OFFSET
1,1
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).
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
Davis Smith, Table of n, a(n) for n = 1..1000
FORMULA
EXAMPLE
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 ...
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Davis Smith, Feb 03 2019
STATUS
approved