A372188 Numbers m such that 18*m + 1, 36*m + 1, 108*m + 1, and 162*m + 1 are all primes.

1, 71, 155, 176, 241, 346, 420, 540, 690, 801, 1145, 1421, 1506, 2026, 2066, 3080, 3235, 3371, 3445, 3511, 3640, 4746, 4925, 5681, 5901, 6055, 6520, 7931, 8365, 8970, 9006, 9556, 9685, 10186, 11396, 11750, 11935, 12055, 12666, 13205, 13266, 13825, 13881, 14606

Offset: 1

Author

Amiram Eldar, Apr 21 2024

Keywords

• nonn
• easy

Comments

If m is a term, then (18*m + 1) * (36*m + 1) * (108*m + 1) * (162*m + 1) is a Carmichael number (A002997). These are the Carmichael numbers of the form W_4(3*m) in Nakamula et al. (2007).

The corresponding Carmichael numbers are 12490201, 288503529142321, 6548129556412321, ...

Examples

			1 is a term since 18*1 + 1 = 19, 36*1 + 1 = 37, 108*1 + 1 = 109, and 162*1 + 1 = 163 are all primes.
71 is a term since 18*71 + 1 = 1279, 36*71 + 1 = 2557, 108*71 + 1 = 7669, and 162*71 + 1 = 11503 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, A318646, A372190.

Similar sequences: A046025, A257035, A206024, A206349, A372186, A372187.

Programs

Mathematica

q[n_] := AllTrue[{18, 36, 108, 162}, PrimeQ[#*n + 1] &]; Select[Range[15000], q]

PARI

is(n) = isprime(18*n + 1) && isprime(36*n + 1) && isprime(108*n + 1) && isprime(162*n + 1);