A031971 -id:A031971 - cp's OEIS frontend

A229481 Final digit of 1^n + 2^n + ... + n^n.

0, 1, 5, 6, 4, 5, 1, 4, 2, 5, 5, 6, 0, 1, 5, 0, 8, 3, 9, 0, 6, 1, 5, 6, 0, 5, 1, 4, 8, 5, 5, 6, 6, 1, 5, 0, 4, 3, 9, 0, 2, 1, 5, 6, 6, 5, 1, 4, 4, 5, 5, 6, 2, 1, 5, 0, 0, 3, 9, 0, 8, 1, 5, 6, 2, 5, 1, 4, 0, 5, 5, 6, 8, 1, 5, 0, 6, 3, 9, 0, 4, 1, 5, 6, 8, 5, 1, 4, 6, 5, 5, 6, 4, 1, 5, 0, 2, 3, 9, 0

Offset: 0

José María Grau Ribas, Sep 24 2013

Cyclic with period 100.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A031971, A056849.

a:= proc(n) local l; l:=[seq(add(k&^i mod 10, k=1..i) mod 10, i=0..99)]:
      proc(n) l[1+irem(n, 100)] end
    end():
seq(a(n), n=0..200);  # Alois P. Heinz, Sep 26 2013

Table[Mod[Sum[PowerMod[i, n, 10], {i, 1, n}], 10], {n, 0, 133}]

a(n)=n%=100;lift(sum(k=1,n,Mod(k,10)^n)) \\ Charles R Greathouse IV, Dec 13 2013

a(n) = a(n-100). - Wesley Ivan Hurt, Jan 02 2024

A263022 a(n) = gcd(n, 1^(n-1) + 2^(n-1) + ... + (n-1)^(n-1)) for n > 1.

Original entry on oeis.org

1, 1, 4, 1, 3, 1, 8, 3, 5, 1, 12, 1, 7, 5, 16, 1, 9, 1, 20, 7, 11, 1, 24, 5, 13, 9, 28, 1, 15, 1, 32, 11, 17, 35, 36, 1, 19, 13, 40, 1, 21, 1, 44, 3, 23, 1, 48, 7, 25, 17, 52, 1, 27, 55, 56, 19, 29, 1, 60, 1, 31, 21, 64, 13, 33, 1, 68, 23, 35, 1, 72, 1, 37, 25, 76, 77, 39, 1, 80, 27, 41, 1, 84, 17, 43, 29, 88, 1, 45, 13, 92, 31, 47, 95, 96

Offset: 2

Thomas Ordowski, Oct 07 2015

nonn

a(n) = 1 if and only if n is a prime or n is a Carmichael number.
a(n) is divisible by 4 if n is divisible by 4, otherwise a(n) is odd. - Robert Israel, Oct 08 2015
a(n) = n iff 4|n or n = 35, 55, 77, 95; A121707 ?
a(5005) = 11: this is the first case where a(n) is prime and A001222(n) > 3. - Altug Alkan, Oct 08 2015

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A002997 (see my Oct 09 2013 comment).

f:= n -> igcd(n, add(j &^(n-1) mod n, j=1..n-1)):
seq(f(n), n=2..1000); # Robert Israel, Oct 08 2015

Table[GCD[n, Total@ Map[#^(n - 1) &, Range[n - 1]]], {n, 2, 96}] (* Michael De Vlieger, Oct 08 2015 *)

vector(100, n, gcd(n+1, sum(k=1, n, k^n))) \\ Altug Alkan, Oct 08 2015

a(4n) = 4n.

a(n) = gcd(A031971(n-1), n). - Michel Marcus, Oct 08 2015

A302353 a(n) = Sum_{k=0..n} k^nbinomial(2n-k,n).

Original entry on oeis.org

1, 1, 7, 69, 936, 16290, 345857, 8666413, 250355800, 8191830942, 299452606190, 12095028921250, 534924268768540, 25710497506696860, 1334410348734174285, 74379234152676275325, 4431350132232658244400, 281020603194039519937590, 18900157831016574533520330, 1343698678390575915132318870

Offset: 0

Ilya Gutkovskiy, Apr 06 2018

nonn

a(n) is the n-th term of the main diagonal of iterated partial sums array of n-th powers (starting with the first partial sums).

			For n = 4 we have:
------------------------
0   1    2    3    [4]
------------------------
0,  1,  17,   98,  354,  ... A000538 (partial sums of fourth powers)
0,  1,  18,  116,  470,  ... A101089 (partial sums of A000538)
0,  1,  19,  135,  605,  ... A101090 (partial sums of A101089)
0,  1,  20,  155,  760,  ... A101091 (partial sums of A101090)
0,  1,  21,  176, [936], ... A254681 (partial sums of A101091)
------------------------
therefore a(4) = 936.

Cf. A002054, A031971, A265612, A293550, A293574, A302352.

