A341662 Primes p such that p^4 - 1 has 160 divisors.

53, 67, 131, 139, 227, 277, 283, 347, 383, 641, 653, 661, 821, 877, 997, 1069, 1181, 1213, 1811, 2083, 2389, 2459, 2819, 3803, 4021, 4253, 4723, 6619, 6829, 7213, 7933, 8069, 9013, 9187, 10589, 11261, 16139, 17827, 18133, 18587, 19309, 19541, 20477, 20947

Offset: 1

Author

Jon E. Schoenfield, Feb 26 2021

Keywords

• nonn

Comments

Conjecture: sequence is infinite.

For every term p, p^4 - 1 is of the form 2^4 * 3 * 5 * q * r * s, where q, r, and s are distinct primes > 5, with three exceptions: p = 53, 383, and 641 (see Example section).

Examples

			      p =
   n  a(n)     p^4 - 1    factorization of p^4 - 1
  --  ----  ------------  -------------------------------
   1    53       7890480  2^4 * 3^3 * 5 * 13 * 281
   2    67      20151120  2^4 * 3 * 5 * 11 * 17 * 449
   3   131     294499920  2^4 * 3 * 5 * 11 * 13 * 8581
   4   139     373301040  2^4 * 3 * 5 * 7 * 23 * 9661
   5   227    2655237840  2^4 * 3 * 5 * 19 * 113 * 5153
   6   277    5887339440  2^4 * 3 * 5 * 23 * 139 * 7673
   7   283    6414247920  2^4 * 3 * 5 * 47 * 71 * 8009
   8   347   14498327280  2^4 * 3 * 5 * 29 * 173 * 12041
   9   383   21517662720  2^9 * 3 * 5 * 191 * 14669
  10   641  168823196160  2^9 * 3 * 5 * 107 * 205441
  11   653  181824635280  2^4 * 3 * 5 * 109 * 163 * 42641
		

Crossrefs

Cf. A000005, A000040, A309906, A341656, A341661.

Programs

Mathematica

Select[Range[21000], PrimeQ[#] && DivisorSigma[0, #^4 - 1] == 160 &] (* Amiram Eldar, Feb 26 2021 *)

PARI

isok(p) = isprime(p) && (numdiv(p^4-1) == 160); \\ Michel Marcus, Feb 26 2021