A396952 Composite numbers k such that d + 1 is not squarefree for all divisors d > 1.

49, 51, 119, 159, 161, 187, 249, 267, 291, 323, 341, 343, 371, 391, 413, 447, 489, 527, 583, 591, 623, 649, 679, 699, 721, 723, 731, 737, 749, 799, 807, 849, 879, 917, 979, 989, 1003, 1007, 1011, 1043, 1047, 1057, 1067, 1077, 1127, 1133, 1139, 1149, 1169, 1207

Offset

1,1

Comments

Definition variant is: numbers k such that mu(d+1) is zero for all divisors d of k except d = 1, where mu is the Möbius function and k is not prime. The primes p where p+1 is not squarefree are at A049098.

Links

David A. Corneth, Table of n, a(n) for n = 1..10011

Example

For k=49 the divisors are 1,7,49. Increment to get 2,8,50, with mu(8)=0 and mu(50)=0, hence k=49 is in the sequence.

Maple

with(NumberTheory): isA := proc(n) local t; t := Divisors(n) minus {1};
nops(t) > 1 and andseq(Möbius(d + 1) = 0, d in t) end:
select(isA, [seq(1..1210)]);  # Peter Luschny, Jun 12 2026

Mathematica

q[k_] := CompositeQ[k] && AllTrue[Divisors[k], # == 1 || !SquareFreeQ[# + 1] &]; Select[Range[1300], q] (* Amiram Eldar, Jun 11 2026 *)

Prog

(PARI) isok(k) = if(isprime(k) || k == 1, return(0)); fordiv(k, d, if ((d!=1) && issquarefree(d+1), return(0))); 1; \\ Michel Marcus, Jun 11 2026

Crossrefs

Cf. A008683, A049098.

Sequence in context: A080201 A042199 A257443 * A020276 A346805 A118073

Adjacent sequences: A396948 A396949 A396950 * A396954 A396955 A396956

Keyword

nonn

Author

Marko Riedel, Jun 11 2026

Status

approved