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.

A107746 Numbers k such that the least prime factor of 6*k+1 > the least prime factor of 6*k-1, A107744(k) > A107745(k).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 16, 17, 18, 20, 21, 23, 25, 26, 27, 30, 31, 32, 33, 35, 36, 37, 38, 40, 41, 45, 46, 47, 48, 51, 52, 55, 56, 58, 61, 62, 63, 66, 68, 70, 71, 72, 73, 76, 77, 81, 83, 86, 87, 88, 90, 91, 95, 96, 97, 100, 101, 102, 103, 105, 106, 107, 110, 111
Offset: 1

Views

Author

Zak Seidov, May 23 2005

Keywords

Comments

Complement of A107747 in the positive integers.
Contains A016861. - Robert Israel, Oct 23 2015

Links

Crossrefs

Cf. A016861, A107744, A107745, A107747.

Programs

  • Maple
    lpf:= n -> min(numtheory:-factorset(n)):
select(n -> lpf(6*n+1) > lpf(6*n-1), [$1..1000]); # Robert Israel, Oct 23 2015
  • Mathematica
    Select[Range@ 111, FactorInteger[6 # + 1][[1, 1]] > FactorInteger[6 # - 1][[1, 1]] &] (* Michael De Vlieger, Oct 23 2015 *)
  • PARI
    isok(n) = vecmin(factor(6*n+1)[,1]) > vecmin(factor(6*n-1)[,1]); \\ Michel Marcus, Feb 04 2014

Extensions

Comments corrected and (at the suggestion of Michel Marcus) moved to Crossrefs by Jason Kimberley, Oct 23 2015

A107747 Numbers k such that the least prime factor of 6*k+1 < the least prime factor of 6*k-1, A107744(k) < A107745(k).

Original entry on oeis.org

4, 8, 9, 14, 15, 19, 22, 24, 28, 29, 34, 39, 42, 43, 44, 49, 50, 53, 54, 57, 59, 60, 64, 65, 67, 69, 74, 75, 78, 79, 80, 82, 84, 85, 89, 92, 93, 94, 98, 99, 104, 108, 109, 113, 114, 117, 119, 120, 124, 127, 129, 130, 133, 134, 139, 140, 144, 148, 149, 150, 152, 154
Offset: 1

Views

Author

Zak Seidov, May 23 2005

Keywords

Links

Crossrefs

Cf. A107744 (smallest prime factor of 6*n+1), A107745 (smallest prime factor of 6*n-1), A107746 (values of k such that A107744(k) > A107745(k)).

Programs

  • Mathematica
    lpfQ[n_]:=Module[{c=6n},FactorInteger[c+1][[1,1]] < FactorInteger [c-1][[1,1]]]; Select[Range[200],lpfQ] (* Harvey P. Dale, Apr 18 2011 *)
  • PARI
    isok(n) = vecmin(factor(6*n+1)[, 1]) < vecmin(factor(6*n-1)[, 1]);
for(n=1, 200,if(isok(n)==1, print1(n", "))) \\ Altug Alkan, Oct 23 2015

Extensions

Comments corrected and (at the suggestion of Michel Marcus) moved to Crossrefs by Jason Kimberley, Oct 23 2015

A107744 Smallest prime factor of 6*n+1.

Original entry on oeis.org

7, 13, 19, 5, 31, 37, 43, 7, 5, 61, 67, 73, 79, 5, 7, 97, 103, 109, 5, 11, 127, 7, 139, 5, 151, 157, 163, 13, 5, 181, 11, 193, 199, 5, 211, 7, 223, 229, 5, 241, 13, 11, 7, 5, 271, 277, 283, 17, 5, 7, 307, 313, 11, 5, 331, 337, 7, 349, 5, 19, 367, 373, 379, 5, 17, 397, 13, 409
Offset: 1

Views

Author

Zak Seidov, May 23 2005

Keywords

Comments

If 6*n+1 is prime, a(n) = 6*n+1.

