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.

A097380 Numbers m such that 1+CubeFreeKernel(m) is prime.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 22, 24, 28, 30, 32, 36, 42, 46, 48, 52, 54, 56, 58, 60, 64, 66, 70, 72, 78, 82, 96, 100, 102, 104, 106, 108, 112, 120, 126, 128, 130, 138, 144, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 200, 208, 210, 216, 222, 224, 226
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 11 2004

Keywords

Examples

			m = 216 = (2*3)^3 -> A097377(216) = 1+(2*3)^2 = 37 = A000040(12), therefore 216 is a term.

Links

Crossrefs

Cf. A007948, A089189, A097377, A097379, A097381.

Programs

  • Mathematica
    f[p_, e_] := p^Min[e, 2]; s[1] = 2; s[n_] := 1 + Times @@ f @@@ FactorInteger[n]; Select[Range[230], PrimeQ[s[#]] &] (* Amiram Eldar, Feb 01 2024 *)
  • PARI
    is(n) = {my(f = factor(n)); isprime(1 + prod(i = 1, #f~, f[i, 1]^min(f[i, 2], 2)));} \\ Amiram Eldar, Feb 01 2024

Formula

A097377(a(n)) = A007948(a(n))+1 is prime.

A097381 Numbers m such that 1+SquareFreeKernel(m)*CubeFreeKernel(m) is prime.

Original entry on oeis.org

1, 2, 6, 10, 12, 14, 18, 24, 26, 48, 54, 60, 66, 74, 84, 94, 96, 98, 110, 120, 130, 132, 134, 146, 162, 168, 170, 192, 204, 206, 210, 230, 234, 240, 264, 300, 314, 326, 336, 372, 384, 386, 406, 408, 430, 466, 470, 474, 480, 486, 490, 528, 570, 588, 600, 634, 646
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 11 2004

Keywords

Links

Crossrefs

Cf. A007947, A007948, A097379, A097380.

Programs

  • Mathematica
    f[p_, e_] := p^(1 + Min[e, 2]); s[1] = 2; s[n_] := 1 + Times @@ f @@@ FactorInteger[n]; Select[Range[650], PrimeQ[s[#]] &] (* Amiram Eldar, Feb 01 2024 *)
  • PARI
    is(n) = {my(f = factor(n)); isprime(1 + prod(i = 1, #f~, f[i, 1]^(1 + min(f[i, 2], 2))));} \\ Amiram Eldar, Feb 01 2024

Formula

A097378(a(n)) is prime.

A359213 Numbers k such that rad(k) - 1 is prime.

Original entry on oeis.org

3, 6, 9, 12, 14, 18, 24, 27, 28, 30, 36, 38, 42, 48, 54, 56, 60, 62, 72, 74, 76, 81, 84, 90, 96, 98, 102, 108, 110, 112, 114, 120, 124, 126, 138, 144, 148, 150, 152, 158, 162, 168, 174, 180, 182, 192, 194, 196, 204, 216, 220, 224, 228, 230, 240, 243, 248, 252
Offset: 1

Views

Author

Arsen Vardanyan, Dec 21 2022

Keywords

Examples

			rad(60) - 1 = 2*3*5 - 1 = 29, so 60 is a term.

Links

Crossrefs

Cf. A007947, A097379.

Programs

  • Mathematica
    Select[Range[250], PrimeQ[Times @@ FactorInteger[#][[;; , 1]] - 1] &] (* Amiram Eldar, Dec 21 2022 *)
  • PARI
    isok(k) = isprime(factorback(factor(k)[, 1]) - 1); \\ Michel Marcus, Dec 22 2022

A374593 Numbers k such that k - rad(k) + 1 is prime, where rad(k) is the radical A007947(k).

Original entry on oeis.org

4, 8, 9, 12, 18, 20, 24, 32, 36, 40, 44, 45, 48, 49, 50, 56, 60, 63, 72, 75, 80, 81, 84, 88, 90, 92, 99, 104, 108, 116, 117, 128, 132, 136, 140, 144, 147, 153, 156, 160, 162, 164, 168, 169, 180, 184, 200, 204, 207, 212, 216, 224, 225, 234, 240, 243, 245, 250
Offset: 1

Views

Author

Arsen Vardanyan, Aug 23 2024

Keywords

Comments

Includes 4*p for p in A005384, 8*p for p in A007693, and 16*p for p in A228857. - Robert Israel, Jun 27 2025

Examples

			12 is a term, because 12 - rad(12) + 1 = 12 - (2*3) + 1 = 12 - 6 + 1 = 7 is prime.

Links

Crossrefs

Cf. A005384, A007693, A007947, A097379, A228857, A359213.

Programs

  • Maple
    rad:= n -> convert(numtheory:-factorset(n),`*`):
select(k -> isprime(k - rad(k)+1), [$1..1000]); # Robert Israel, Jun 27 2025
  • Mathematica
    rad[n_] := Times @@ (First@# & /@ FactorInteger@ n);Select[Range[250],PrimeQ[#-rad[#]+1]&] (* James C. McMahon, Sep 27 2024 *)
  • PARI
    isok(k) = isprime(k - (factorback(factor(k)[, 1])) + 1);

A375777 Numbers k such that k - rad(k) - 1 is prime, where rad(k) is A007947(k).

Original entry on oeis.org

8, 9, 12, 16, 18, 24, 25, 27, 28, 32, 36, 40, 45, 48, 49, 54, 56, 60, 63, 64, 75, 76, 84, 90, 96, 98, 100, 108, 112, 120, 121, 124, 126, 136, 144, 148, 152, 153, 160, 171, 175, 180, 184, 189, 196, 198, 204, 207, 208, 220, 228, 232, 243, 250, 261, 264, 270, 276
Offset: 1

Views

Author

Arsen Vardanyan, Aug 27 2024

Keywords

Examples

			12 is a term, because 12 - rad(12) - 1 = 12 - (2 * 3) - 1 = 12 - 6 - 1 = 5 is prime.

Crossrefs

Cf. A007947, A097379, A359213, A374593.

Programs

  • Mathematica
    Select[Range[300], PrimeQ[# - Times @@ FactorInteger[#][[;; , 1]] - 1] &] (* Amiram Eldar, Aug 27 2024 *)
  • PARI
    isok(k) = isprime(k - (vecprod(factor(k)[, 1])) - 1);
Showing 1-5 of 5 results.

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

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