A034602 -id:A034602 - cp's OEIS frontend

A222207 Morley quotients: (2^(2p-2) - (-1)^((p-1)/2)binomial(p-1,(p-1)/2)) / p^3, where p = prime(n) and n >= 3.

2, 12, 788, 7636, 874202, 10018884, 1445893544, 2954512034024, 38700329118256, 93229749133527532, 17540746936557672236, 243284404062970619608, 47694250379410432495952, 136236017676683906365850456, 404504597532158799519693872144, 5856120097210409121404621878992, 18102352585707069737371994385420772, 3894254646848417473467131712404310728

Offset: 3

Jonathan Sondow, Feb 22 2013

nonn

Morley (1894/95) proved 2^(2*p-2) == (-1)^((p-1)/2)*binomial(p-1,(p-1)/2) mod p^3 for all primes p > 3.

Morley quotients are even, since 2^(2*p-2) and binomial(p-1,(p-1)/2) are even and p^3 is odd.

			prime(3) = 5, so a(3) = (2^(2*5-2) - (-1)^((5-1)/2)*binomial(5-1,(5-1)/2))/5^3 = (2^8 - binomial(4,2))/5^3 = (256-6)/125 = 2.

Vincenzo Librandi, Table of n, a(n) for n = 3..200
C. Aebi, G. Cairns, Morley’s other miracle, Math. Mag., 85 (2012), 205-211.
F. Morley, Note on the Congruence 2^4n == (-1)^n*(2n)!/(n!)^2 where 2n+1 is a prime, Annals of Mathematics, Vol. 9 (1894 - 1895), pp. 168-170.

Cf. A007619, A007663, A034602, A197630, A197633.

m[p_] := (2^(2*p-2) - (-1)^((p-1)/2)*Binomial[p-1, (p-1)/2])/p^3; Table[ m[ Prime[n]], {n, 3, 20}]

A244919 For odd prime p, largest k such that binomial(2p-1, p-1) is congruent to 1 modulo p^k.

Original entry on oeis.org

2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 2

Felix Fröhlich, Jul 08 2014

nonn

Wolstenholme's theorem implies that k >= 3 for all p > 3. The prime p is a Wolstenholme prime if and only if k > 3. For the primes up to 10^9 this holds only for p = 16843 and p = 2124679, where in each case a(n) = 4 (i.e. a(1944) = 4 and a(157504) = 4).

R. J. McIntosh, On the converse of Wolstenholme's theorem, Acta Arith., Volume 71, Issue 4 (1995), 381-389.
R. J. McIntosh and E. L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Math. Comp., 76 (2007), 2087-2094.

Cf. A034602, A088164.

forprime(p=3, 10^3, k=1; while(Mod(binomial(2*p-1, p-1), p^k)==1, j=k; k++); if(Mod(binomial(2*p-1, p-1), p^k)!=1, print1(j, ", ")))

A125551 As p runs through primes >= 5, sequence gives { numerator of Sum_{k=1..p-1} 1/k^2 } / p.

Original entry on oeis.org

41, 767, 178939, 18500393, 48409924397, 12569511639119, 15392144025383, 358066574927343685421, 282108494885353559158399, 911609127797473147741660153, 1128121200256091571107985892349

Offset: 3

Artur Jasinski, Jan 03 2007

nonn

This is an integer by a theorem of Waring and Wolstenholme.

R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057, 2011

Cf. A061002, A034602, A186720, A186722.

f1:=proc(n) local p;
p:=ithprime(n);
(1/p)*numer(add(1/i^2,i=1..p-1));
end proc;
[seq(f1(n),n=3..20)];

a = {}; Do[AppendTo[a, (1/(Prime[x]))Numerator[Sum[1/x^2, {x, 1, Prime[x] - 1}]]], {x, 3, 50}]; a
Table[Sum[1/k^2,{k,p-1}]/p,{p,Prime[Range[3,20]]}]//Numerator (* Harvey P. Dale, Nov 20 2019 *)

A217772 a(n) = ((p+1)(3p)!/((2p-1)!(p+1)!*2p) - 3)/(3p^3), where p is the n-th prime.

