A099905 -id:A099905 - cp's OEIS frontend

A228562 Composite numbers k that are not prime powers such that binomial(2k-1, k-1) is congruent to 1 (mod k).

27173, 2001341, 16024189487

Offset: 1

Felix Fröhlich, Aug 25 2013

Richard J. McIntosh, On the converse of Wolstenholme's Theorem, Acta Arithmetica, 71 (1995), 381-389.

The odd terms of A328497.

Select[Range[30000], PrimeNu[#] > 1 && Mod[Binomial[2# - 1, # - 1], #] == 1 &] (* Alonso del Arte, May 11 2014 *)

N=10^9; for(n=2, N, if(Mod(binomial(2*n-1, n-1), n)==1 && !ispower(n) && !isprime(n), print1(n, ", "))); \\ Felix Fröhlich, May 11 2014

vp(n,p)=my(s); while(n\=p, s+=n); s
is(n)=my(f=factor(n)[,1],G); if(#f==1, return(0)); for(i=1,#f, if(vp(2*n-1,f[i]) > vp(n,f[i])+vp(n-1,f[i]), return(0))); G=prod(i=1,#f,f[i]^(log(n)\log(f[i]))); prod(i=n+1,2*n-1, i/gcd(i,G), Mod(1,n))/prod(i=2,n-1, i/gcd(i,G), Mod(1,n))==1
forcomposite(n=4,1e9, if(is(n), print1(n", "))) \\ Charles R Greathouse IV, May 12 2014

A099905(a(n)) = 1. - Jonathan Sondow, Jan 24 2016

A099906 a(n) = binomial(2n-1,n-1) mod n^2.

Original entry on oeis.org

0, 3, 1, 3, 1, 30, 1, 35, 10, 78, 1, 62, 1, 52, 135, 35, 1, 138, 1, 10, 402, 124, 1, 270, 126, 172, 253, 476, 1, 812, 1, 291, 978, 870, 616, 674, 1, 364, 10, 410, 1, 756, 1, 1124, 1260, 532, 1, 1422, 1716, 1128, 2322, 1556, 1, 1920, 1941, 2172, 1815, 844, 1, 3528, 1, 964

Offset: 1

Henry Bottomley, Oct 29 2004

nonn

For odd primes p, Charles Babbage showed in 1819 that a(p) = 1.

			a(11) = binomial(21,10) mod 11^2 = 352716 mod 121 = 1.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A088218, A099905, A099907, A099908.

[Binomial(2*n-1, n-1) mod(n^2): n in [1..65]]; // Vincenzo Librandi, Jul 29 2015

Table[ Mod[ Binomial[2n - 1, n - 1], n^2], {n, 60}] (* Robert G. Wilson v, Dec 14 2004 *)

A099906(n)=binomial(2*n-1,n-1)%n^2 \\ M. F. Hasler, Jul 30 2015

from _future_ import division
A099906_list, b = [], 1
for n in range(1,10001):
    A099906_list.append(b % n**2)
    b = b*2*(2*n+1)//(n+1) # Chai Wah Wu, Jan 26 2016

A099908 C(2n-1,n-1) mod n^4.

Original entry on oeis.org

0, 3, 10, 35, 126, 462, 1716, 2339, 4627, 2378, 1332, 4238, 2198, 5148, 1260, 57635, 14740, 85026, 61732, 64410, 100509, 163716, 158172, 171918, 93876, 309780, 148969, 444220, 268280, 370712, 29792, 532771, 652200, 938386, 816466, 907874

Offset: 1

Henry Bottomley, Oct 29 2004

nonn

a(16843)=a(2124679)=1 meaning that 16843 and 2124679 are Wolstenholme primes A088164.

			a(11) =352716 mod 1461 =1332.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A088164, A088218, A099905, A099906, A099907.

