A031971 -id:A031971 - cp's OEIS frontend

A249564 a(n) = Sum_{k = 0..n} (k*(k+1)/2)^n.

1, 1, 10, 244, 11378, 867395, 98204132, 15475158552, 3239399341956, 869652788703285, 291315412833808702, 119114020598815073524, 58386684085633233147478, 33797341113242898165287495, 22810507257314647778044971848, 17755122836243141585656207243952

Offset: 0

Vaclav Kotesovec, Nov 01 2014

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A000217, A031971, A177385.

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

a(n) = sum(k=0, n, (k*(k+1)/2)^n); \\ Michel Marcus, Aug 24 2023

E.g.f.: Sum_{n>=0} exp(x*n*(n+1)/2).

a(n) ~ exp(3) * n^(2*n) / ((exp(2)-1) * 2^n).

New name from Peter Bala, Aug 18 2023

A276487 Denominator of Sum_{k=1..n} 1/k^n.

Original entry on oeis.org

1, 4, 216, 20736, 777600000, 46656000000, 768464444160000000, 247875891108249600000000, 4098310578334288576512000000000, 413109706296096288512409600000000, 7425496288284402957501110551810198732800000000000

Offset: 1

Ilya Gutkovskiy, Sep 05 2016

Also denominator 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) = 216, 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, A276485 (numerators).

A276487:=n->denom(add(1/k^n, k=1..n)): seq(A276487(n), n=1..12); # Wesley Ivan Hurt, Sep 07 2016

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

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

A308481 a(n) = Sum_{k=1..n, gcd(n,k) = 1} k^n.

Original entry on oeis.org

1, 1, 9, 82, 1300, 15626, 376761, 6161988, 176787117, 3769318700, 142364319625, 3152513804548, 154718778284148, 4340009120036086, 210971169748692000, 7281661100510001416, 435659030617933827136, 14181101408561469791694, 1052864393300587929716721, 41673894815421072916530408

Offset: 1

Ilya Gutkovskiy, May 30 2019

nonn

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

First superdiagonal of A308477.

Cf. A031971.

a[n_] := Sum[If[GCD[n, k] == 1, k^n, 0], {k, 1, n}]; Table[a[n], {n, 1, 20}]

a(n) = sum(k=1, n, (gcd(n,k)==1)*k^n) \\ Felix Fröhlich, May 30 2019

A332655 a(n) = Sum_{k=1..n} (k/gcd(n, k))^n.

Original entry on oeis.org

1, 2, 10, 84, 1301, 15693, 376762, 6168552, 176787631, 3770427352, 142364319626, 3152758480715, 154718778284149, 4340093860950619, 210971170836848270, 7281694486114555088, 435659030617933827137, 14181121059071691716406, 1052864393300587929716722, 41673907052879908244100770

Offset: 1

Ilya Gutkovskiy, Feb 18 2020

nonn

Cf. A031971, A057661, A226561, A332517, A332621, A332652, A332653, A332654.

[&+[(k div Gcd(n,k))^n:k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 18 2020

Table[Sum[(k/GCD[n, k])^n, {k, 1, n}], {n, 1, 20}]
Table[Sum[Sum[If[GCD[k, d] == 1, k^n, 0], {k, 1, d}], {d, Divisors[n]}], {n, 1, 20}]

a(n) = Sum_{k=1..n} (lcm(n, k)/n)^n.

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

A345100 a(n) = Sum_{k=1..n} k^floor(n/k).

Original entry on oeis.org

1, 3, 6, 12, 17, 33, 40, 68, 95, 141, 152, 328, 341, 461, 738, 1130, 1147, 2159, 2178, 4068, 5841, 6997, 7020, 18198, 20723, 25001, 38798, 61546, 61575, 137445, 137476, 223252, 342593, 408435, 485376, 1213988, 1214025, 1476549, 2541498, 4202810, 4202851, 8777205

Offset: 1

Seiichi Manyama, Jun 08 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A031971, A055225, A344551, A345098.

a[n_] := Sum[k^Floor[n/k], {k, 1, n}]; Array[a, 40] (* Amiram Eldar, Jun 08 2021 *)

PARI
```
a(n) = sum(k=1, n, k^(n\k));
```

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, k*x^k*(1-x^k)/(1-k*x^k))/(1-x))

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

a(n) ~ 3^((n - mod(n,3))/3). - Vaclav Kotesovec, Jun 11 2021

A359659 a(n) = Sum_{k=0..n} k^(k * (n-k+1)).

