A260210 -id:A260210 - cp's OEIS frontend

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

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.

A260209 Values A such that p=prime(n) satisfies binomial(2p-1, p-1) == 1 + A*p (mod p^4).

Original entry on oeis.org

1, 3, 25, 245, 121, 169, 867, 3249, 6877, 9251, 961, 15059, 57154, 61017, 68479, 106742, 201898, 208376, 107736, 176435, 330398, 237158, 158447, 213867, 903264, 856884, 21218, 755634, 1259386, 944906, 161290, 531991, 150152, 656914, 1287658, 592826, 640874

Offset: 1

Felix Fröhlich, Jul 19 2015

nonn

p is a Wolstenholme prime (A088164) iff a(n) == 0. This holds for n = 1944 and n = 157504.
When performing a search for Wolstenholme primes, one can choose an integer constant c >= 0 and record all primes p with A <= c in order to get a larger data set.
The values here asymptotically appear to grow more quickly than those in A260210.
It appears that a(n)/A260210(n) = A001248(n) for all n.

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Cf. A088164, A241014, A244801, A250406, A258367, A260210.

f[n_] := Block[{p = Prime@ n}, (Mod[ Binomial[2p - 1, p - 1], p^4] - 1)/p]; Array[f, 37] (* Robert G. Wilson v, Jul 29 2015 *)

a(n) = p=prime(n); (lift(Mod(binomial(2*p-1, p-1), p^4))-1)/p

A338558 Absolute value q such that tau(p) == q (mod p), where p = prime(n) and tau(i) = A000594(i).

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 7, 7, 1, 5, 10, 6, 11, 12, 20, 24, 14, 12, 3, 19, 6, 37, 20, 33, 20, 27, 50, 34, 36, 29, 18, 64, 4, 2, 66, 32, 3, 64, 61, 51, 60, 84, 95, 83, 63, 97, 42, 28, 61, 67, 32, 10, 29, 73, 37, 92, 16, 120, 31, 107, 120, 141, 145, 39, 12, 74, 150

Offset: 1

Felix Fröhlich, Dec 21 2020

nonn

These are essentially the values that can be used to define "near-misses" in a search of terms for A007659, similar to how "near-Wieferich primes", "near-Wilson primes" and "near-Wall-Sun-Sun primes" are defined in searches for Wieferich primes (A001220), Wilson primes (A007540) and Wall-Sun-Sun (Fibonacci-Wieferich) primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Nik Lygeros and Olivier Rozier, A new solution for the equation tau(p)=0 (mod p), Journal of Integer Sequences 13 (2010), Article 10.7.4.

Cf. A001220, A007540, A007659, A295645.

A-values: A258367 (near-Wieferich), A250406 (near-Wilson), A244801 and A241014 (near-Wall-Sun-Sun), A260209 and A260210 (near-Wolstenholme).

a[n_] := Module[{p = Prime[n]}, Min[Abs[Mod[RamanujanTau[p], {-p, p}]]]]; Array[a, 100] (* Amiram Eldar, Jan 10 2025 *)

a(n) = my(p=prime(n)); abs(centerlift(Mod(ramanujantau(p), p)))

a(n) = 0 iff prime(n) is a term of A007659.

A352858 a(n) = abs(E_{p-3} (mod p)) for p = prime(n), where E_i is the i-th Euler number (A000364).

Original entry on oeis.org

1, 2, 1, 3, 8, 7, 1, 3, 9, 4, 4, 4, 14, 7, 12, 16, 25, 22, 25, 4, 23, 33, 42, 15, 46, 18, 23, 38, 58, 2, 6, 55, 0, 37, 74, 63, 10, 61, 21, 38, 92, 89, 70, 79, 69, 59, 85, 22, 27, 69, 0, 45, 58, 96, 106, 6, 50, 28, 91, 133, 46, 147, 133, 38, 29, 128, 167, 116

Offset: 3

Felix Fröhlich, Apr 06 2022

nonn

a(n) = 0 iff p is a term of A198245.

These are the absolute values of the "A-values" that can be used to define "near-misses" in a search for terms of A198245 (cf. Mestrovic, 2014).

Romeo Mestrovic, A search for primes p such that Euler number E_p-3 is divisible by p, Mathematics of Computation 83 (2014), 2967-2976.

Cf. A000364, A198245.

A-values: A258367 (near-Wieferich), A250406 (near-Wilson), A244801 and A241014 (near-Wall-Sun-Sun), A260209 and A260210 (near-Wolstenholme), A338558 (near-misses for A007659).

eulmod(n) = abs(centerlift(Mod(eulerfrac(n-3), n)))
a(n) = my(p=prime(n)); eulmod(p)

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

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A260209 Values A such that p=prime(n) satisfies binomial(2p-1, p-1) == 1 + A*p (mod p^4).

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A338558 Absolute value q such that tau(p) == q (mod p), where p = prime(n) and tau(i) = A000594(i).

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A352858 a(n) = abs(E_{p-3} (mod p)) for p = prime(n), where E_i is the i-th Euler number (A000364).

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