Join[{1}, Table[Sum[k^n Binomial[2 n - k, n], {k, 0, n}], {n, 19}]]
Table[SeriesCoefficient[HurwitzLerchPhi[x, -n, 0]/(1 - x)^(n + 1), {x, 0, n}], {n, 0, 19}]

a(n) ~ c * (r * (2-r)^(2-r) / (1-r)^(1-r))^n * n^n, where r = 0.69176629470097668698335106516328398961170464277337300459988208658267146... is the root of the equation (2-r) = (1-r) * exp(1/r) and c = 0.96374921279011282619632879505754646526289414675402231447188230355850496... - Vaclav Kotesovec, Apr 08 2018

A332624 a(n) = Sum_{k=1..n} ceiling(n/k)^n.

Original entry on oeis.org

1, 5, 36, 289, 3433, 47578, 842499, 16850338, 389415029, 10010878371, 285679026506, 8918295095267, 302973286652448, 11112691430262573, 437929106387544254, 18447028378472722051, 827256956775203666857, 39346558275376372606086, 1978429667078835508142129

Offset: 1

Ilya Gutkovskiy, Feb 17 2020

nonn

Cf. A006590, A031971, A332469, A332490, A332623.

[&+[Ceiling(n/k)^n:k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 17 2020

Table[Sum[Ceiling[n/k]^n, {k, 1, n}], {n, 1, 19}]
Table[n + Sum[Sum[(d + 1)^n - d^n, {d, Divisors[k]}], {k, 1, n - 1}], {n, 1, 19}]
Table[SeriesCoefficient[x/(1 - x)^2 + x/(1 - x) Sum[((k + 1)^n - k^n) x^k/(1 - x^k), {k, 1, n}], {x, 0, n}], {n, 1, 19}]

a(n) = [x^n] x/(1 - x)^2 + (x/(1 - x)) * Sum_{k>=1} ((k + 1)^n - k^n) * x^k / (1 - x^k).

a(n) = n + Sum_{k=1..n-1} Sum_{d|k} ((d + 1)^n - d^n).

A341331 a(n) = n^n - (n-1)^n - (n-2)^n - ... - 1^n.

Original entry on oeis.org

1, 3, 18, 158, 1825, 26141, 446782, 8869820, 200535993, 5085658075, 142947350986, 4410243535402, 148156328308105, 5382924338773177, 210309307208574750, 8791961076113491704, 391581231268402937041, 18510377905675629883959, 925555262359725659407258

Offset: 1

Seiichi Manyama, Feb 09 2021

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

Cf. A000312, A031971, A080956, A121706, A179297, A179298, A173142, A191686, A290844.

f:= proc(n) local k;  n^n - add(k^n,k=1..n-1) end proc:
map(f, [$1..30]); # Robert Israel, Feb 10 2021

a[n_] := n^n - Sum[k^n, {k, 0, n - 1}]; Array[a, 20] (* Amiram Eldar, Apr 28 2021 *)

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

a(n) = A000312(n) - A121706(n).

a(n) = - A290844(n-1,n) for n > 1.

A344260 a(n) is the number of relations from an n-element set into a set of at most n elements.

Original entry on oeis.org

1, 3, 21, 585, 69905, 34636833, 69810262081, 567382630219905, 18519084246547628289, 2422583247133816584929793, 1268889750375080065623288448001, 2659754699919401766201267083003561985, 22306191045953951743035482794815064402563073, 748380193317489370459454048174977015562807531282433

Offset: 0

Stefano Spezia, May 13 2021

nonn

Symmetrically, also the number of relations from a set of at most n elements into an n-element set.

Row sums of A344110.

Cf. A000079, A002416, A031971, A089072, A117401, A275779.

Join[{1},Table[(2^(n+n^2)-1)/(2^n-1),{n,13}]]

a(n) = (2^(n+n^2) - 1)/(2^n - 1) for n > 0 and a(0) = 1.
a(n) ~ 2^(n^2).
a(n) = A275779(n) + 1. - Hugo Pfoertner, May 14 2021

A366329 a(n) = Product_{k=1..n} Sum_{j=1..k} j^n.

Original entry on oeis.org

1, 5, 324, 589764, 52393770000, 347773153451938500, 244632735619259069507040000, 24547871392966749661547369532868031040, 455140097017244017295446005144727669016636127744000, 1960564895414510364772369567330640938816177001699555385515625000000

Offset: 1

Vaclav Kotesovec, Oct 07 2023

nonn

Cf. A031971, A036740, A366305, A366342.

Table[Product[Sum[j^n, {j, 1, k}], {k, 1, n}], {n, 1, 12}]
Table[Product[HarmonicNumber[k, -n], {k, 1, n}], {n, 1, 12}] // FunctionExpand

a(n) = A036740(n) * Product_{k=1..n} Sum_{j=1..k} (j/k)^n.
a(n) ~ n!^n * c * d^n, where d = exp(-Integral_{x=0..1} log(1 - exp(-1/x)) dx) = 1.187538543919977798892363400109897833660222697152558038684860736484... and c = exp(1 - 1/(exp(1) - 1)) / (exp(1) - 1) = 0.88399704290317414073109479991305699114875723090346..., updated Apr 19 2024
a(n) ~ c * d^n * (2*Pi)^(n/2) * n^(n*(2*n+1)/2) / exp(n^2 - 1/12).