Original entry on oeis.org

1, 2, 6, 45, 1051, 88602, 27121964, 37004504305, 198705527223757, 5595513387083114570, 686714367475480207331582, 468422339816915120237104999421, 1664212116512828935888786624225704855

Offset: 0

Seiichi Manyama, Jan 10 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..51

Cf. A026898, A349893, A359658.
Cf. A031971, A349836, A349883.
Cf. A003101, A349882.

PARI
```
a(n) = sum(k=0, n, k^(k*(n-k+1)));
```

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-k^k*x)))

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(k+1)^(k+1)*x)))

G.f.: Sum_{k>=0} (k * x)^k/(1 - k^k * x).
G.f.: Sum_{k>=0} x^k/(1 - (k+1)^(k+1) * x).
a(n) = A349893(n+1) - 1.

A074309 Sum of next n terms of the form i^i.

Original entry on oeis.org

4, 283, 50037, 17650540, 10405067904, 9211817140115, 11424093748466841, 18896062057822100616, 40192544399240309019728, 106876212200059543898143707, 347377340594805599176614321101

Offset: 1

Zak Seidov, Sep 22 2002

nonn

Sum of next n terms of the form i^n is A074209. Sum of first n terms of the form i^n is A031971. Sum of first n terms of the form i^i is A001923.

			a(2) = 283 = 3^3 + 4^4, a(3) = 50037 = 4^4 + 5^5 + 6^6, a(4) = 17650540 = 5^5 + 6^6 + 7^7 + 8^8, a(5) = 10405067904 = 6^6 + 7^7 + 8^8 + 9^9 + 10^10.

Seiichi Manyama, Table of n, a(n) for n = 1..193

Cf. A001923, A031971, A074209.

Mathematica
```
Table[Sum[i^i, {i, n+1, 2n}], {n, 20}]
```

a(n) = sum(k=n+1, 2*n, k^k); \\ Seiichi Manyama, Dec 05 2021

a(n) = Sum_{i=n+1..2n} i^i.

a(n) ~ (2*n)^(2*n). - Vaclav Kotesovec, Dec 06 2021

A220779 Exponent of highest power of 2 dividing the sum 1^n + 2^n + ... + n^n.

Original entry on oeis.org

0, 0, 2, 1, 0, 0, 4, 2, 0, 0, 2, 1, 0, 0, 6, 3, 0, 0, 2, 1, 0, 0, 4, 2, 0, 0, 2, 1, 0, 0, 8, 4, 0, 0, 2, 1, 0, 0, 4, 2, 0, 0, 2, 1, 0, 0, 6, 3, 0, 0, 2, 1, 0, 0, 4, 2, 0, 0, 2, 1, 0, 0, 10, 5, 0, 0, 2, 1, 0, 0, 4, 2, 0, 0, 2, 1, 0, 0, 6, 3, 0, 0, 2, 1, 0, 0, 4

Offset: 1

Jonathan Sondow, Dec 20 2012

2-adic valuation of Sum_{k = 1..n} k^n.

Omitting the zero terms (for n == 1 or 2 mod 4) gives A220780.

			1^3 + 2^3 + 3^3 = 1 + 8 + 27 = 36 = 2^2 * 9, so a(3) = 2.

T. D. Noe, Table of n, a(n) for n = 1..1024
T. Lengyel, On divisibility of some power sums, INTEGERS, 7(2007), A41, 1-6.
K. MacMillan and J. Sondow, Problem 11546: 2-adic Valuation of Bernoulli-style Sums, Amer. Math. Monthly, 118 (2011), 84 (proposal); 119 (2012), 886-887 (solution).
K. MacMillan and J. Sondow, Divisibility of power sums and the generalized Erdos-Moser equation, arXiv:1010.2275 [math.NT], 2010-2011; Elemente der Mathematik, 67 (2012), 182-186.

