A133906 -id:A133906 - cp's OEIS frontend

A317058 a(n) is the smallest composite k such that 1^(k-1) + 2^(k-1) + ... + n^(k-1) == n (mod k).

4, 341, 473, 4, 4, 133, 497, 4, 4, 15, 9, 4, 4, 143, 35, 4, 4, 51, 57, 4, 4, 77, 253, 4, 4, 65, 9, 4, 4, 115, 155, 4, 4, 187, 35, 4, 4, 9, 247, 4, 4, 287, 2051, 4, 4, 15, 33, 4, 4, 35, 85, 4, 4, 9, 9, 4, 4, 551, 1711, 4, 4, 713, 21, 4, 4, 55, 77, 4, 4, 35, 35, 4

Offset: 1

Thomas Ordowski, Jul 26 2018

nonn

According to the Agoh-Giuga conjecture, a(n) <> n+1.

a(n) = 4 if and only if n == {0, 1} (mod 4).

Chai Wah Wu, Table of n, a(n) for n = 1..9661 (n = 1..500 from Seiichi Manyama)
Wikipedia, Agoh-Giuga conjecture

Cf. A000790, A133906, A133907, A317357, A317358.

a[n_] := Block[{k = 4}, While[PrimeQ[k] || Mod[Sum[PowerMod[j, k-1, k], {j, n}], k] != Mod[n, k], k++]; k]; Array[a, 72] (* Giovanni Resta, Jul 26 2018 *)

a(n) = forcomposite(k=1,, if (sum(j=1,n, Mod(j,k)^(k-1)) == n, return (k));); \\ Michel Marcus, Jul 26 2018

from sympy import isprime
def g(n,p,q): # compute (-n + sum_{k=1,n} k^p)  mod q
    c = (-n) % q
    for k in range(1,n+1):
        c = (c+pow(k,p,q)) % q
    return c
def A317058(n):
    k = 2
    while isprime(k) or g(n,k-1,k):
        k += 1
    return k # Chai Wah Wu, Jul 30 2018

More terms from Giovanni Resta, Jul 26 2018

A317358 a(n) is the smallest number k > 1 such that 1^(k-1) + 2^(k-1) + ... + n^(k-1) == n (mod k).

Original entry on oeis.org

2, 3, 5, 2, 2, 7, 11, 2, 2, 3, 3, 2, 2, 17, 17, 2, 2, 3, 3, 2, 2, 23, 29, 2, 2, 5, 3, 2, 2, 31, 37, 2, 2, 37, 35, 2, 2, 3, 41, 2, 2, 43, 47, 2, 2, 3, 3, 2, 2, 5, 5, 2, 2, 3, 3, 2, 2, 59, 61, 2, 2, 67, 3, 2, 2, 55, 71, 2, 2, 35, 35, 2, 2, 3, 5, 2, 2, 5, 5, 2, 2

Offset: 1

Thomas Ordowski, Jul 26 2018

nonn

a(n) = 2 if and only if n == {0, 1} (mod 4).
a(n) <= A151800(n).
A133906(n) <= a(n) <= A133907(n).
The sequence is unbounded.
Numbers n such that a(n-1) = n are 2, 3, 7, 23, 31, 43, 59, 139, 283, ...
By the Agoh-Giuga conjecture, if a(n-1) = n, then n is a prime.
It seems that if a(n) > n, then a(n) is a prime (the next prime after n).
If a(n) = n, then n is in A121707. These numbers are 35, 143, 187, 215, ...
Conjecture: all composite terms of the sequence are A121707.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1000 from Seiichi Manyama)
Wikipedia, Agoh-Giuga conjecture

Cf. A133906, A133907, A151800, A121707, A317058, A317357.

a[n_] := Block[{k=2}, While[Mod[Sum[PowerMod[j, k-1, k], {j, n}], k] != Mod[n, k], k++]; k]; Array[a, 81] (* Giovanni Resta, Jul 29 2018 *)

a(n) = for(k=2,oo, if (sum(j=1,n, Mod(j,k)^(k-1)) == n, return (k));); \\ Michel Marcus, Jul 26 2018

def g(n,p,q): # compute (-n + sum_{k=1,n} k^p)  mod q
    c = (-n) % q
    for k in range(1,n+1):
        c = (c+pow(k,p,q)) % q
    return c
def A317358(n):
    k = 2
    while g(n,k-1,k):
        k += 1
    return k # Chai Wah Wu, Jul 30 2018

More terms from Michel Marcus, Jul 26 2018

A133907 Least prime number p such that binomial(n+p, p) mod p = 1.