Table[Mod[Binomial[2n-1,n-1],n^4],{n,40}] (* Harvey P. Dale, Dec 12 2021 *)

from _future_ import division
A099908_list, b = [], 1
for n in range(1,10001):
    A099908_list.append(b % n**4)
    b = b*2*(2*n+1)//(n+1) # Chai Wah Wu, Jan 26 2016

A099907 a(n) = C(2n-1,n-1) mod n^3.

Original entry on oeis.org

0, 3, 10, 35, 1, 30, 1, 291, 253, 378, 1, 782, 1, 2404, 1260, 291, 1, 3378, 1, 410, 7899, 3996, 1, 6030, 126, 10988, 11188, 5180, 1, 19712, 1, 8483, 5334, 34394, 1841, 21410, 1, 20580, 39556, 38810, 1, 64260, 1, 35972, 66060, 36504, 1, 61326, 1716, 123628

Offset: 1

Henry Bottomley, Oct 29 2004

nonn

For p prime > 3, Joseph Wolstenholme showed in 1862 that a(p)=1. - corrected by Jonathan Sondow, Jan 24 2016

			a(11) = 352716 mod 1331 = 1.

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

Cf. A088218, A099905, A099906, A099908, A244214.

[Binomial(2*n-1, n-1) mod n^3: n in [1..50]]; // Vincenzo Librandi, Jan 24 2016

seq(binomial(2*n-1,n-1) mod n^3, n=1..100); # Robert Israel, Jan 24 2016

Table[Mod[Binomial[2 n - 1, n - 1], n^3], {n, 1, 50}] (* Vincenzo Librandi, Jan 24 2016 *)

a(n) = binomial(2*n-1, n-1) % n^3; \\ Michel Marcus, Jan 24 2016

A136327 Numbers k > 1 such that binomial(2k-1, k-1) == 1 (mod k).

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251

Offset: 1

Franz Vrabec, Mar 26 2008

nonn

k such that A099905(k) = 1.
Contains primes, squares of odd primes and cubes of primes >= 5.
See A228562 for terms that are neither primes nor prime powers. [Joerg Arndt, Aug 27 2013]

			a(3) = 5 because binomial(9, 4) = 126 == 1 (mod 5).

Robert Israel, Table of n, a(n) for n = 1..10000
McIntosh, R. J. (1995), On the converse of Wolstenholme's theorem, Acta Arithm., LXXI.4 (1995), 381-389.

Cf. A099905.

filter:= k -> binomial(2*k-1,k-1) mod k = 1:
select(filter, [$1..1000]); # Robert Israel, Feb 11 2025

Select[Range[300], Mod[Binomial[2# - 1, # - 1], #] == 1 &] (* Alonso del Arte, May 11 2014 *)

isok(n) = (binomial(2*n-1, n-1) % n) == 1; \\ Michel Marcus, Aug 26 2013

Name corrected by Robert Israel, Feb 11 2025

A267824 Composite numbers n such that binomial(2n-1, n-1) == 1 (mod n^2).

Original entry on oeis.org

283686649, 4514260853041

Offset: 1

Jonathan Sondow, Jan 25 2016

Babbage proved the congruence holds if n > 2 is prime.
See A088164 and A263882 for references, links, and additional comments.
Conjecture: n is a term if and only if n = A088164(i)^2 for some i >= 1 (cf. McIntosh, 1995, p. 385). - Felix Fröhlich, Jan 27 2016
The "if" part of the conjecture is true: see the McIntosh reference. - Jonathan Sondow, Jan 28 2016
The above conjecture implies that this sequence and A228562 are disjoint. - Felix Fröhlich, Jan 27 2016
Composites c such that A281302(c) > 1. - Felix Fröhlich, Feb 21 2018

			a(1) = 16843^2 and a(2) = 2124679^2 are squares of Wolstenholme primes A088164.

Richard J. McIntosh, On the converse of Wolstenholme's Theorem, Acta Arithmetica, 71 (1995), 381-389.
J. Sondow, Extending Babbage's (non-)primality tests, in Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, 269-277, CANT 2015 and 2016, New York, 2017; arXiv:1812.07650 [math.NT], 2018.

