A104238 - OEIS

A104238

Positive integers n such that n^5 + 1 is semiprime.

2, 10, 12, 16, 22, 126, 136, 180, 256, 268, 276, 366, 388, 396, 438, 462, 606, 642, 652, 658, 676, 738, 760, 768, 982, 1012, 1068, 1116, 1230, 1276, 1320, 1452, 1488, 1530, 1618, 1692, 1698, 1752, 1846, 1948, 1996, 2080, 2112, 2160, 2332, 2392, 2440, 2520

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OFFSET

1,1

COMMENTS

n^5+1 can only be prime when n = 1, n^5+1 = 2. This is because of the polynomial factorization n^5+1 = (n+1) * (n^4 - n^3 + n^2 - n + 1) = (n+1)*A060884(n). Hence after the initial n=1 prime, the binomial can at best be semiprime and that only when both (n+1) and (n^4 - n^3 + n^2 - n + 1) are primes.

LINKS

Robert Price, Table of n, a(n) for n = 1..1414

FORMULA

a(n)^5 + 1 is semiprime. a(n)+1 is prime and a(n)^4 - a(n)^3 + a(n)^2 - a(n) + 1 is prime.

EXAMPLE

n n^5+1 = (n+1) * (n^4 - n^3 + n^2 - n + 1)

2 33 = 3 * 11

10 100001 = 11 * 9091

12 248833 = 13 * 19141

16 1048577 = 17 * 61681

MATHEMATICA

Select[Range[2600], PrimeOmega[#^5+1]==2&] (* Harvey P. Dale, May 20 2011 *)