Original entry on oeis.org

2, 3, 5, 2, 2, 7, 11, 2, 2, 3, 3, 2, 2, 17, 17, 2, 2, 3, 3, 2, 2, 23, 29, 2, 2, 5, 3, 2, 2, 31, 37, 2, 2, 37, 37, 2, 2, 3, 41, 2, 2, 43, 47, 2, 2, 3, 3, 2, 2, 5, 5, 2, 2, 3, 3, 2, 2, 59, 61, 2, 2, 67, 3, 2, 2, 67, 71, 2, 2, 71, 73, 2, 2, 3, 5, 2, 2, 5, 5, 2, 2, 3, 3, 2, 2, 89, 89, 2, 2, 3, 3, 2, 2, 97

Offset: 1

Hieronymus Fischer, Oct 20 2007

nonn

Also the least prime number p such that p divides floor(n/p) or p > n.
a(n) = 2 if and only if n is in A042948. - Robert Israel, May 11 2017
Conjecture: a(n) is the smallest prime p such that Sum_{k=1..n} k^(p-1) == n (mod p). Thus a(n) >= A317358(n). - Thomas Ordowski, Jul 29 2018

			a(2)=3, since binomial(2+3,3) mod 3 = 10 mod 3 = 1 and 3 is the minimal prime number with this property.
a(7)=11 because of binomial(7+11, 11) = 31824 = 2893*11 + 1, but binomial(7+k, k) mod k <> 1 for all primes < 11.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A042948, A133620, A133621, A133623, A133630, A133635.

Cf. A133872, A133880, A133890, A133900, A133906, A133910, A317358.

f:= proc(n) local m;
  m:= 2:
  while floor(n/m) mod m <> 0 do m:= nextprime(m) od:
  m
end proc:
map(f, [$1..100]); # Robert Israel, May 11 2017

a[n_] := Module[{p}, For[p = 2, True, p = NextPrime[p], If[Mod[Binomial[n+p, p], p] == 1, Return[p]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 05 2023 *)

a(n) = my(p=2); while (binomial(n+p, p) % p != 1, p = nextprime(p+1)); p; \\ Michel Marcus, Dec 17 2022

from sympy import nextprime, ff
def A133907(n):
    p, m = 2, (n+2)*(n+1)>>1
    while m%p != 1:
        q = nextprime(p)
        m = m*ff(n+q,q-p)//ff(q,q-p)
        p = q
    return p # Chai Wah Wu, Feb 22 2023

A317872 a(n) is the number of times that binomial(n+m, m) mod m = 1, for 0 < m <= n.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 0, 1, 3, 2, 2, 1, 1, 0, 0, 2, 2, 1, 1, 3, 2, 0, 0, 1, 2, 2, 4, 4, 4, 0, 0, 3, 4, 1, 1, 3, 3, 1, 0, 1, 1, 2, 2, 1, 2, 1, 1, 2, 4, 2, 2, 4, 4, 6, 4, 3, 2, 2, 2, 1, 1, 0, 1, 7, 6, 1, 1, 1, 1, 1, 1, 3, 3, 2, 2, 2, 2, 1, 1, 3, 7, 4, 4, 4, 4, 0, 1, 2, 2, 2, 2, 2, 1, 0, 0, 3, 3, 3, 4, 5, 5, 2, 2, 3, 1

Offset: 1

Hieronymus Fischer and Robert G. Wilson v, Aug 09 2018

nonn

Inspired by A133906.
First occurrence of k with k = 0, 1, 2,...: 1, 4, 10, 9, 27, 100, 54, 64, 176, 544, 352, 648, 649, 129, 128, 1378, 513, 729, 7776, 1377, 5832, 1701, 3728, 13312, 13825, ...
Records: 0, 1, 3, 4, 6, 7, 14, 16, 17, 19, 21, 22, ..., ; and they occur at: 1, 4, 9, 27, 54, 64, 128, 513, 729, 1377, 1701, 3728, 6656, ...

			a(9) = 3 because binomial(9+2,2) mod 2 = binomial(9+3,3) mod 3 = binomial(9+6,6) mod 6 = 1. [Corrected by _Jon E. Schoenfield_, Aug 28 2018]

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A133906.

a[n_] := Block[{c = 0, m = 1}, While[m < n + 1, If[ Mod[ Binomial[n + m, m], m] == 1, c++]; m++]; c]; Array[a, 105]
Table[Count[Table[Mod[Binomial[n+m,m],m],{m,n}],1],{n,120}] (* Harvey P. Dale, Aug 18 2022 *)

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A317058 a(n) is the smallest composite k such that 1^(k-1) + 2^(k-1) + ... + n^(k-1) == n (mod k).

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A317358 a(n) is the smallest number k > 1 such that 1^(k-1) + 2^(k-1) + ... + n^(k-1) == n (mod k).

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A133907 Least prime number p such that binomial(n+p, p) mod p = 1.

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A317872 a(n) is the number of times that binomial(n+m, m) mod m = 1, for 0 < m <= n.

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