Cf. A031971, A001511, A007814, A220780.

Table[ IntegerExponent[ Sum[ k^n, {k, 1, n}], 2], {n, 150}]

a(n) = valuation(sum(k=1, n, k^n), 2); \\ Michel Marcus, Jul 09 2022

from sympy import harmonic
def A220779(n): return (~(m:=int(harmonic(n,-n)))&m-1).bit_length() # Chai Wah Wu, Jul 08 2022

a(n) = d - 1 or 2*(d - 1), according as n or n+1 = 2^d * odd, with d > 0.

a(n) = A007814(A031971(n)).

A220780 Nonzero terms of A220779: exponent of highest power of 2 dividing an even sum 1^n + 2^n + ... + n^n.

Original entry on oeis.org

2, 1, 4, 2, 2, 1, 6, 3, 2, 1, 4, 2, 2, 1, 8, 4, 2, 1, 4, 2, 2, 1, 6, 3, 2, 1, 4, 2, 2, 1, 10, 5, 2, 1, 4, 2, 2, 1, 6, 3, 2, 1, 4, 2, 2, 1, 8, 4, 2, 1, 4, 2, 2, 1, 6, 3, 2, 1, 4, 2, 2, 1, 12, 6, 2, 1, 4, 2, 2, 1, 6, 3, 2, 1, 4, 2, 2, 1, 8, 4, 2, 1, 4, 2, 2, 1

Offset: 1

Jonathan Sondow, Dec 20 2012

nonn

2-adic valuation of Sum_{k=1..n} k^n for n == 0 or 3 mod 4.

See references, links, formulas, and example in A220779.

Cf. A001511, A031971, A220779.

Table[n = 2*k + Mod[k, 2]; IntegerExponent[ Sum[a^n, {a, 1, n}], 2], {k, 150}]

from sympy import harmonic
def A220780(n): return (~(m:=int(harmonic(k:=(n<<1)+(n&1),-k)))&m-1).bit_length() # Chai Wah Wu, Jul 11 2022

A225578 Sum of first (prime(n) - 1) (prime(n) - 1)th powers.

Original entry on oeis.org

1, 5, 354, 67171, 14914341925, 13421957361110, 28101527071305611528, 60182438244917445266889, 525344775209112229247070397995, 51296981152155330485450049059398345004638, 319099356359853147544285512855368258519442575

Offset: 1

Alonso del Arte, May 10 2013

It follows from Fermat's little theorem that a(n) is congruent to -1 mod the n-th prime.

			a(2) = 5 because, since 3 is the second prime, we have 1^2 + 2^2 = 1 + 4 = 5.
a(3) = 354 because, since 5 is the third prime, we have 1^4 + 2^4 + 3^4 + 4^4 = 1 + 4 + 81 + 256 = 354.

R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section A17.
Paulo Ribemboim, The Little Book of Big Primes, New York, Springer-Verlag (1991): 17.

Seiichi Manyama, Table of n, a(n) for n = 1..76

Cf. A055030, A031971, A204187.

Table[Sum[i^(Prime[n] - 1), {i, Prime[n] - 1}], {n, 15}]

a(n) = Sum_{i=1..prime(n)-1} i^(prime(n) - 1).

cp's OEIS Frontend

A249564 a(n) = Sum_{k = 0..n} (k*(k+1)/2)^n.

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A276487 Denominator of Sum_{k=1..n} 1/k^n.

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Maple

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A308481 a(n) = Sum_{k=1..n, gcd(n,k) = 1} k^n.

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A332655 a(n) = Sum_{k=1..n} (k/gcd(n, k))^n.

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Magma

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A345100 a(n) = Sum_{k=1..n} k^floor(n/k).

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A359659 a(n) = Sum_{k=0..n} k^(k * (n-k+1)).

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PARI

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A074309 Sum of next n terms of the form i^i.

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A220779 Exponent of highest power of 2 dividing the sum 1^n + 2^n + ... + n^n.

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