A285483 - OEIS

A285483

a(n) is the smallest positive integer that makes a(n)*A007694(n)-1 a prime number, while a(n) and A007694(n) are coprimes.

7, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 17, 1, 5, 1, 1, 5, 5, 1, 7, 5, 5, 5, 1, 1, 13, 7, 5, 1, 11, 1, 1, 5, 1, 17, 19, 17, 19, 5, 25, 5, 1, 7, 5, 13, 11, 5, 1, 5, 5, 7, 1, 1, 19, 1, 1, 17, 5, 7, 29, 1, 5, 1, 5, 7, 7, 17, 1, 7, 7, 1, 7, 49, 5, 13, 13, 25, 5, 5, 23

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OFFSET

2,1

COMMENTS

All terms are elements of A007310, which are free of prime factor 2 and 3, since if a(n) has a factor of 2, 2*A007694(n) is also an element of A007694. Ditto for a(n) is divisible by 3 cases.

a(1) is not defined since any odd number greater than 3 minus 1 is an even nonprime number.

LINKS

Lei Zhou, Table of n, a(n) for n = 2..10001

EXAMPLE

For n = 2, A007694(2) = 2, testing k*2-1 for k in set {1, 5, 7, 11, 13, 17, 19... }, we find that 7*2 - 1 = 13 is the first prime number found. So a(2) = 7;

In the similar way, 1*A007694(3) - 1 = 1*4 - 1 = 3 is the first prime number found for n = 3, so a(3) = 1.

For n = 7, A007694(7) = 16, 5*16 - 1 = 89 is the smallest prime found, so a(7) = 5.

MATHEMATICA

b = 2; a = {b}; sm = {}; r = a; While[Length[sm] < 81, f = 0;

While[f++; (fc = FactorInteger[f];

MemberQ[{2, 3}, fc[[1, 1]]]) || (! PrimeQ[f*a[[Length[a]]] - 1])];

AppendTo[sm, f]; c = r*2; d = r*3; e = Sort[Union[c, d]]; i = 1;

While[e[[i]] <= a[[Length[a]]], i++]; AppendTo[a, e[[i]]];

AppendTo[r, e[[i]]];

While[(3*r[[1]]) < r[[Length[r]]], r = Delete[r, 1]]]; sm

CROSSREFS

Cf. A003586, A007310, A007694.