A121706 -id:A121706 - cp's OEIS frontend

A121707 Numbers n > 1 such that n^3 divides Sum_{k=1..n-1} k^n = A121706(n).

35, 55, 77, 95, 115, 119, 143, 155, 161, 187, 203, 209, 215, 221, 235, 247, 253, 275, 287, 295, 299, 319, 323, 329, 335, 355, 371, 377, 391, 395, 403, 407, 413, 415, 437, 455, 473, 475, 493, 497, 515, 517, 527, 533, 535, 539, 551, 559, 575, 581, 583, 589, 611

Offset: 1

Alexander Adamchuk, Aug 16 2006

nonn

All terms belong to A038509 (Composite numbers with smallest prime factor >= 5). Many but not all terms belong to A060976 (Odd nonprimes, c, which divide Bernoulli(2*c)).
Many terms are semiprimes:
- the non-semiprimes are {275, 455, 475, 539, 575, 715, 775, 875, 935, ...}: see A321487;
- semiprime terms that are multiples of 5 have indices {7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, ...} = A002145 (Primes of form 4*k + 3, except 3, or k > 0; or Primes which are also Gaussian primes);
- semiprime terms that are multiples of 7 have indices {5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, ...} = A003627 (Primes of form 3*k - 1, except 2, or k > 1);
- semiprime terms that are multiples of 11 have indices {5, 7, 13, 17, 19, 23, 37, 43, 47, 53, 59, 67, 73, 79, 83, ...} = Primes of the form 4*k + 1 and 4*k - 1. [Edited by Michel Marcus, Jul 21 2018, M. F. Hasler, Nov 09 2018]
Conjecture: odd numbers n > 1 such that n divides Sum_{k=1..n-1} k^(n-1). - Thomas Ordowski and Robert Israel, Oct 09 2015. Professor Andrzej Schinzel (in a letter to me, dated Dec 29 2015) proved this conjecture. - Thomas Ordowski, Jul 20 2018
Note that n^2 divides Sum_{k=1..n-1} k^n for every odd number n > 1. - Thomas Ordowski, Oct 30 2015
Conjecture: these are "anti-Carmichael numbers" defined; n > 1 such that p - 1 does not divide n - 1 for every prime p dividing n. Equivalently, odd numbers n > 1 such that n is coprime to A027642(n-1). A number n > 1 is an "anti-Carmichael" if and only if gcd(n, b^n - b) = 1 for some integer b. - Thomas Ordowski, Jul 20 2018
It seems that these numbers are all composite terms of A317358. - Thomas Ordowski, Jul 30 2018
a(62) = 697 is the first term not in A267999: see A306097 for all these terms. - M. F. Hasler, Nov 09 2018
If the conjecture from Thomas Ordowski is true, then no term is a multiple of 2 or 3. - Jianing Song, Jan 27 2019
Conjecture: an odd number n > 1 is a term iff gcd(n, A027642(n-1)) = 1. - Thomas Ordowski, Jul 19 2019
Conjecture: Sequence consists of numbers n > 1 such that r = b^n + n - b will produce a prime for one or more integers b > 1. Only when n is in this sequence do one or more prime factors of n fail to divide r for all b.  Also, n and b must be coprime for r to be prime. The above also applies to r = b^n - n - b, ignoring n=3, b=2. - Richard R. Forberg, Jul 18 2020
Odd numbers n > 1 such that Sum_{k(even)=2..n-1}2*k^(n-1) == 0 (mod n). - Davide Rotondo, Oct 28 2020
What is the asymptotic density of these numbers? The numbers A267999 have a slightly lower density. The difference between the densities is equal to the density of the numbers A306097. - Thomas Ordowski, Feb 15 2021
The asymptotic density of this sequence is in the interval (0.253, 0.265) (Ordowski, 2021). - Amiram Eldar, Feb 26 2021

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1371 terms from Robert Israel)
T. Ordowski, Density of anti-Carmichael numbers, SeqFan Mailing List, Feb 17 2021.
Don Reble, Comments on A121707

Cf. A000312, A002145, A002997, A027642, A031971, A038509, A060976, A121706, A267999 (probably a subsequence).
Cf. A306097 for terms of this sequence A121707 not in sequence A267999, A321487 for terms which are not semiprimes.
Cf. A191677 (n divides Sum_{kCf. A326478 for a conjectured connection with the Bernoulli numbers.

