A099906 -id:A099906 - cp's OEIS frontend

A099905 a(n) = binomial(2n-1, n-1) mod n.

0, 1, 1, 3, 1, 0, 1, 3, 1, 8, 1, 2, 1, 10, 0, 3, 1, 12, 1, 10, 3, 14, 1, 6, 1, 16, 10, 0, 1, 2, 1, 3, 21, 20, 21, 26, 1, 22, 10, 10, 1, 0, 1, 24, 0, 26, 1, 30, 1, 28, 27, 48, 1, 30, 16, 44, 48, 32, 1, 48, 1, 34, 6, 35, 35, 0, 1, 18, 33, 20, 1, 18, 1, 40, 60, 16, 0, 72, 1, 10, 10, 44, 1, 56, 75

Offset: 1

Henry Bottomley, Oct 29 2004

For p prime, a(p)=1. For n in A058008, a(n)=0.
For n the square of a prime p>=3 or the cube of a prime p>=5, a(n)=1. - Franz Vrabec, Mar 26 2008
For n in A228562, a(n)=1. - Felix Fröhlich, Oct 17 2015

			a(11) = 352716 mod 11 = 1.

Felix Fröhlich, Table of n, a(n) for n = 1..10000
R. J. McIntosh, On the converse of Wolstenholme's theorem, Acta Arithmetica 71 (4): 381-389, (1995).

Cf. A058008, A088218, A099906, A099907, A099908, A228562.

[Binomial(2*n-1,n-1) mod n : n in [1..100]]; // Wesley Ivan Hurt, Oct 17 2015

A099905:=n->binomial(2*n-1,n-1) mod n: seq(A099905(n), n=1..100); # Wesley Ivan Hurt, Oct 17 2015

Table[Mod[Binomial[2n-1,n-1],n],{n,90}] (* Harvey P. Dale, Dec 12 2011 *)

a(n) = lift(Mod(binomial(2*n-1, n-1), n)) \\ Felix Fröhlich, Oct 17 2015

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

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.

A263882 Babbage quotients b_p = (binomial(2p-1, p-1) - 1)/p^2 with p = prime(n).

Original entry on oeis.org

1, 5, 35, 2915, 30771, 4037381, 48954659, 7782070631, 17875901604959, 242158352370063, 637739431824553035, 126348774791431208099, 1794903484322270273951, 367972191114796344623951, 1116504994413003106003899551, 3498520498083111051973370669639

Offset: 2

Jonathan Sondow, Nov 22 2015

nonn

Charles Babbage proved in 1819 that b_p is an integer for prime p > 2. In 1862 Wolstenholme proved that the Wolstenholme quotient W_p = b_p / p is an integer for prime p > 3; see A034602.

The quotient b_n is an integer for composite n in A267824. No composite n is known for which W_n is an integer.

			a(2) = (binomial(2*3-1,3-1) - 1)/3^2 = (binomial(5,2) - 1)/9 = (10-1)/9 = 1.

R. K. Guy, Unsolved Problems in Number Theory, Sect. B31.

Robert Israel, Table of n, a(n) for n = 2..260
C. Babbage, Demonstration of a theorem relating to prime numbers, Edinburgh Philosophical Journal, 1 (1819), 46-49.
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
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.
Wikipedia, Wolstenholme's theorem
J. Wolstenholme, On certain properties of prime numbers, Quarterly Journal of Pure and Applied Mathematics, 5 (1862), 35-39.

Cf. A034602, A088164, A099906, A267824.

[(Binomial(2*NthPrime(n)-1, NthPrime(n)-1)-1)/NthPrime(n)^2: n in [2..20]]; // Vincenzo Librandi, Nov 25 2015

map(p -> (binomial(2*p-1,p-1)-1)/p^2, select(isprime,[seq(i,i=3..100,2)])); # Robert Israel, Nov 24 2015

Table[(Binomial[2*Prime[n] - 1, Prime[n] - 1] - 1)/Prime[n]^2, {n, 2, 17}]
Table[(Binomial[2p-1,p-1]-1)/p^2,{p,Prime[Range[2,20]]}] (* Harvey P. Dale, Jul 20 2019 *)

a(n) = prime(n)*A034602(n) for n > 2.

a(PrimePi(A088164(n))) == 0 mod A088164(n)^2.

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

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A099905 a(n) = binomial(2n-1, n-1) mod n.

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

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A263882 Babbage quotients b_p = (binomial(2p-1, p-1) - 1)/p^2 with p = prime(n).

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

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