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 1-3 of 3 results.

A074175 Duplicate of A074174.

Original entry on oeis.org

44, 18, 172, 171, 45, 12, 45, 12, 236, 18, 52, 63, 50, 261, 524, 12, 28, 45, 44, 637, 404
Offset: 1

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Author

Keywords

A074172 Smaller of two consecutive numbers of the form p^2*q where p and q are distinct primes.

Original entry on oeis.org

44, 75, 98, 116, 147, 171, 244, 332, 387, 507, 548, 603, 604, 724, 844, 908, 931, 963, 1075, 1083, 1251, 1324, 1412, 1467, 1556, 1587, 1675, 1772, 2523, 2524, 2636, 2644, 2763, 3283, 3356, 3411, 3508, 3788, 3987, 4075, 4203, 4204, 4418, 4491, 4804, 4868
Offset: 1

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Author

Amarnath Murthy, Aug 30 2002

Keywords

Comments

From Robert Israel, Dec 06 2018: (Start)
There are four forms of terms, for odd primes p,q,r:
4*p where 4*p+1 = q^2*r, r == 1 (mod 4)
2*p^2 where 2*p^2+1 = q^2*r, r == 3 (mod 4)
p^2*q where p^2*q+1 = 2*r^2, q == 1 (mod 4)
p^2*q where p^2*q+1 = 4*r, q == 3 (mod 4).
Are there infinitely many terms of each type?
(End)

Examples

			44 is a member as 44 = 2^2*11 and 45 = 3^2*5.
		

Crossrefs

Programs

  • Maple
    filter:= proc(n) local F;
    F:= map(t -> t[2],ifactors(n)[2]);
    F = [2,1] or F = [1,2]
    end proc:
    A054753:= select(filter, {$1..10000}):
    sort(convert(A054753 intersect map(`-`,A054753,1),list)); # Robert Israel, Dec 06 2018
  • Mathematica
    lst={}; Do[f1=FactorInteger[n]; If[Sort[Transpose[f1][[2]]]=={1, 2}, f2=FactorInteger[n+1]; If[Sort[Transpose[f2][[2]]]=={1, 2}, AppendTo[lst, n]]], {n, 3, 10000}]; lst
  • PARI
    isok1(n) = vecsort(factor(n)[,2]) == [1,2]~;
    isok(n) = isok1(n) && isok1(n+1); \\ Michel Marcus, Sep 20 2017

Extensions

More terms from T. D. Noe, Oct 04 2004
Name clarified by Sean A. Irvine, Jan 13 2025

A074173 Numbers n such that n and n+2 are of the form p^2*q where p and q are distinct primes.

Original entry on oeis.org

18, 50, 242, 423, 475, 603, 637, 722, 845, 925, 1682, 1773, 2007, 2523, 2525, 2527, 3175, 3177, 4203, 4475, 4525, 4923, 5823, 6725, 6811, 6962, 7299, 7442, 7675, 8425, 8957, 8973, 9457, 9925, 10051, 10082, 10467, 11673, 11709, 12427, 12482, 12591
Offset: 1

Views

Author

Amarnath Murthy, Aug 30 2002

Keywords

Examples

			18 is a member as 18 = 3^2*2 and 20 = 2^2*5.
		

Crossrefs

Programs

  • Mathematica
    lst={}; Do[f1=FactorInteger[n]; If[Sort[Transpose[f1][[2]]]=={1, 2}, f2=FactorInteger[n+2]; If[Sort[Transpose[f2][[2]]]=={1, 2}, AppendTo[lst, n]]], {n, 3, 10000}]; lst

Formula

Even terms in sequence are 2*A048161(n)^2. - Ray Chandler, Jun 24 2019

Extensions

More terms from T. D. Noe, Oct 04 2004
Showing 1-3 of 3 results.