Original entry on oeis.org

1, 8, 113, 48469, 1232351, 1002175798, 30956114561, 32956274508457, 1386101220044940571, 50017672586399947073, 2548160990547719392420658, 3694160975065765801289789930, 142486973648670437070915061157

Offset: 2

Alexander R. Povolotsky, Mar 24 2013

nonn

This sequence is based on Gary Detlefs's conjecture, which he posted as a comment to A005809. His conjecture is equivalent to the conjecture that the Diophantine equation ((n+1)*(3*n)!/((2*n-1)!*(n+1)!*2*n)-3)/n^3 = m has integer solutions m for all odd primes n.
Additionally I conjecture that all m are divisible by 3, therefore terms of this sequence a(n) = m/3.
It is also notable that for quite a few values of n (2, 3, 4, 5, 6, 7, 17, 19, 21, 22, 23, 24, 25, 26, 35, 39, 43, ...) a(n+1) = a(n) mod 7.
The values of this sequence's terms are replicated by conjectured general formula, given in A223886 (and also added to the formula section here) for k=3, j=1 and n>=2. - Alexander R. Povolotsky, Apr 18 2013
For n>=3 and k>=2 ((binomial(k*p,p)-k)/p^3)/k is an integer. For k=2 this is the Wolstenholme quotient (A034602) and for k=3 the current sequence. - Peter Luschny, Feb 09 2016

Cf. A005809, A034602, A223886.

WQ := proc(n,k) local p; p := ithprime(n); ((binomial(k*p,p)-k)/p^3)/k end:
seq(WQ(n,3), n=2..14); # Peter Luschny, Feb 09 2016

a(n)=my(p=prime(n)); (binomial(3*p,p+1)*(p+1)/(2*p)-3)/(3*p^3) \\ Charles R Greathouse IV, Mar 26 2013

a(n) = (binomial(j*k*prime(n), j*prime(n)) - binomial(k*j, j))/(k*prime(n)^3) for k=3, j=1 and n>=2 (conjectured). - Alexander R. Povolotsky, Apr 18 2013

A223886 Numbers (binomial(jkprime(n), jprime(n)) - binomial(kj, j))/(k*prime(n)^3) for k=4, j=3 and n>=2.

Original entry on oeis.org

871695, 106388178385, 23847838715080655, 2591856748839247419391825095, 1049841259371423735816549330164685, 216822871259048720341882553570648156557191421

Offset: 2

Alexander R. Povolotsky, Mar 28 2013

nonn

This sequence (together with already present in the OEIS A034602 and A217772) is based on Gary Detlefs' conjecture, which he disclosed to me in a private communication on 3/29/13 and recently he gave me permission to make it public. Specifically he wrote to me the following: "I have a conjecture which is broader than the one I submitted, having to do with binomial(k*n,n) mod n^3. It appears that binomial(j*k*n,j*n) mod n^3 will be binomial(k*j,j) for n sufficiently large."
In effect above conjecture further extends Wolstenholme's and Ljunggren's ideas and could also be expressed as follows: starting with some specific (for any given unchanged values of integers k>0 and j>0) sufficiently large value of n=N and further on for n>N it is true that (binomial(j*k*prime(n), j*prime(n)) - binomial(k*j, j))/k/(prime(n))^3 = m(j, k, n ), where m(j, k, n ) are integer values.
Note that the values of A034602 are replicated by above general formula for k=2, j=1 and n>=3 and the values of A217772 are replicated by the same formula for k=3, j=1 and n>=2.

R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv preprint arXiv:1111.3057 [math.NT], 2011.

Cf. A034602, A217772.

A333593 a(n) = Sum_{k = 0..n} (-1)^(n + k)*binomial(n + k - 1, k)^2.

Original entry on oeis.org

1, 0, 6, 72, 910, 12000, 163086, 2266544, 32043726, 459167040, 6651400756, 97214919648, 1431514320886, 21213380196736, 316072831033350, 4731683468079072, 71128104013487310, 1073134004384407680, 16243463355081280080, 246585461357885877920