Cf. A000984, A034602, A082180, A088164, A099905, A099906, A099907, A099908, A136327, A177783, A212557, A228562, A242473, A244214, A244919, A246130, A246132, A246133, A246134, A260209, A260210, A263429, A263882, A281302.

A242473 Binomial(2p-1,p-1) modulo p^4, with p=prime(n).

Original entry on oeis.org

3, 10, 126, 1716, 1332, 2198, 14740, 61732, 158172, 268280, 29792, 557184, 2343315, 2623732, 3218514, 5657327, 11911983, 12710937, 7218313, 12526886, 24119055, 18735483, 13151102, 19034164, 87616609, 86545285, 2185455, 80852839, 137273075, 106774379, 20483831, 69690822, 20570825

Offset: 1

Felix Fröhlich, May 26 2014

nonn

A value of 1 indicates a Wolstenholme prime.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A088164, A099905, A099906, A099907. Subsequence of A099908.

Table[Mod[Binomial[2p-1,p-1],p^4],{p,Prime[Range[30]]}] (* Harvey P. Dale, Jun 26 2017 *)

forprime(n=2, 10^2, m=(binomial(2*n-1, n-1)%n^4); print1(m, ", "));

from _future_ import division
from sympy import isprime
A242473_list, b = [], 1
for n in range(1,10**4):
    if isprime(n):
        A242473_list.append(b % n**4)
    b = b*2*(2*n+1)//(n+1) # Chai Wah Wu, Jan 26 2016

from sympy import Mod, binomial, prime
def A242473(n): return int(Mod(binomial(2*(p:=prime(n))-1,p-1,evaluate=False),p**4)) # Chai Wah Wu, Apr 24 2025

A383505 Least integer k >= 0 such that binomial(k*n,k+1) = -1 mod n, or -1 if no such integer exists.

Original entry on oeis.org

0, 1, 2, 3, 4, 341, 6, 79, 8, 19599, 10, 3937027727, 12, 2841, 22679, 47, 16, 18459448019, 18, 179, 146, 4003647, 22, 77934182399, 24, 299519, 80, 29952579, 28

Offset: 1

Jason Bard, May 05 2025

Conjecture: a(A000430(n)) = A000430(n)-1.
It is not hard to see that conjecture is true. - Max Alekseyev, Jul 24 2025
Other values calculated: a(32) = 1343, a(34) = 1121, a(38) = 417.
If exists, a(30) > 10^13. - Max Alekseyev, Jul 24 2025

			a(6) = 341 because binomial(341*6, 341+1) = 5 mod 6, and no smaller nonnegative integer satisfies this.

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: Binomial coefficients modulo integers (binomod.gp).

Cf. A000430, A059288, A099905.

f = {}; Do[k = 0; While[! Mod[Binomial[k*n, k + 1], n] == n - 1, k++]; f = Join[f, {k}], {n, 1, 11}]

a(n) = my(k=0); while (binomod(k*n,k+1, n) != Mod(-1, n), k++); k; \\ Michel Marcus, May 10 2025

If p is prime, then a(p) = p-1 by Lucas' theorem. - Chai Wah Wu, Jul 21 2025

a(12) from Chai Wah Wu, Jul 21 2025

a(18), a(22), a(24), a(26), a(28) from Max Alekseyev, Jul 24 2025

cp's OEIS Frontend

A228562 Composite numbers k that are not prime powers such that binomial(2k-1, k-1) is congruent to 1 (mod k).

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A099906 a(n) = binomial(2n-1,n-1) mod n^2.

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A099908 C(2n-1,n-1) mod n^4.

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A099907 a(n) = C(2n-1,n-1) mod n^3.

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A136327 Numbers k > 1 such that binomial(2k-1, k-1) == 1 (mod k).

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A267824 Composite numbers n such that binomial(2n-1, n-1) == 1 (mod n^2).

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A242473 Binomial(2p-1,p-1) modulo p^4, with p=prime(n).

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