A372187 Numbers m such that 72*m + 1, 576*m + 1, 648*m + 1, 1296*m + 1, and 2592*m + 1 are all primes.

95, 890, 3635, 8150, 9850, 12740, 13805, 18715, 22590, 23591, 32526, 36395, 38571, 49016, 49456, 57551, 58296, 61275, 80756, 81050, 84980, 99940, 104346, 115361, 116761, 121055, 122550, 129320, 140331, 142625, 149431, 153505, 159306, 159730, 169625, 173485, 181661

Offset: 1

Author

Amiram Eldar, Apr 21 2024

Keywords

• nonn
• easy

Comments

If m is a term, then (72*m + 1) * (576*m + 1) * (648*m + 1) * (1296*m + 1) * (2592*m + 1) is a Carmichael number (A002997). These are the Carmichael numbers of the form U_{5,5}(m) in Nakamula et al. (2007).

The corresponding Carmichael numbers are 698669495582067436250881, 50411423376758357271937215361, 57292035175893741987253427965441, ...

Examples

			95 is a term since 72*95 + 1 = 6841, 576*95 + 1 = 54721, 648*95 + 1 = 61561, 1296*95 + 1 = 123121, and 2592*95 + 1 = 246241 are all primes.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Ken Nakamula, Hirofumi Tsumura, and Hiroaki Komai, New polynomials producing absolute pseudoprimes with any number of prime factors, arXiv:math/0702410 [math.NT], 2007.

Crossrefs

Cf. A002997, A126797, A318646.

Similar sequences: A046025, A257035, A206024, A206349, A372186, A372188.

Programs

Mathematica

q[n_] := AllTrue[{72, 576, 648, 1296, 2592}, PrimeQ[#*n + 1] &]; Select[Range[200000], q]

PARI

is(n) = isprime(72*n + 1) && isprime(576*n + 1) && isprime(648*n + 1) && isprime(1296*n + 1) && isprime(2592*n + 1);