A244214 -id:A244214 - cp's OEIS frontend

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

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.

A298946 a(n) = binomial(2*c-1, c-1) (mod c^4), where c is the n-th composite number.

Original entry on oeis.org

35, 462, 2339, 4627, 2378, 4238, 5148, 1260, 57635, 85026, 64410, 100509, 163716, 171918, 93876, 309780, 148969, 444220, 370712, 532771, 652200, 938386, 816466, 907874, 569300, 1107298, 2470810, 2953692, 887812, 1341810, 2956584, 1941390, 589961, 6248628

Offset: 1

Felix Fröhlich, Jan 30 2018

nonn

Composites c where a(n) = 1 could be called "Wolstenholme pseudoprimes". Do any such composites exist?

A necessary condition for c to be a "Wolstenholme pseudoprime" would be that it is a term of A228562 or A267824.

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

Cf. A088164, A228562, A244214, A267824, A281302, A298944, A298945.

R:= NULL:
count:= 0: F:= 10;
for n from 4 while count < 100 do
  F:= F * (4*n-2)/n;
  if not isprime(n) then
     count:= count+1;
     R:= R, F mod (n^4);
  fi
od:
R; # Robert Israel, Feb 02 2018

Table[Mod[Binomial[2 c - 1, c - 1], c^4], {c, Select[Range@ 50, CompositeQ]}] (* Michael De Vlieger, Feb 01 2018 *)

forcomposite(c=1, 200, print1(lift(Mod(binomial(2*c-1, c-1), c^4)), ", "))

from sympy import binomial, composite
def A298946(n):
    c = composite(n)
    return binomial(2*c-1,c-1) % c**4 # Chai Wah Wu, Feb 02 2018

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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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A298946 a(n) = binomial(2*c-1, c-1) (mod c^4), where c is the n-th composite number.

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