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 1-4 of 4 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

A107745 Smallest prime factor of 6*n-1.

Original entry on oeis.org

5, 11, 17, 23, 29, 5, 41, 47, 53, 59, 5, 71, 7, 83, 89, 5, 101, 107, 113, 7, 5, 131, 137, 11, 149, 5, 7, 167, 173, 179, 5, 191, 197, 7, 11, 5, 13, 227, 233, 239, 5, 251, 257, 263, 269, 5, 281, 7, 293, 13, 5, 311, 317, 17, 7, 5, 11, 347, 353, 359, 5, 7, 13, 383, 389, 5, 401, 11
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. A020639, A016969, A111863.
Cf. A107744 (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
    a[n_]:= FactorInteger[6*n - 1][[1, 1]]; Array[a, 100]
  • PARI
    vector(100, n, vecmin(factor(6*n-1)[, 1])) \\ Altug Alkan, Oct 23 2015

Formula

a(n) = A020639(6n-1). - R. J. Mathar, Jan 23 2007
5 <= a(n) <= 6n - 1; both bounds are sharp. - Charles R Greathouse IV, Sep 02 2024

Extensions

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

A306289 The smallest prime factor of numbers greater than 1 and coprime to 6.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 5, 29, 31, 5, 37, 41, 43, 47, 7, 53, 5, 59, 61, 5, 67, 71, 73, 7, 79, 83, 5, 89, 7, 5, 97, 101, 103, 107, 109, 113, 5, 7, 11, 5, 127, 131, 7, 137, 139, 11, 5, 149, 151, 5, 157, 7, 163, 167, 13, 173, 5, 179, 181, 5, 11, 191, 193
Offset: 1

Views

Author

Davis Smith, Feb 03 2019

Keywords

Comments

a(n) is the least prime factor of the n-th number that is greater than 1 and congruent to 1 or 5 (mod 6).
a(n) = 5 when n is congruent to {1, 8} (mod 10) (n is a term in A017281, A017365, or A306277). a(n) = 7 when n is congruent to {2, 11} (mod 14) but not {1, 8} (mod 10). a(n) = 11 when n is congruent to {3, 18} (mod 22) but not a case where it equals 5 or 7. a(n) = 13 when n is congruent to {4, 21} (mod 26) (n is a term in A306285) but not a case where it equals 5, 7, or 11. a(n) = 17 when n is congruent to {5, 28} (mod 34) but not a case where it equals 5, 7, 11, or 13. a(n) = 19 when n is congruent to {6, 31} (mod 38) (n is a term in A306331) but not a case where it equals 5, 7, 11, 13, or 17.
Conjecture: This pattern continues indefinitely. a(n) = A007310(m + 1) when n is congruent to {m, A306277(m + 1)} (mod A091999(m + 1)) but not congruent to {k, A306277(k + 1)} (mod A091999(k + 1)), m > k >= 1. The indices of the first appearance of a number in this sequence supports this conjecture in that they are never, for m > 0, congruent to A306277(m + 1) mod A091999(m + 1).

Examples

			a(n) is the least term, other than 0, in n-th row of the array A(m,n), where A(m,n) is A007310(m + 1) when A007310(n + 1) mod A007310(m + 1) is congruent to 0, otherwise 0.
Table begins
  \m  1 2  3  4  5  6  7  8  9  10  11  12  13  14  15  16 ...
  n\
   1| 5 0  0  0  0  0  0  0  0   0   0   0   0   0   0   0 ...
   2| 0 7  0  0  0  0  0  0  0   0   0   0   0   0   0   0 ...
   3| 0 0 11  0  0  0  0  0  0   0   0   0   0   0   0   0 ...
   4| 0 0  0 13  0  0  0  0  0   0   0   0   0   0   0   0 ...
   5| 0 0  0  0 17  0  0  0  0   0   0   0   0   0   0   0 ...
   6| 0 0  0  0  0 19  0  0  0   0   0   0   0   0   0   0 ...
   7| 0 0  0  0  0  0 23  0  0   0   0   0   0   0   0   0 ...
   8| 5 0  0  0  0  0  0 25  0   0   0   0   0   0   0   0 ...
   9| 0 0  0  0  0  0  0  0 29   0   0   0   0   0   0   0 ...
  10| 0 0  0  0  0  0  0  0  0  31   0   0   0   0   0   0 ...
  11| 5 7  0  0  0  0  0  0  0   0  35   0   0   0   0   0 ...
  12| 0 0  0  0  0  0  0  0  0   0   0  37   0   0   0   0 ...
  13| 0 0  0  0  0  0  0  0  0   0   0   0  41   0   0   0 ...
  14| 0 0  0  0  0  0  0  0  0   0   0   0   0  43   0   0 ...
  15| 0 0  0  0  0  0  0  0  0   0   0   0   0   0  47   0 ...
  16| 0 7  0  0  0  0  0  0  0   0   0   0   0   0   0  49 ...
For the n-th row of this square array, the leftmost terms, other than 0, are the factors of A(n,n). A(n,n) = A007310(n + 1). If for every m, m < n, A(m,n) = 0, then a(n) = A007310(n + 1) and A007310(n + 1) is prime.

References

  • G. Pólya and G. Szegő, Problems and Theorems in Analysis II (Springer 1924, reprinted 1976), Part Eight, Chap. 2, Section 2, Problems 96 and 105.

Links

Crossrefs

Cf. A000034, A007310, A010729, A017281, A017365, A020639, A091999, A273669, A306277, A306285, A306331.
Cf. A107744, A111863 (bisections).

Programs

  • Maple
    seq(min(op(numtheory[factorset] (6*ceil(n/2)+(-1)^n))), n=1..64) ;
  • Mathematica
    FactorInteger[Rest@ Flatten@ Array[6 # + {1, 5} &, 33, 0]][[All, 1, 1]] (* Michael De Vlieger, Feb 15 2019 *)
FactorInteger[#][[1,1]]&/@Select[Range[2,200],CoprimeQ[#,6]&] (* Harvey P. Dale, Jul 10 2020 *)
  • PARI
    for(n=2, 211, if((n%6==1)||(n%6==5), print1(factor(n)[1,1], ", ")))
  • PARI
    vector(64,n,factor(6*ceil(n/2)+(-1)^n)[1,1])
  • PARI
    a(n) = n++; factor(n\2*6-(-1)^n)[1,1]; \\ Michel Marcus, Feb 06 2019

Formula

a(n) = A020639(A007310(n + 1)).
a(n) = A020639(3n + A000034(n + 1)).
a(n) = A020639(6*ceiling(n/2) + (-1)^n).
a(floor(prime(n + 2)/3)) = prime(n + 2).
Showing 1-4 of 4 results.

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