A378634 Numbers k such that for each odd prime dividing k, the previous prime divides k-1.

1, 2, 3, 4, 8, 9, 10, 16, 21, 22, 25, 27, 32, 40, 56, 64, 81, 99, 100, 115, 116, 128, 160, 171, 176, 196, 243, 250, 256, 400, 424, 441, 484, 507, 512, 531, 625, 640, 686, 729, 783, 896, 1000, 1024, 1246, 1331, 1408, 1450, 1600, 1660, 1701, 1863, 2048, 2080, 2086, 2109, 2187, 2366, 2401, 2432

Offset

1,2

Comments

If k is a term, then so are all powers of k.

If p is an odd prime, then p^k is a term where k = A226367(A000720(p)-1) is the multiplicative order of p modulo A151799(p).

Links

Robert Israel, Table of n, a(n) for n = 1..500

Example

a(7) = 10 is a term because the only odd prime dividing 10 is 5, and the previous prime 3 divides 10 - 1 = 9.
a(8) = 16 is a term because 16 is not divisible by any odd prime.
a(9) = 21 is a term because the odd primes dividing 21 are 3 and 7, and 2 (the prime previous to 3) and 5 (the prime previous to 7) both divide 21 - 1 = 20.

Maple

filter:= n -> andmap(p -> n-1 mod prevprime(p) = 0, numtheory:-factorset(n) minus {2}):
select(filter, [$1..3000]);

Mathematica

Select[Range[2450], Function[k, Or[IntegerQ@ Log2[k], AllTrue[FactorInteger[k/2^IntegerExponent[k, 2] ][[All, 1]], Divisible[k - 1, NextPrime[#, -1] ] &] ] ] ] (* Michael De Vlieger, Dec 03 2024 *)

Prog

(PARI) isok(k) = my(f=factor(k)); for (i=1, #f~, if ((f[i, 1] % 2) && ((k-1) % precprime(f[i, 1]-1)), return(0))); return(1); \\ Michel Marcus, Dec 02 2024

Crossrefs

Cf. A000720, A151799, A163619, A226367.

Sequence in context: A364381 A246444 A117847 * A045577 A111020 A002971

Adjacent sequences: A378631 A378632 A378633 * A378636 A378637 A378638

Keyword

nonn,new

Author

Robert Israel, Dec 02 2024

Status

approved