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A306289
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The smallest prime factor of numbers greater than 1 and coprime to 6.
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3
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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
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OFFSET
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1,1
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COMMENTS
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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).
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REFERENCES
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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.
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LINKS
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FORMULA
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a(n) = A020639(6*ceiling(n/2) + (-1)^n).
a(floor(prime(n + 2)/3)) = prime(n + 2).
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EXAMPLE
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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.
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MAPLE
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seq(min(op(numtheory[factorset] (6*ceil(n/2)+(-1)^n))), n=1..64) ;
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000034, A007310, A010729, A017281, A017365, A020639, A091999, A273669, A306277, A306285, A306331.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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