filter:= n -> add(k &^ n mod n^3, k=1..n-1) mod n^3 = 0:
select(filter, [$2..1000]); # Robert Israel, Oct 08 2015

fQ[n_] := Mod[Sum[PowerMod[k, n, n^3], {k, n - 1}], n^3] == 0; Select[
Range[2, 611], fQ] (* Robert G. Wilson v, Apr 04 2011 and slightly modified Aug 02 2018 *)

is(n)=my(n3=n^3);sum(k=1,n-1,Mod(k,n3)^n)==0 \\ Charles R Greathouse IV, May 09 2013

for(n=2, 1000, if(sum(k=1, n-1, k^n) % n^3 == 0, print1(n", "))) \\ Altug Alkan, Oct 15 2015

# after Andrzej Schinzel
def isA121707(n):
    if n == 1 or is_even(n): return False
    return n.divides(sum(k^(n-1) for k in (1..n-1)))
[n for n in (1..611) if isA121707(n)] # Peter Luschny, Jul 18 2019

Sequence corrected by Robert G. Wilson v, Apr 04 2011

A228919 Numbers n such that 1^(n+1) + 2^(n+1) + ... + n^(n+1) == 0 (mod n).

Original entry on oeis.org

1, 4, 5, 7, 8, 11, 12, 13, 16, 17, 19, 20, 23, 24, 25, 28, 29, 31, 32, 36, 37, 40, 41, 43, 44, 47, 48, 49, 52, 53, 56, 59, 60, 61, 64, 65, 67, 68, 71, 72, 73, 76, 79, 80, 83, 84, 85, 88, 89, 91, 92, 96, 97, 100, 101, 103, 104, 107, 108, 109, 112, 113, 116

Offset: 1

José María Grau Ribas, Sep 08 2013

nonn

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

Cf. A191677, A121706.

f[n_] := Mod[Sum[PowerMod[i, n + 1, n], {i, 1, n}], n]; Select[Range[1000], f[#] == 0 &]

is(n)=my(m=n+1);sum(k=1,n,Mod(k,n)^m)==0 \\ Charles R Greathouse IV, Nov 20 2013

A321487 Numbers in A121707 (n^3 > 1 divides Sum_{k=1..n-1} k^n) which are not semiprimes.

Original entry on oeis.org

275, 455, 475, 539, 575, 715, 775, 875, 935, 1075, 1127, 1175, 1235, 1295, 1375, 1421, 1463, 1475, 1495, 1547, 1595, 1615, 1675, 1715, 1775, 1859, 1955, 1975, 2009, 2015, 2035, 2057, 2075, 2093, 2135, 2255, 2261, 2299, 2303, 2375, 2387, 2555, 2575, 2597, 2635, 2639, 2675, 2717, 2783

Offset: 1

M. F. Hasler, Nov 11 2018

nonn

Most terms of A121707 and its (conjectured) subsequence A267999 are semiprimes. This sequence lists the exceptions.
At first, it looked as if most terms were multiples of 5. The first exceptions are a({4, 11, 16}) = {539, 1127, 1421}. However, after the first 30 terms, almost every other term is not divisible by 5.
The first terms not in A267999 are {2057, 2873, 3689, 4015, 4991, 6137, ...}, cf. A321488.

M. F. Hasler, Table of n, a(n) for n = 1..2500

Cf. A121706, A267999, A319386, A306097.

A342395 a(n) = Sum_{k=1..n} k^(n/gcd(k,n)).

Original entry on oeis.org

1, 3, 12, 90, 1305, 15713, 376768, 6163324, 176787369, 3769360335, 142364319636, 3152514811878, 154718778284161, 4340009168261557, 210971169749009040, 7281661102087491416, 435659030617933827153, 14181101408651996188995

Offset: 1

Seiichi Manyama, Mar 10 2021

Cf. A121706, A332620, A342389, A342396.

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

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

If p is prime, a(p) = A121706(p) + p.

A356100 a(n) = Sum_{k=1..n} (k - 1)^n * floor(n/k).

Original entry on oeis.org

0, 1, 9, 99, 1301, 20581, 376891, 7914216, 186905206, 4915451602, 142368695176, 4506118905870, 154720069309364, 5729167232515112, 227585086051159866, 9654819212943764500, 435659280972794395356, 20836049921760968809231, 1052864549462731148832219

Offset: 1

Seiichi Manyama, Jul 26 2022

nonn

Cf. A121706, A236632, A319194, A332469.

