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%I #122 Jul 10 2020 13:23:32
%S 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,
%T 79,83,5,89,7,5,97,101,103,107,109,113,5,7,11,5,127,131,7,137,139,11,
%U 5,149,151,5,157,7,163,167,13,173,5,179,181,5,11,191,193
%N The smallest prime factor of numbers greater than 1 and coprime to 6.
%C 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).
%C 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.
%C 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).
%D 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.
%H Davis Smith, <a href="/A306289/b306289.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = A020639(A007310(n + 1)).
%F a(n) = A020639(3n + A000034(n + 1)).
%F a(n) = A020639(6*ceiling(n/2) + (-1)^n).
%F a(floor(prime(n + 2)/3)) = prime(n + 2).
%e 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.
%e Table begins
%e \m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...
%e n\
%e 1| 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...
%e 2| 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...
%e 3| 0 0 11 0 0 0 0 0 0 0 0 0 0 0 0 0 ...
%e 4| 0 0 0 13 0 0 0 0 0 0 0 0 0 0 0 0 ...
%e 5| 0 0 0 0 17 0 0 0 0 0 0 0 0 0 0 0 ...
%e 6| 0 0 0 0 0 19 0 0 0 0 0 0 0 0 0 0 ...
%e 7| 0 0 0 0 0 0 23 0 0 0 0 0 0 0 0 0 ...
%e 8| 5 0 0 0 0 0 0 25 0 0 0 0 0 0 0 0 ...
%e 9| 0 0 0 0 0 0 0 0 29 0 0 0 0 0 0 0 ...
%e 10| 0 0 0 0 0 0 0 0 0 31 0 0 0 0 0 0 ...
%e 11| 5 7 0 0 0 0 0 0 0 0 35 0 0 0 0 0 ...
%e 12| 0 0 0 0 0 0 0 0 0 0 0 37 0 0 0 0 ...
%e 13| 0 0 0 0 0 0 0 0 0 0 0 0 41 0 0 0 ...
%e 14| 0 0 0 0 0 0 0 0 0 0 0 0 0 43 0 0 ...
%e 15| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 47 0 ...
%e 16| 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 49 ...
%e 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.
%p seq(min(op(numtheory[factorset] (6*ceil(n/2)+(-1)^n))), n=1..64) ;
%t FactorInteger[Rest@ Flatten@ Array[6 # + {1, 5} &, 33, 0]][[All, 1, 1]] (* _Michael De Vlieger_, Feb 15 2019 *)
%t FactorInteger[#][[1,1]]&/@Select[Range[2,200],CoprimeQ[#,6]&] (* _Harvey P. Dale_, Jul 10 2020 *)
%o (PARI) for(n=2, 211, if((n%6==1)||(n%6==5), print1(factor(n)[1,1], ", ")))
%o (PARI) vector(64,n,factor(6*ceil(n/2)+(-1)^n)[1,1])
%o (PARI) a(n) = n++; factor(n\2*6-(-1)^n)[1,1]; \\ _Michel Marcus_, Feb 06 2019
%Y Cf. A000034, A007310, A010729, A017281, A017365, A020639, A091999, A273669, A306277, A306285, A306331.
%Y Cf. A107744, A111863 (bisections).
%K nonn
%O 1,1
%A _Davis Smith_, Feb 03 2019