cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A328497 Terms of A082180 that are not squares or cubes of primes.

Original entry on oeis.org

418, 27173, 2001341, 16024189487
Offset: 1

Views

Author

Felix Fröhlich, Oct 17 2019

Keywords

Comments

Complement of A168363 in A082180.
Are there any other terms like 418 that are not in A228562?
No other terms below 2*10^10. - Max Alekseyev, Dec 15 2023

Crossrefs

The odd terms form A228562.
Cf. A082180, A168363.

Programs

  • Mathematica
    Select[Range[30000], CompositeQ[#] && Not[IntegerQ[Sqrt[#]]] && Not[IntegerQ[#^(1/3)]] && Mod[Binomial[2*#, #], #] == 2 &] (* Vaclav Kotesovec, Oct 17 2019 *)
  • PARI
    forcomposite(c=1, , if(!setintersect(Set(isprimepower(c)), [2, 3]), if(Mod(binomial(2*c, c), c)==2, print1(c, ", "))))

Extensions

a(4) from Max Alekseyev, Dec 15 2023

A246131 Duplicate of A082180.

Original entry on oeis.org

4, 9, 25, 49, 121, 125, 169, 289, 343, 361, 418, 529, 841, 961, 1331, 1369, 1681, 1849, 2197, 2209, 2809, 3481, 3721, 4489, 4913, 5041, 5329, 6241, 6859, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24389, 24649, 26569, 27173
Offset: 1

Views

Author

Stanislav Sykora, Aug 16 2014

Keywords

Programs

  • Maple
    select(n -> not isprime(n) and coeff(Power(1+x,2*n) mod n, x, n) = 2, [$4 .. 10000]); # Robert Israel, Sep 22 2014
  • PARI
    a(file,nmax)={my(n=0,p=1);
  while(1,p++;if(((binomial(2*p,p)-2)%p)==0,
    if(!isprime(p),n++;write(file,n," ",p);if(n==nmax,break)));
  );}

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

Original entry on oeis.org

283686649, 4514260853041
Offset: 1

Views

Author

Jonathan Sondow, Jan 25 2016

Keywords

Comments

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

Examples

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

Links

Crossrefs

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.

A084699 Composite integers j such that binomial(2*j,j) == 2^j (mod j).

Original entry on oeis.org

12, 30, 56, 424, 992, 16256, 58288, 119984, 356992, 1194649, 9973504, 12327121, 13141696, 22891184, 67100672, 233850649
Offset: 1

Views

Author

Benoit Cloitre, Oct 15 2003

Keywords

Comments

If p is prime, binomial(2*p,p) == 2^p (mod p).
a(17) > 10^9.
From Gabriel Guedes and Ricardo Machado, Nov 16 2023: (Start)
Theorem. Let j = (2^k)*p, where p is an odd prime and k is in N; then binomial(2*j,j) == 2^j (mod j) if and only if p satisfies the following conditions:
a) p divides binomial(2^(k+1),2^k) - 2^(2^k);
b) p has at least k 1's in its binary expansion.
Theorem. If m is an even perfect number then j = 2m satisfies the congruence binomial(2*j,j) == 2^j (mod j). See A000396.
Theorem. Let j = p^2 with p a prime number. Then p is a Wieferich prime if and only if binomial(2*j,j) == 2^j (mod j). See A001220. (End)
Contains 17179738112 and 274877382656 (from Guedes-Machado paper). - Michael De Vlieger, Nov 22 2023
Contains 3386741824, 750984028672, 33029195197184, 1145067923695616, 422612863956511744. - Ricardo Machado, Nov 23 2023
Contains 84385517065596416, 62648180117928433664, 273984397779878971648, 36506097537257040703232. - Max Alekseyev, Dec 07 2023

Links

Crossrefs

Contains A139256 as a subsequence.
Cf. A015910, A059288, A080469, A082180, A109760, A109769, A228562, A328497.

Programs

  • PARI
    lista(nn) = {forcomposite(n=1, nn, if (binomod(2*n, n, n) == Mod(2, n)^n, print1(n, ", ")));} \\ Michel Marcus, Dec 06 2013 and Dec 03 2023

Extensions

More terms from David Wasserman, Jan 03 2005
a(11)-a(16) from Max Alekseyev, Aug 05 2011

A331343 a(n) = lcm(1,2,...,n) * Sum_{k=1..n} (2^(k-1) - 1) / k.

Original entry on oeis.org

0, 1, 9, 39, 375, 685, 8575, 30485, 162855, 291627, 5785857, 10514427, 250200951, 461037291, 854622483, 3185234481, 101381371377, 190598779657, 6833215763803, 12935721409039, 24559552771039, 46750514134519, 2051664357879617, 3923102768811707, 37581323659852375
Offset: 1

Views

Author

Amiram Eldar and Thomas Ordowski, Jan 14 2020

Keywords

Comments

By Wolstenholme's theorem, if p > 3 is a prime, then p^3 | a(p).
Conjecture: for n > 3, if n^3 | a(n), then n is prime. If so, there are no such pseudoprimes.
Problem: are there weak pseudoprimes m such that m^2 | a(m)? None up to 5*10^4.
Composite numbers m such that m | a(m) are 9, 25, 49, 99, 121, 125, 169, 221, 289, 343, 357, 361, 399, 529, 665, 841, 961, 1331, 1369, 1443, 1681, 1849, 2183, ... Cf. A082180.
Prime numbers p such that p^4 | a(p) are probably only the Wolstenholme primes A088164.

Links

Crossrefs

Cf. A003418, A025529, A082180, A088164, A330718, A330719.

Programs

  • Magma
    [Lcm([1..n])*&+[(2^(k-1)-1)/k:k in [1..n]]:n in [1..25]]; // Marius A. Burtea, Jan 14 2020
  • Mathematica
    a[n_] := LCM @@ Range[n] * Sum[(2^(k-1) - 1) / k, {k, 1, n}]; Array[a, 25]
  • PARI
    a(n) = lcm([1..n])*sum(k=1, n, (2^(k-1) - 1) / k); \\ Michel Marcus, Jan 14 2020

Formula

a(n) = A003418(n) * A330718(n) / A330719(n).
Showing 1-5 of 5 results.

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