Table[Sum[(k-1)^n Floor[n/k],{k,n}],{n,20}] (* Harvey P. Dale, Dec 14 2024 *)

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

a(n) = sum(k=1, n, sigma(k, n)-(n\k)^n);

a(n) = sum(k=1, n, sumdiv(k, d, (d-1)^n));

def A356100(n): return sum((k-1)**n*(n//k) for k in range(2,n+1)) # Chai Wah Wu, Jul 26 2022

a(n) = A319194(n) - A332469(n).
a(n) = Sum_{k=1..n} Sum_{d|k} (d - 1)^n.
a(n) = [x^n] (1/(1-x)) * Sum_{k>=1} (k - 1)^n * x^k/(1 - x^k).

A228926 Sum(m^(n+1), m=1...n-1) modulo n.

Original entry on oeis.org

0, 1, 2, 0, 0, 3, 0, 0, 6, 5, 0, 0, 0, 7, 7, 0, 0, 9, 0, 0, 14, 11, 0, 0, 0, 13, 18, 0, 0, 15, 0, 0, 22, 17, 23, 0, 0, 19, 26, 0, 0, 21, 0, 0, 30, 23, 0, 0, 0, 25, 34, 0, 0, 27, 44, 0, 38, 29, 0, 0, 0, 31, 42, 0, 0, 33, 0, 0, 46, 35, 0, 0, 0, 37, 35, 0, 66

Offset: 1

José María Grau Ribas, Sep 08 2013

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A121706, A228919, A204187.

f[n_] := Mod[Sum[PowerMod[i,n+1,n], {i, 1, n}], n]; Table[f[n], {n, 100}]

a(n)=lift(sum(m=1,n-1,Mod(m,n)^(n+1))) \\ Charles R Greathouse IV, Dec 27 2013

a(n) = A121706(n) mod (n-1). - T. D. Noe, Sep 16 2013

A340806 a(n) = Sum_{k=1..n-1} (k^n mod n).

Original entry on oeis.org

0, 1, 3, 2, 10, 13, 21, 4, 27, 45, 55, 38, 78, 77, 105, 8, 136, 93, 171, 146, 210, 209, 253, 172, 250, 325, 243, 294, 406, 365, 465, 16, 528, 561, 595, 402, 666, 665, 741, 372, 820, 673, 903, 726, 945, 897, 1081, 536, 1029, 1125, 1275, 1170, 1378, 765, 1485

Offset: 1

Sebastian Karlsson, Jan 22 2021

nonn

Sebastian Karlsson, Table of n, a(n) for n = 1..10000

Cf. A010848, A048153, A048155, A094264, A121706, A195637, A195812, A202034.

a:= n-> add(k&^n mod n, k=1..n-1):
seq(a(n), n=1..55);  # Alois P. Heinz, Feb 13 2021

a(n) = sum(k=1, n-1, lift(Mod(k, n)^n)); \\ Michel Marcus, Jan 22 2021

def a(n):
    return sum([pow(k,n,n) for k in range(1, n)])
for n in range(1, 56):
    print(a(n), end=', ')

a(n) = n*A010848(n)/2, if n is odd.
a(n) = n*(n-1)/2, if n is both odd and squarefree.
a(p^e) = (1/2)*(p-1)*p^(2*e-1), if p is an odd prime.
a(2^e) = 2^(e-1).

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.

Showing 1-8 of 8 results.

cp's OEIS Frontend

A121707 Numbers n > 1 such that n^3 divides Sum_{k=1..n-1} k^n = A121706(n).

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A228919 Numbers n such that 1^(n+1) + 2^(n+1) + ... + n^(n+1) == 0 (mod n).

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A321487 Numbers in A121707 (n^3 > 1 divides Sum_{k=1..n-1} k^n) which are not semiprimes.

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A342395 a(n) = Sum_{k=1..n} k^(n/gcd(k,n)).

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A356100 a(n) = Sum_{k=1..n} (k - 1)^n * floor(n/k).

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A228926 Sum(m^(n+1), m=1...n-1) modulo n.

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A340806 a(n) = Sum_{k=1..n-1} (k^n mod n).

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

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