Links

Crossrefs

Cf. A016921, A020639.
Cf. A107745 (smallest prime factor of 6*n-1), A107746 (values of k such that A107744(k) > A107745(k)), A107747 (values of k such that A107744(k) < A107745(k)).

Programs

  • Magma
    [Min(PrimeFactors(6*n+1)):n in[1..68]]; // Jason Kimberley, Oct 28 2015
  • Mathematica
    A107744[n_]:= FactorInteger[6 n + 1][[1, 1]]
  • PARI
    vector(100, n, vecmin(factor(6*n+1)[, 1])) \\ Altug Alkan, Oct 23 2015

Formula

a(n) = A020639(A016921(n)). - Amiram Eldar, Nov 03 2024

Extensions

Comments corrected and (at the suggestion of Michel Marcus) moved to Crossrefs by Jason Kimberley, Oct 23 2015

A111863 a(n) is the smallest prime factor of 6*n-1 that is congruent to 5 modulo 6.

Original entry on oeis.org

5, 11, 17, 23, 29, 5, 41, 47, 53, 59, 5, 71, 11, 83, 89, 5, 101, 107, 113, 17, 5, 131, 137, 11, 149, 5, 23, 167, 173, 179, 5, 191, 197, 29, 11, 5, 17, 227, 233, 239, 5, 251, 257, 263, 269, 5, 281, 41, 293, 23, 5, 311, 317, 17, 47, 5, 11, 347, 353, 359, 5, 53, 29, 383, 389, 5
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 20 2005

Keywords

Comments

From Robert Israel, Jan 18 2023: (Start)
a(n) = 5 if n == 1 (mod 5).
a(n) = 6*n - 1 if n is in A024898. (End)

Examples

			For n = 13, 6*n - 1 = 77 = 7*11; 7 == 1 (mod 6), but 11 == 5 (mod 6), so a(13) = 11.

References

  • G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part Eight, Chap. 2, Section 2, Problem 96.

Links

Crossrefs

Cf. A007528, A016969, A024898, A107745.

Programs

  • Maple
    f:= n -> min(select(p -> p mod 6 = 5, numtheory:-factorset(6*n-1))):
map(f, [$1..100]); # Robert Israel, Jan 18 2023
  • PARI
    for(k=1,60,my(f=factor(6*k-1)[,1]);for(j=1,#f,if(f[j]%6==5,print1(f[j],", ");break))) \\ Hugo Pfoertner, Dec 25 2019

A125255 Smallest prime divisor of 4n-1.

Original entry on oeis.org

3, 7, 11, 3, 19, 23, 3, 31, 5, 3, 43, 47, 3, 5, 59, 3, 67, 71, 3, 79, 83, 3, 7, 5, 3, 103, 107, 3, 5, 7, 3, 127, 131, 3, 139, 11, 3, 151, 5, 3, 163, 167, 3, 5, 179, 3, 11, 191, 3, 199, 7, 3, 211, 5, 3, 223, 227, 3, 5, 239, 3, 13, 251, 3, 7, 263, 3, 271, 5, 3, 283, 7, 3, 5, 13, 3, 307
Offset: 1

Views

Author

Nick Hobson, Nov 26 2006

Keywords

Comments

All divisors of 4n-1 are odd.

Examples

			The prime divisors of 4*9 - 1 = 35 are 5 and 7, so a(9) = 5.

Links

Crossrefs

Cf. A004767, A107745, A125254.

Programs

  • Mathematica
    Table[FactorInteger[4n-1][[1,1]],{n,77}] (* James C. McMahon, Dec 16 2024 *)
  • PARI
    vector(77, n, factor(4*n-1)[1,1])

Formula

a(3k+1) = 3, k = 0,1,2,... a(5k+4) = 5 if k is not a multiple of 3; k = 1,2,4,5,7,8,... - Alexander Adamchuk, Nov 28 2006
a(n) = A020639(4n-1). - R. J. Mathar, Jan 23 2007
Showing 1-5 of 5 results.

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