau Ribas, Jun 20 2013

nonn

For n>1, a(n) is even. Alternatively, the even terms of this sequence can be characterized in any of the following ways: (i) even integers n such that n*B(n) == n/2 (mod n), where B(n) is the n-th Bernulli number; OR (ii) integers n such that gcd(n,A027642(n)) = 2; OR (iii) even integers n such that (p-1) does not divide n for every odd prime p dividing n (cf. A124240). - Max Alekseyev, Sep 05 2013

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A014117, A031971, A228869, A228870.

Join[{1}, Select[Range[200], Mod[Sum[PowerMod[k, #, #], {k, #}], #] == #/2 &]] (* T. D. Noe, Sep 04 2013 *)

is(n)=if(n%2,return(n==1));my(f=factor(n)[,1]);for(i=2,#f,if(n%(f[i]-1)==0,return(0)));1 \\ Charles R Greathouse IV, Sep 04 2013

cp's OEIS Frontend

A226964 Numbers n such that 1^n + 2^n + 3^n + ... + n^n == 6 (mod n).

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A226965 Numbers n such that 1^n + 2^n + 3^n +...+ n^n == 7 (mod n).

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A226966 Numbers n such that 1^n + 2^n + 3^n + ... + n^n == 8 (mod n).

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A231409 Least k with 1^(k*m) + 2^(k*m) + ... + (k*m)^(k*m) == k (mod k*m) for m in A230311.

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A121706 a(n) = Sum_{k=1..n-1} k^n.

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A344110 Triangle read by rows: T(n,k) = 2^(n*k), n >= 0, 0 <= k <= n.

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A349886 a(n) = Sum_{k=0..n} k^(k*n).

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A332408 a(n) = Sum_{k=0..n} binomial(n,k) * k! * k^n.

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A231409 Least k with 1^(km) + 2^(km) + ... + (km)^(km) == k (mod k*m) for m in A230311.