Offset: 0

Peter Bala, Mar 27 2020

It is known that Sum_{k = 0..2*n} (-1)^(n+k)*C(2*n,k)^2 = C(2*n,n). Here, we consider by analogy Sum_{k = 0..n} (-1)^(n+k)*C(-n,k)^2, where C(-n,k) = (-1)^k * C(n+k-1,k) for integer n and nonnegative integer k.
The sequence b(n) = C(2*n,n) of central binomial coefficients satisfies the supercongruences b(n*p^k) = b(n*p^(k-1)) ( mod p^(3*k) ) for all prime p >= 5 and any positive integers n and k -  see Mestrovic. We conjecture that the present sequence also satisfies these congruences. Some examples of the congruences are given below. For a proof of the particular case of these congruences, a(p) == 0 ( mod p^3 ) for prime p >= 5, see the Bala link. The sequence a(p)/(2*p^3) for prime p >= 5 begins [48, 304, 36519504, 4827806144, 109213719151680, 17975321574419440, ...]. Cf. A034602.
More generally, calculation suggests that for positive integer A and integer B, the sequence a(A,B;n) := Sum_{k = 0..A*n} (-1)^(n+k)* C(-B*n,k)^2 = Sum_{k = 0..A*n} (-1)^(n+k)*C(B*n+k-1,k)^2 may also satisfy the above congruences for all prime p >= 5.

			Examples of congruences:
a(11)  = 97214919648 = (2^5)*3*(7^2)*(11^3)*15527 == 0 ( mod 11^3 ).
a(2*7) - a(2) = 316072831033350 - 6 = (2^13)*3*(7^3)*11*691*4933 == 0 ( mod 7^3 ).
a(5^2) - a(5) = 3164395891098711251676512000 - 12000 = (2^5)*(5^6)*29* 124891891*1747384859327 == 0 ( mod 5^6 ).

Seiichi Manyama, Table of n, a(n) for n = 0..500
Peter Bala, Notes on A333593
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Cf. A000984, A034602, A333592.

seq( add( (-1)^(n+k)*binomial(n+k-1,k)^2, k = 0..n ), n = 0..25);

Table[Binomial[2*n-1, n]^2 * HypergeometricPFQ[{1, -n, -n}, {1 - 2 n, 1 - 2 n}, -1], {n, 0, 20}] (* Vaclav Kotesovec, Mar 28 2020 *)

a(n) = sum(k=0, n, (-1)^(n+k)*binomial(n+k-1, k)^2); \\ Michel Marcus, Mar 29 2020

a(n) ~ 2^(4*n) / (5*Pi*n). - Vaclav Kotesovec, Mar 28 2020

A127045 Primes p such that denominator of Sum_{k=1..p-1} 1/k^9 is a 9th power.

Original entry on oeis.org

2, 3, 5, 11, 13, 17, 29, 31, 37, 97, 127, 131, 251, 257, 263, 293, 431, 433, 439, 443, 449, 457, 461, 463, 467, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3797, 3803, 3821, 3823, 3833, 3907, 3911, 3917

Offset: 1

Artur Jasinski, Jan 04 2007

nonn

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

Cf. A061002, A034602, A127029, A127042, A127043, A127044, A127046, A127047, A127048, A127049, A127051.

d[n_] := Module[{}, su = 0; a = {}; For[i = 1, i <= n, i++, su = su + 1/ i^9; If[PrimeQ[i + 1], If[IntegerQ[(Denominator[su])^(1/9)], AppendTo[a, i + 1]]]]; a] d[2000]
Select[Flatten[Position[Denominator[Accumulate[1/Range[4000]^9]],?(IntegerQ[ Surd[ #,9]]&)]]+1,PrimeQ] (* _Harvey P. Dale, Aug 06 2022 *)

A127052 Primes p such that denominator of Sum_{k=1..p-1} 1/k^8 is an eighth power.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 41, 53, 67, 71, 73, 97, 101, 127, 131, 197, 199, 211, 251, 367, 373, 379, 773, 787, 797, 809, 811, 1373, 1433, 1439, 2027, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A061002, A034602, A127029, A127042, A127046, A127047, A127048, A127051.

d[n_] := Module[{}, su = 0; a = {}; For[i = 1, i <= n, i++, su = su + 1/ i^8; If[PrimeQ[i + 1], If[IntegerQ[(Denominator[su])^(1/8)], AppendTo[a, i + 1]]]]; a]; d[2000]

A127062 Primes p such that denominator of Sum_{k=1..p-1} 1/k^2 is a square and denominator Sum_{k=1..p-1} 1/k^3 is a cube and denominator Sum_{k=1..p-1} 1/k^4 is a fourth power.

