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.

A097379 Numbers m such that 1+SquareFreeKernel(m) is prime.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 30, 32, 36, 40, 42, 44, 46, 48, 50, 54, 58, 60, 64, 66, 70, 72, 78, 80, 82, 84, 88, 90, 92, 96, 100, 102, 106, 108, 116, 120, 126, 128, 130, 132, 138, 140, 144, 150, 156, 160, 162, 164, 166, 168, 176, 178, 180, 184, 190, 192
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 11 2004

Keywords

Examples

			m = 100 = (2*5)^2 -> A076618(100) = 1+2*5 = 11 = A000040(5), therefore 100 is a term.

Links

Crossrefs

Cf. A007947, A076618, A039787, A097380, A097381, A359213.

Programs

  • Mathematica
    Select[Range[200], PrimeQ[1 + Times @@ FactorInteger[#][[;; , 1]]] &] (* Amiram Eldar, Feb 01 2024 *)
  • PARI
    is(n) = isprime(1 + vecprod(factor(n)[, 1])); \\ Amiram Eldar, Feb 01 2024

Formula

A076618(a(n)) = A007947(a(n))+1 is prime.

A089632 1 + product of prime factors of n is a perfect square.

Original entry on oeis.org

3, 9, 15, 27, 35, 45, 75, 81, 135, 143, 175, 195, 225, 243, 245, 255, 323, 375, 399, 405, 483, 585, 675, 729, 765, 875, 899, 975, 1023, 1125, 1155, 1197, 1215, 1225, 1275, 1295, 1443, 1449, 1573, 1599, 1715, 1755, 1763, 1859, 1875, 2025, 2187, 2295, 2535
Offset: 1

Views

Author

Joseph L. Pe, Jan 04 2004

Keywords

Comments

From Robert Israel, Apr 14 2019: (Start)
Numbers k such that A076618(k) is a square.
All terms are odd.
Squarefree terms are k^2-1 for k in A067874.
(End)

Examples

			The prime factors of 35 are 5 and 7 and 5 * 7 + 1 = 36 is a square; so 35 belongs to the sequence.

Links

Crossrefs

Cf. A089653. A091278 gives squares, A091279 gives square roots.
Cf. A067874, A076618.

Programs

  • Maple
    filter:= n -> issqr(1+convert(numtheory:-factorset(n),`*`)):
select(filter, [$1..10000]); # Robert Israel, Apr 14 2019
  • Mathematica
    ppf[n_] := Apply[Times, Transpose[FactorInteger[n]][[1]]]; Select[Range[2, 10^3], IntegerQ[Sqrt[ppf[ # ] + 1]] &]
  • PARI
    isok(n) =  my(f=factor(n)); issquare(1+prod(k=1, #f~, f[k,1])); \\ Michel Marcus, Apr 15 2019

Extensions

More terms from Ray Chandler, Jan 05 2004

A076619 Least x>1 such that x^d == 1 (mod d) for each divisor d of n, for all nonsquarefree numbers n (cf. A013929).

Original entry on oeis.org

3, 3, 4, 7, 3, 7, 11, 7, 6, 4, 15, 3, 7, 11, 23, 16, 7, 8, 11, 27, 7, 15, 31, 22, 3, 35, 7, 16, 39, 11, 4, 43, 23, 31, 47, 7, 15, 34, 11, 27, 7, 15, 59, 40, 31, 12, 63, 6, 43, 3, 67, 16, 35, 71, 7, 22, 75, 31, 39, 52, 79, 11, 7, 83, 43, 14, 58, 87, 36, 23, 31, 47, 95, 22, 7, 15, 67
Offset: 1

Views

Author

Benoit Cloitre, Oct 22 2002

Keywords

Comments

If n is squarefree (cf. A005117), then the least x>1 such that x^d == 1 (mod d) (for each divisor d of n) equals n+1.

Crossrefs

Cf. A013929, A076333, A076618 (sequence for all integers).

Programs

  • Mathematica
    f[n_] := If[(r = Times @@ FactorInteger[n][[;; , 1]]) < n, r, 0]; Select[f /@ Range[200], # > 0 &] + 1 (* Amiram Eldar, Feb 11 2021 *)
  • PARI
    lista(nn) = {for(n=1, nn, if (!issquarefree(n), print1(A076618(n), ", ");););} \\ Michel Marcus, Jul 13 2013

Formula

a(p^m) = p+1 for p prime and m>1.
a(n) = A076618(A013929(n)). - Michel Marcus, Jul 13 2013
a(n) = A076333(n) + 1. - Amiram Eldar, Feb 11 2021
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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