A190105 a(n) = (3*A002145(n) - 1)/4.

2, 5, 8, 14, 17, 23, 32, 35, 44, 50, 53, 59, 62, 77, 80, 95, 98, 104, 113, 122, 125, 134, 143, 149, 158, 167, 170, 179, 188, 197, 203, 212, 230, 233, 248, 260, 269, 275, 284, 287, 314, 323, 329, 332, 347, 350, 359, 365, 368, 374, 377, 392, 410, 422, 428, 440

Offset: 1

Author

J. M. Bergot, May 04 2011

Keywords

• nonn
• easy

Comments

For primes p of the form 4n+3, in the order of A002145, let us seek solutions for prime p|(a^x + b^y) or p|(a^y + b^x) subject to the conditions p = a+b = x+y and 0 < a,b,x,y < p. The larger of the two exponents x and y is inserted into the sequence.

If either of (a,b) is a primitive root of p, there is a unique solution, either p|(a^x + b^y) or p|(a^y + b^x). If neither is a primitive root (see A060749), there are multiple solutions and p|(a^x + b^y) or p|(a^y + b^x) will always be one of them for some of the given exponents (x,y) contributing to the sequence.

Examples

			For p=43=A002145(7), (x,y)=(11,32) because 43-(43+1)/4=32; hence x=43-32.  With (a,b)=(12,31) the unique solution is 43|(12^11 + 31^32) because 12 is a primitive root of 43. The larger of 11 and 32 is a(7)=32 in the sequence. For 43 multiple solutions occur when neither of the pairs (a,b) is a primitive root of 43: p divides each of 11^4 + 32^39, 11^18 + 32^25, 11^32 + 32^11; note that the exponents (11,32) occur in the last entry.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005099 is the list of x in (x,y).

Programs

Maple

for n from 1 to 200 do p:=4*n-1: if(isprime(p))then printf("%d, ", (3*p-1)/4); fi: od: # Nathaniel Johnston, May 18 2011

Mathematica

A002145 := Select[4 Range[300] - 1, PrimeQ]; Table[(3*A002145[[n]] - 1)/4, {n, 1, 60}] (* G. C. Greubel, Nov 07 2018 *)