A031971 -id:A031971 - cp's OEIS frontend

A233045 1^m + 2^m + ... + m^m (mod m) for primary pseudoperfect numbers m.

1, 1, 1, 1, 5797, 272753965, 8749232767, 1045741078641946876220133713545

Offset: 1

Jonathan Sondow, Dec 10 2013

A031971(m) (mod m) for m in A054377 = 2, 6, 42, 1806, 47058, 2214502422, 52495396602, 8490421583559688410706771261086. The known values of m for which 1^m + 2^m + ... + m^m == 1 (mod m) are m = 1, 2, 6, 42, 1806.

For any m and prime p | m, use Sum_{j=1..m} j^m == -m/p (mod p) if p-1 | m or == 0 (mod p) otherwise (see Lemma 3 in Grau et al.) and the Chinese Remainder Theorem.

			The 1st primary pseudoperfect number is 2, and 1^2 + 2^2 = 5 == 1 (mod 2), so a(1) = 1.

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, A054377, A231409.

ps={2, 6, 42, 1806, 47058, 2214502422,52495396602, 8490421583559688410706771261086}; fa = FactorInteger; VonStaudt[n_] := Mod[n - Sum[If[IntegerQ[n/(fa[n][[i, 1]] - 1)], n/fa[n][[i, 1]], 0], {i, Length[fa[n]]}], n]; Table[VonStaudt[ps[[i]]], {i, 1, 8}]

a(n) = 1 for n = 1, 2, 3, 4.

A276485 Numerator of Sum_{k=1..n} 1/k^n.

Original entry on oeis.org

1, 5, 251, 22369, 806108207, 47464376609, 774879868932307123, 248886558707571775009601, 4106541588424891370931874221019, 413520574906423083987893722912609, 7429165883912264897181708263009894640627544300697

Offset: 1

Ilya Gutkovskiy, Sep 05 2016

Also numerators of zeta(n) - Hurwitz zeta(n,n+1), where zeta(s) is the Riemann zeta function and Hurwitz zeta(s,a) is the Hurwitz zeta function.

Sum_{k>=1} 1/k^n = zeta(n).

			1, 5/4, 251/216, 22369/20736, 806108207/777600000, 47464376609/46656000000, 774879868932307123/768464444160000000, ...
a(3) = 251, because 1/1^3 + 1/2^3 + 1/3^3 = 251/216.

Eric Weisstein's World of Mathematics, Harmonic Number
Eric Weisstein's World of Mathematics, Riemann Zeta Function
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function

Cf. A001008, A002805, A007406, A007407, A031971, A276487 (denominators).

Table[Numerator[HarmonicNumber[n, n]], {n, 1, 11}]

a(n) = numerator(sum(k=1, n, 1/k^n)); \\ Michel Marcus, Sep 06 2016

A291140 Sum of the n-th powers of the first n primes.

Original entry on oeis.org

2, 13, 160, 3123, 181258, 6732437, 493478344, 24995572327, 2255433009730, 470444892889497, 38714638073629150, 7749166585021832891, 1203906832960860262108, 121893712541593098356317, 17161342484454585041813494

Offset: 1

Vojtech Strnad, Aug 18 2017

nonn

			a(3) = 2^3 + 3^3 + 5^3 = 160.

Vojtech Strnad, Table of n, a(n) for n = 1..200

Cf. A031971, A062457, A087480.

f:= n -> add(ithprime(i)^n,i=1..n):
map(f, [$1..20]); # Robert Israel, Aug 20 2017

Table[Total[Prime[Range@ n]^n], {n, 15}] (* Michael De Vlieger, Aug 19 2017 *)

a(n) = sum(i=1, n, prime(i)^n) \\ Felix Fröhlich, Aug 18 2017

a(n) = prime(1)^n + prime(2)^n + ... + prime(n)^n.

A342396 a(n) = Sum_{k=1..n} k^(n/gcd(k,n) - 1).

Original entry on oeis.org

1, 2, 6, 31, 355, 3150, 67172, 904085, 22998481, 427799450, 14914341926, 287337926355, 13421957361111, 339940911160914, 15434209582905140, 493467700905592777, 28101527071305611529, 836396358233559195382

Offset: 1

Seiichi Manyama, Mar 10 2021

Cf. A031971, A332621, A342394, A342395.

a[n_] := Sum[k^(n/GCD[k, n] - 1), {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Mar 10 2021 *)

PARI
```
a(n) = sum(k=1, n, k^(n/gcd(k, n)-1));
```

If p is prime, a(p) = A031971(p-1) + 1.

