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.

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A231814 Squarefree numbers (from A005117) with prime divisors in a 2p-1 progression.

Original entry on oeis.org

6, 15, 30, 91, 703, 1891, 2701, 12403, 18721, 38503, 49141, 51319, 79003, 88831, 104653, 146611, 188191, 218791, 226801, 269011, 286903, 385003, 497503, 597871, 665281, 721801, 736291, 765703, 873181, 954271, 1056331, 1314631, 1373653, 1537381, 1755001, 1869211
Offset: 1

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Author

Jaroslav Krizek, Nov 13 2013

Keywords

Comments

Squarefree numbers with k >= 2 prime factors of the form p_1 * p_2 * ... * p_k, where p_1 < p_2 < ... < p_k = primes with p_k = 2 * p_(k-1) - 1.
Each of these numbers is divisible by the arithmetic mean of its proper divisors.
Supersequence of A129521 (numbers of the form p*q, p and q prime with q=2*p-1; see A005382 and A005383).

Examples

			51319 = 19*37*73 where 37 = 2*19 - 1, 73 = 2*37 - 1.

Links

Crossrefs

Cf. A000040, A005117, A005382, A005383, A129521, A231815, A231816.
Cf. A057330 (first prime for such numbers that has n factors).

Programs

  • Maple
    N:= 10^7: # for terms <= N
p:= 1: S:= NULL: count:= 0:
do
  p:= nextprime(p);
  if p*(2*p-1) > N then break fi;
  q:= p; x:= p;
  do
    q:= 2*q-1;
    if not isprime(q) then break fi;
    x:= x*q;
    if x > N then break fi;
    S:= S,x; count:= count+1;
  od;
od:
sort([S]); # Robert Israel, Mar 24 2023
  • Mathematica
    geomQ[lst_] := Module[{x = lst - 1}, x = x/x[[1]]; Log[2, x] + 1 == Range[Length[x]]]; Select[Range[2, 1000000], ! PrimeQ[#] && SquareFreeQ[#] && geomQ[Transpose[FactorInteger[#]][[1]]] &] (* T. D. Noe, Nov 14 2013 *)

A231815 Squarefree numbers (A005117) of the form p*q*r with prime factors p, q, r with q = 2*p-1 and r = 2*q-1.

Original entry on oeis.org

30, 51319, 3882139, 289022911, 674910259, 991523479, 1893583519, 4550912389, 9761467669, 16721570539, 28685399311, 72886214809, 77372307511, 82720376839, 98685849571, 173850108931, 220038912319, 229352039821, 240313142749, 257401051861, 428178002569
Offset: 1

Views

Author

Jaroslav Krizek, Nov 13 2013

Keywords

Comments

Squarefree numbers of the form p*q*r, where p < q < r = primes with q = 2*p - 1 and r = 2*q - 1; that is, r = 4*p - 3.
These numbers are divisible by the arithmetic mean of their proper divisors.

Examples

			3882139 = 79*157*313; 157 = 2*79 - 1; 313 = 2*157 - 1.

Links

Crossrefs

Cf. A005117, A057326, A000040, A231814, A231816.
Cf. A057326 (first member of a prime triple in a 2p-1 progression).

Programs

  • Mathematica
    t = {}; p = 1; Do[While[p = NextPrime[p]; ! (PrimeQ[p2 = 2 p - 1] && PrimeQ[p3 = 2 p2 - 1])]; AppendTo[t, p*p2*p3], {30}]; t (* T. D. Noe, Nov 15 2013 *)
3#-10#^2+8#^3&/@Select[Prime[Range[600]],AllTrue[{2#-1,4#-3},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 02 2016 *)
Showing 1-2 of 2 results.

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