A396952 - OEIS

%I #44 Jun 12 2026 18:04:17

%S 49,51,119,159,161,187,249,267,291,323,341,343,371,391,413,447,489,

%T 527,583,591,623,649,679,699,721,723,731,737,749,799,807,849,879,917,

%U 979,989,1003,1007,1011,1043,1047,1057,1067,1077,1127,1133,1139,1149,1169,1207

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

%C 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.

%H David A. Corneth, <a href="/A396952/b396952.txt">Table of n, a(n) for n = 1..10011</a>

%e 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.

%p with(NumberTheory): isA := proc(n) local t; t := Divisors(n) minus {1};

%p nops(t) > 1 and andseq(Möbius(d + 1) = 0, d in t) end:

%p select(isA, [seq(1..1210)]);  # _Peter Luschny_, Jun 12 2026

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

%o (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

%Y Cf. A008683, A049098.

%K nonn

%O 1,1

%A _Marko Riedel_, Jun 11 2026