A349964 a(n) = Sum_{k=0..n} (k*n)^n.

Original entry on oeis.org

1, 1, 20, 972, 90624, 13828125, 3133930176, 988501957072, 414139067400192, 222497518123837665, 149143419250000000000, 122020951254446884154196, 119671520043865789861724160, 138593796657903100873209121453

Offset: 0

Seiichi Manyama, Dec 07 2021

nonn

Cf. A031971, A155956, A185393, A349966, A349969,

a[n_] := Sum[If[k == n == 0, 1, (k*n)^n], {k, 0, n}]; Array[a, 14, 0] (* Amiram Eldar, Dec 07 2021 *)

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

a(n) = n^n * [x^n] Sum_{k>=0} (k * x)^k/(1 - k * x) = n^n * A031971(n).

a(n) ~ c * n^(2*n), where c = 1/(1 - 1/exp(1)) = A185393. - Vaclav Kotesovec, Dec 07 2021

A068477 a(n) is the digital sum of 1^n + 2^n + ... + n^n.

Original entry on oeis.org

0, 1, 5, 9, 12, 15, 22, 10, 33, 45, 43, 60, 44, 79, 70, 72, 65, 90, 111, 91, 125, 117, 132, 168, 133, 127, 171, 189, 172, 195, 218, 232, 217, 234, 221, 243, 240, 280, 290, 261, 315, 348, 352, 325, 345, 351, 346, 303, 338, 367, 373, 396, 404, 414, 495, 424, 428

Offset: 0

Francois Jooste (phukraut(AT)hotmail.com), Mar 10 2002

G. C. Greubel, Table of n, a(n) for n = 0..1000

dig := X->convert((convert(X,base,10)),`+`); a := n->dig(sum(m^n,m=1..n));

Table[Sum[DigitCount[Sum[k^n, {k, 1, n}]][[i]]*i, {i, 9}], {n, 0, 100}] (* G. C. Greubel, Oct 13 2018 *)

a(n) = sumdigits(sum(k=1, n, k^n)); \\ Michel Marcus, Oct 14 2018

a(n) = A007953(A031971(n))

A096141 a(n) = sum of n n-th powers starting from n^n.

Original entry on oeis.org

1, 13, 216, 4578, 119525, 3729451, 135771160, 5658574916, 265921407297, 13918657338925, 803220053336096, 50674352524725590, 3470170166345203477, 256369124879898560271, 20325382637400264402000

Offset: 1

Amarnath Murthy, Jul 16 2004

nonn

			a(4) = 4^4 +5^4 + 6^4 +7^4 = 4578.

Harvey P. Dale, Table of n, a(n) for n = 1..351

Cf. A031971.

Table[Total[Range[n,2n-1]^n],{n,20}] (* Harvey P. Dale, Aug 23 2019 *)

PARI
```
a(n)=sum(k=n,2*n-1,k^n)
```

a(n) = n! * [x^n] exp(n*x)*(exp(n*x) - 1)/(exp(x) - 1). - Ilya Gutkovskiy, Apr 07 2018

Extended by Ray Chandler, Jul 17 2004

A119758 Numerator of Sum_{k=1..n} k^n/n^k.

Original entry on oeis.org

1, 3, 20, 225, 3789, 89341, 2821552, 115377921, 5939637425, 375840753541, 28641787322796, 2583828842108449, 271949027324094925, 32986652806128680205, 4563200871898056653504, 713455071424061222336513

Offset: 1

Alexander Adamchuk, Jun 18 2006, Jun 25 2006

a(p-1) is divisible by prime p>2. a(p) is divisible by ((p+1)/2)^2 for prime p>2.

Denominator of Sum[k^n/n^k,{k,1,n}] is equal to n^(n-1) = A000169(n). - Alexander Adamchuk, Jun 27 2006

Cf. A023037, A031971.

Cf. A000169.