Original entry on oeis.org

2, 3, 5, 17, 29, 31, 97, 439, 443, 449, 457, 461, 463, 1009, 1013, 24391, 24407, 24413, 24419, 24421, 24439, 24443, 24469, 24473, 24481, 117659, 117671, 117673, 117679, 117701, 117703, 117709, 117721, 117727, 117731, 117751, 117757, 117763, 117773

Offset: 1

Artur Jasinski, Jan 04 2007

nonn

Subsequence of A127061. - Max Alekseyev, Feb 08 2007

Cf. A061002, A034602, A127029, A127042, A127043, A127044, A127046, A127047, A127048, A127049, A127051, A127061.

pdenQ[n_]:=Module[{c=Denominator[Table[Sum[1/k^i,{k,n-1}],{i,2,4}]]}, AllTrue[{ Surd[c[[1]],2], Surd[c[[2]],3],Surd[c[[3]],4]},IntegerQ]]; Select[Prime[Range[12000]],pdenQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jun 06 2015 *)

lista(nn) = {forprime(p = 2, nn, if (issquare(denominator(sum(k=1, p-1, 1/k^2))) && ispower(denominator(sum(k=1, p-1, 1/k^3)),3) && ispower(denominator(sum(k=1, p-1, 1/k^4)),4), print1(p, ", ")););} \\ Michel Marcus, Nov 05 2013

Intersection of A127042, A127046 and A127047. - Michel Marcus, Nov 05 2013

More terms from Max Alekseyev, Feb 08 2007

A127063 Primes p such that denominator of Sum_{k=1..p-1} 1/k^2 is a square and denominator Sum_{k=1..p-1} 1/k^3 is a cube and denominator Sum_{k=1..p-1} 1/k^4 is a fourth power and denominator Sum_{k=1..p-1} 1/k^5 is a fifth power.

Original entry on oeis.org

2, 3, 5, 17, 439, 443, 16400183, 16400191, 16400201, 16400203, 16400221, 16400231, 16400233, 16400269, 16400273, 16400299, 16400309, 16400317, 16400347, 16400383, 16400387, 16400389, 16400411, 16400413, 16400429, 16400431

Offset: 1

Artur Jasinski, Jan 04 2007

nonn

Subsequence of A127062 and of A127061. - Max Alekseyev, Feb 08 2007

Cf. A061002, A034602, A127029, A127042, A127043, A127044, A127046, A127047, A127048, A127049, A127051, A127061, A127062.

More terms from Max Alekseyev, Feb 08 2007

cp's OEIS Frontend

A222207 Morley quotients: (2^(2*p-2) - (-1)^((p-1)/2)*binomial(p-1,(p-1)/2)) / p^3, where p = prime(n) and n >= 3.

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A244919 For odd prime p, largest k such that binomial(2p-1, p-1) is congruent to 1 modulo p^k.

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A125551 As p runs through primes >= 5, sequence gives { numerator of Sum_{k=1..p-1} 1/k^2 } / p.

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A217772 a(n) = ((p+1)*(3p)!/((2p-1)!*(p+1)!*2p) - 3)/(3p^3), where p is the n-th prime.

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A223886 Numbers (binomial(j*k*prime(n), j*prime(n)) - binomial(k*j, j))/(k*prime(n)^3) for k=4, j=3 and n>=2.

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A333593 a(n) = Sum_{k = 0..n} (-1)^(n + k)*binomial(n + k - 1, k)^2.

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A127045 Primes p such that denominator of Sum_{k=1..p-1} 1/k^9 is a 9th power.

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Mathematica

A127052 Primes p such that denominator of Sum_{k=1..p-1} 1/k^8 is an eighth power.

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Mathematica

A127062 Primes p such that denominator of Sum_{k=1..p-1} 1/k^2 is a square and denominator Sum_{k=1..p-1} 1/k^3 is a cube and denominator Sum_{k=1..p-1} 1/k^4 is a fourth power.

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A222207 Morley quotients: (2^(2p-2) - (-1)^((p-1)/2)binomial(p-1,(p-1)/2)) / p^3, where p = prime(n) and n >= 3.

A217772 a(n) = ((p+1)(3p)!/((2p-1)!(p+1)!*2p) - 3)/(3p^3), where p is the n-th prime.

A223886 Numbers (binomial(jkprime(n), jprime(n)) - binomial(kj, j))/(k*prime(n)^3) for k=4, j=3 and n>=2.