A306277 -id:A306277 - cp's OEIS frontend

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

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

Davis Smith, Feb 03 2019

nonn

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).

			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.

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.

Davis Smith, Table of n, a(n) for n = 1..1000

Cf. A000034, A007310, A010729, A017281, A017365, A020639, A091999, A273669, A306277, A306285, A306331.

Cf. A107744, A111863 (bisections).

seq(min(op(numtheory[factorset] (6*ceil(n/2)+(-1)^n))), n=1..64) ;

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 *)

for(n=2, 211, if((n%6==1)||(n%6==5), print1(factor(n)[1,1], ", ")))

vector(64,n,factor(6*ceil(n/2)+(-1)^n)[1,1])

a(n) = n++; factor(n\2*6-(-1)^n)[1,1]; \\ Michel Marcus, Feb 06 2019

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).

A306278 Numbers congruent to 2 or 11 mod 14.

Original entry on oeis.org

2, 11, 16, 25, 30, 39, 44, 53, 58, 67, 72, 81, 86, 95, 100, 109, 114, 123, 128, 137, 142, 151, 156, 165, 170, 179, 184, 193, 198, 207, 212, 221, 226, 235, 240, 249, 254, 263, 268, 277, 282, 291, 296, 305, 310, 319, 324, 333, 338, 347, 352, 361, 366, 375, 380, 389, 394

Offset: 1

Davis Smith, Feb 02 2019

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A007310, A010703, A020639, A047470, A091999, A273669, A306277, A306289.

Primes greater than 2 in this sequence: A045471.

Maple
```
seq(seq(14*i+j, j=[2, 11]), i=0..28);
```

Flatten[Table[{14n + 2, 14n + 11}, {n, 0, 28}]]
LinearRecurrence[{1,1,-1},{2,11,16},60] (* Harvey P. Dale, Jan 16 2023 *)

for(n=2, 394, if((n%14==2) || (n%14==11), print1(n, ", ")))

PARI
```
for(n=1,57,print1(7*n-4+(-1)^n,", "))
```

for(n=1,500,if(n%14==2,print1(n,", "));if(n%14==11,print1(n,", "))) \\ Jinyuan Wang, Feb 03 2019

Vec(x*(2 + 9*x + 3*x^2) / ((1 - x)^2*(1 + x)) + O(x^40)) \\ Colin Barker, Mar 14 2019

upto(n) = forstep(i = 2, n, [9, 5], print1(i", ")) \\ David A. Corneth, Mar 27 2019

a(n) = 7*n - A010703(n).
a(n) = 7*n - 4 + (-1)^n.
a(n) = a(n - 1) + a(n - 2) - a(n - 3) for n > 3.
A007310(a(n) + 1) = 7*A007310(n)
From Jinyuan Wang, Feb 03 2019: (Start)
For odd number k, a(k) = 7*k - 5.
For even number k, a(k) = 7*k - 3.
(End)
G.f.: x*(2 + 9*x + 3*x^2) / ((1 - x)^2*(1 + x)). - Colin Barker, Mar 14 2019
E.g.f.: 3 + (7*x - 4)*exp(x) + exp(-x). - David Lovler, Sep 07 2022

A306279 Numbers congruent to 3 or 18 mod 22.

Original entry on oeis.org

3, 18, 25, 40, 47, 62, 69, 84, 91, 106, 113, 128, 135, 150, 157, 172, 179, 194, 201, 216, 223, 238, 245, 260, 267, 282, 289, 304, 311, 326, 333, 348, 355, 370, 377, 392, 399, 414, 421, 436, 443, 458, 465, 480, 487, 502, 509, 524, 531, 546, 553, 568

Offset: 1

Davis Smith, Feb 02 2019

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A007310, A020639, A042948, A091999, A105398, A131555, A141850, A158459, A273669, A306277, A306289.

Primes in this sequence: A141850.

Maple
```
seq(seq(22*i+j, j=[3, 18]), i=0..200);
```

Select[Range[200], MemberQ[{3, 18}, Mod[#, 22]] &]
Flatten[Table[{22n + 3, 22n + 18}, {n, 0, 43}]] (* Alonso del Arte, Feb 18 2019 *)

for(n=3, 678, if((n%22==3) || (n%22==18), print1(n, ", ")))

PARI
```
vector(62,n,11*n-6+2*(-1)^n)
```

Vec(x*(3 + 15*x + 4*x^2) / ((1 - x)^2*(1 + x)) + O(x^40)) \\ Colin Barker, Feb 07 2019

(3 to 949 by 22).union(18 to 942 by 22).sorted // Alonso del Arte, Feb 18 2019

a(n) = 11*n - 6 + 2*(-1)^n.
a(n) = 11*n - A105398(n + 4).
A007310(a(n) + 1) = 11*A007310(n).
From Colin Barker, Feb 07 2019: (Start)
G.f.: x*(3 + 15*x + 4*x^2) / ((1 - x)^2*(1 + x)).
a(n) = a(n - 1) + a(n - 2) - a(n - 3) for n > 3. (End)
E.g.f.: 4 + (11*x - 6)*exp(x) + 2*exp(-x). - David Lovler, Sep 08 2022

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

A306278 Numbers congruent to 2 or 11 mod 14.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

PARI

PARI

Formula

A306279 Numbers congruent to 3 or 18 mod 22.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Scala

Formula