Table[Numerator[Sum[k^n/n^k,{k,1,n}]],{n,1,20}]
Table[Sum[k^n/n^k,{k,1,n}]*n^(n-1),{n,1,50}] (* Alexander Adamchuk, Jun 27 2006 *)

a(n) = numerator(prod(k=2, n, 1-1/(prime(k)-1)^2)); \\ Michel Marcus, May 31 2022

a(n) = numerator(Sum_{k=1..n} k^n/n^k).
a(n) = n^(n-1)*(Sum_{k=1..n} k^n/n^k). - Alexander Adamchuk, Jun 27 2006
a(2m) is divisible by 2m+1 for integer m>0. a(2m-1) is divisible by m^2 for integer m>0. - Alexander Adamchuk, Jun 27 2006

A120487 Denominator of 1^n/n + 2^n/(n-1) + 3^n/(n-2) + ... + (n-1)^n/2 + n^n/1.

Original entry on oeis.org

1, 2, 3, 12, 5, 20, 35, 280, 63, 2520, 385, 27720, 6435, 8008, 45045, 720720, 85085, 4084080, 969969, 739024, 29393, 5173168, 7436429, 356948592, 42902475, 2974571600, 717084225, 80313433200, 215656441, 2329089562800, 4512611027925

Offset: 1

Alexander Adamchuk, Jul 22 2006

Numerator is A115071(n).

Also a(n) is denominator of (n+1)^(n+1) * (H(n+1) - 1), where H(k) is harmonic number, H(k) = Sum_{i=1..k} 1/i = A001008(k)/A002805(k). - Alexander Adamchuk, Jan 02 2007

Cf. A115071, A027612, A031971, A002805.

Cf. A001008, A002805.

Denominator[Table[Sum[k^n/(n-k+1),{k,1,n}],{n,1,50}]]
Table[ Denominator[ (n+1)^(n+1) * Sum[ 1/i,{i,2,n+1} ] ], {n,1,40} ] (* Alexander Adamchuk, Jan 02 2007 *)

a(n) = denominator(Sum_{k=1..n} k^n/(n-k+1)).

a(n) = denominator((n+1)^(n+1) * Sum_{i=2..n+1} 1/i). - Alexander Adamchuk, Jan 02 2007

A225821 a(n) = Product_{p | p is prime and p, p-1 both divide n}.

Original entry on oeis.org

1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 10, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 10, 1, 42, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 30, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 10, 1, 2, 1, 42

Offset: 1

José María Grau Ribas, Jul 30 2013

nonn

a(n) = 2 iff n is even and is a term of A226872. - Daniel Suteu, Jul 28 2019
From Bernard Schott, Jul 30 2019: (Start)
a(n) = n if n = 1, 2, 6, 42, 1806.
a(n) = 6 if n is of the form 2^i*3^j, i and j >= 1, so if n is a term of A033845.
a(n) = 10 if n is of the form 2^i*5^j, i >= 2 and j >= 1.
a(n) = 30 if n is of the form 2^i*3^j*5^k, i >=2, j >= 1 and k >= 1. (End)

Eric M. Schmidt, Table of n, a(n) for n = 1..10000

Cf. A173557, A027760.

fa=FactorInteger; d[m_]:= Product[If[IntegerQ[m/(fa[m][[i, 1]]-1)],fa[m][[i, 1]], 1], {i, Length@fa@m}]; Table[d[n], {n, 1, 333}]

a(n)=my(f=factor(n)[,1]); prod(i=1,#f,if(n%(f[i]-1)==0,f[i],1)) \\ Charles R Greathouse IV, Nov 13 2013

def A225821(n) : return prod(p for (p,m) in factor(n) if n%(p-1)==0) # Eric M. Schmidt, Jul 31 2013

a(n) = denominator(A031971(n)/n) = gcd(n, A027642(n)). - Daniel Suteu, Jul 28 2019

cp's OEIS Frontend

A233045 1^m + 2^m + ... + m^m (mod m) for primary pseudoperfect numbers m.

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A276485 Numerator of Sum_{k=1..n} 1/k^n.

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A291140 Sum of the n-th powers of the first n primes.

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A342396 a(n) = Sum_{k=1..n} k^(n/gcd(k,n) - 1).

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A349964 a(n) = Sum_{k=0..n} (k*n)^n.

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A068477 a(n) is the digital sum of 1^n + 2^n + ... + n^n.

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Maple

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A096141 a(n) = sum of n n-th powers starting from n^n.

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A119758 Numerator of Sum_{k=1..n} k^n/n^k.

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