A308779 Almost Carmichael numbers: Squarefree composite n for which omega(n) = k >= 3 and there are exactly k-1 prime divisors p of n such that p-1 | n-1.

105, 165, 231, 285, 345, 385, 465, 645, 705, 805, 885, 1005, 1045, 1065, 1185, 1221, 1245, 1545, 1551, 1605, 1645, 1653, 1771, 1885, 1905, 1965, 2065, 2085, 2193, 2211, 2265, 2445, 2485, 2505, 2553, 2665, 2685, 2737, 2865, 2905, 2985, 3009, 3165, 3201, 3345, 3405, 3445

Offset

1,1

Comments

Numbers satisfying the p-1 | n-1 criterion to the greatest extent possible without actually being Carmichael numbers (semiprimes excluded).

All terms are odd, since if not there is an even term n divisible by at least one odd prime p such that p-1 | n-1; contradiction since p-1 is even and n-1 is odd.

Subsequence of first terms with k=3,4,5,... begins: 105, 5005, 185185, ... (always divisible by 5?).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

105 = 3*5*7; 2|104 and 4|104 but 6 does not divide 104.

Mathematica

acQ[n_] := Module[{f = FactorInteger[n]}, p = f[[;; , 1]]; e = f[[;; , 2]]; om = Length[p]; om > 2 && Max[e] == 1 && Length@Select[p, Divisible[n-1, #-1]&] == om - 1]; Select[Range[1, 3500, 2], acQ] (* Amiram Eldar, Jul 04 2019 *)

Prog

(PARI) isok(n) = if ((n>1) && (n % 2) && issquarefree(n) && (f = factor(n)) && (#f~ >= 3), return (sum(k=1, #f~, !((n-1) % (f[k, 1]-1))) == #f~ - 1)); \\ Michel Marcus, Jun 24 2019

Crossrefs

Cf. A002997 (Carmichael numbers).

Sequence in context: A119434 A091495 A256673 * A046405 A128278 A234103

Adjacent sequences: A308776 A308777 A308778 * A308780 A308781 A308782

Keyword

nonn

Author

David James Sycamore, Jun 23 2019

Extensions

More terms from Michel Marcus, Jun 24 2019

Status

approved