A366342 a(n) = Product_{k=1..n} Sum_{j=1..k} j^k.

Original entry on oeis.org

1, 5, 180, 63720, 281961000, 18939602331000, 22733280436308624000, 561162207057469095693888000, 322278252906706683140441912431680000, 4806568058842248598039183477606983722184000000, 2055653754202086984879290521714456895014175320595424000000

Offset: 1

Vaclav Kotesovec, Oct 07 2023

nonn

Cf. A002109, A031971, A074962, A185393, A366329.

Table[Product[Sum[j^k, {j, 1, k}], {k, 1, n}], {n, 1, 12}]
Table[Product[HarmonicNumber[k, -k], {k, 1, n}], {n, 1, 12}] // FunctionExpand

a(n) = A002109(n) * Product_{k=1..n} Sum_{j=1..k} (j/k)^k.
a(n) ~ A002109(n) * c * d^n / n^f, where
d = 1/(1 - exp(-1)) = A185393
f = (exp(1) + 1) / (2*(exp(1) - 1)^2) = 0.629685240773129106752912520161993823...
c = 1.038111196610478473178942324022485064169644880240145128332184584611...
a(n) ~ A * c * d^n * n^(n*(n+1)/2 + 1/12 - f) / exp(n^2/4), where A is the Glaisher-Kinkelin constant A074962.

A368466 a(n) = Sum_{k=0..n} 2^k * k^n.

Original entry on oeis.org

1, 2, 18, 250, 4810, 118458, 3557610, 126109562, 5153959338, 238596116794, 12340467941098, 705262375055610, 44135963944338474, 3001795007526424250, 220466095716711140202, 17389850740043552754298, 1466156761178169939270826, 131580021359494993268692026

Offset: 0

Seiichi Manyama, Dec 25 2023

Main diagonal of A368479.

Cf. A031971, A120485, A349963.

PARI
```
a(n) = sum(k=0, n, 2^k*k^n);
```

a(n) ~ 2^n * n^n / (1 - exp(-1)/2). - Vaclav Kotesovec, Dec 26 2023

A264748 a(n) = Sum_{k = 1..n} (k^n - n^k).

Original entry on oeis.org

0, -1, -3, 14, 520, 11185, 239505, 5510652, 138456936, 3803230815, 113833152565, 3695302326650, 129479186068128, 4874312730972685, 196306448145080385, 8425000059348756472, 383956514250037779376, 18521535576956405481147, 942952190208348285876501

Offset: 1

Ilya Gutkovskiy, Nov 23 2015

sign

			a(1) = 1^1 - 1^1 = 0;
a(2) = 1^2 - 2^1 + 2^2 - 2^2 = -1;
a(3) = 1^3 - 3^1 + 2^3 - 3^2 + 3^3 - 3^3 = -3;
a(4) = 1^4 - 4^1 + 2^4 - 4^2 + 3^4 - 4^3 + 4^4 - 4^4 = 14;
a(5) = 1^5 - 5^1 + 2^5 - 5^2 + 3^5 - 5^3 + 4^5 - 5^4 + 5^5 - 5^5 = 520, etc.

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

Cf. A031971, A031972.

Table[Sum[k^n - n^k, {k, 1, n}], {n, 1, 20}]
Join[{0}, Table[HarmonicNumber[n, -n] - n (n^n - 1)/(n - 1), {n, 2, 20}]]

a(n) = sum(k=1, n, k^n - n^k); \\ Altug Alkan, Nov 23 2015

a(n) =  A031971(n) - A031972(n).
a(n) = ((1 - n)*zeta(-n, n + 1) - n*(n^n - 1) + (n - 1)*zeta(-n))/(n - 1) for n>1, where zeta(s) is the Riemann zeta function and zeta(s, a) is the Hurwitz zeta function.
a(n) ~ n^n / (exp(1) - 1). - Vaclav Kotesovec, Jul 16 2025

cp's OEIS Frontend

A229481 Final digit of 1^n + 2^n + ... + n^n.

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A263022 a(n) = gcd(n, 1^(n-1) + 2^(n-1) + ... + (n-1)^(n-1)) for n > 1.

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A302353 a(n) = Sum_{k=0..n} k^n*binomial(2*n-k,n).

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A332624 a(n) = Sum_{k=1..n} ceiling(n/k)^n.

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A341331 a(n) = n^n - (n-1)^n - (n-2)^n - ... - 1^n.

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A344260 a(n) is the number of relations from an n-element set into a set of at most n elements.

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A366329 a(n) = Product_{k=1..n} Sum_{j=1..k} j^n.

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A366342 a(n) = Product_{k=1..n} Sum_{j=1..k} j^k.

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A302353 a(n) = Sum_{k=0..n} k^nbinomial(2n-k,n).