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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.

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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).

A306331 Numbers congruent to 6 or 31 mod 38.

Original entry on oeis.org

6, 31, 44, 69, 82, 107, 120, 145, 158, 183, 196, 221, 234, 259, 272, 297, 310, 335, 348, 373, 386, 411, 424, 449, 462, 487, 500, 525, 538, 563, 576, 601, 614, 639, 652, 677, 690, 715, 728, 753, 766, 791, 804, 829, 842, 867, 880, 905
Offset: 1

Views

Author

Davis Smith, Feb 07 2019

Keywords

Comments

A007310(a(n) + 1) is always a multiple of 19.
A020639(A007310(a(n) + 1)) = 5, 7, 11, 13, 17, or 19.
It equals 5 when n is a term in A273669.
It equals 7 when n is congruent to 3 or 12 (mod 14) but not a term in A273669.
It equals 11 when n is congruent to 4 or 19 (mod 22) but not a case where it equals 5 or 7.
It equals 13 when n is congruent to 5 or 22 (mod 26) (one more than a term in A306285) but not a case where it equals 5, 7, or 11.
It equals 17 when n is congruent to 6 or 29 (mod 34) but not a case where it equals 5, 7, 11, or 13.
For all other cases, it equals 19.
a(n) and (n - 1) have the same remainder (mod 6) (see A010875).

Links

Crossrefs

Cf. A005408, A005843, A007310, A010875, A020639, A040031, A273669, A306285.

Programs

  • Maple
    seq(seq(38*i+j, j=[6, 31]), i=0..200);
  • Mathematica
    Select[Range[200], MemberQ[{6, 31}, Mod[#, 38]] &]
Union[38Range[30] - 32, 38Range[30] - 7] (* Alonso del Arte, Feb 08 2019 *)
  • PARI
    for(n=6, 905, if((n%38==6) || (n%38==31), print1(n, ", ")))
  • PARI
    Vec(x*(6 + 25*x + 7*x^2) / ((1 - x)^2*(1 + x)) + O(x^40)) \\ Colin Barker, Feb 09 2019
  • Scala
    (6 to 1108 by 38).union(31 to 1133 by 38).sorted // Alonso del Arte, Feb 08 2019

Formula

G.f.: x*(6 + 25*x + 7*x^2) / ((1 - x)^2*(1 + x)). - Colin Barker, Feb 09 2019
a(n) = a(n - 1) + a(n - 2) - a(n - 3) for n > 3.
a(n) = 19*n - 10 + 3*(-1)^n. - Wesley Ivan Hurt, Mar 10 2019
a(n) = 19*n - 13 when n is odd and 19*n - 7 when n is even.
a(n) = 19*n - (A040031(n + 1) + 1).
E.g.f.: 7 + (19*x - 10)*exp(x) + 3*exp(-x). - David Lovler, Sep 10 2022
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