A340268 Composite numbers k>1 such that (s-1) | (d-1) for each d | k, where s = lpf(k) = A020639(k).

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 96

Offset: 1

Author

Maxim Karimov, Jan 02 2021

Keywords

• nonn

Comments

Not a duplicate of A340058 because the complements A335902 and A340269 differ. - R. J. Mathar, Feb 16 2021

Crossrefs

Cf. A000010, A000961, A020639, A340058, A335902, A340269 (complement).

Contains all composite terms of at least A003586, A003591, A003592, A003593, A003596.

Programs

MATLAB

n=300; % gives all terms of the sequence not exceeding n
A=[];
for i=2:n
    lpf=2;
    while mod(i,lpf)~=0
        lpf=lpf+1;
    end
    for d=1:floor(i/2)
        if mod(i,d)==0 && mod(d-1,lpf-1)~=0
            break
        elseif d==floor(i/2)
            A=[A i];
        end
    end
end

Maple

with(numtheory):
q:= n-> (f-> andmap(d-> irem(d-1, f)=0, divisors(n)))(min(factorset(n))-1):
select(not isprime and q, [$2..96])[];  # Alois P. Heinz, Feb 12 2021

Mathematica

Select[Range[2, 96], Function[{n, s}, And[! PrimeQ@ n, AllTrue[Divisors[n] - 1, Mod[#, s] == 0 &]]] @@ {#, FactorInteger[#][[1, 1]] - 1} &] (* Michael De Vlieger, Feb 12 2021 *)

PARI

isok(c) = if ((c>1) && !isprime(c), my(f=factor(c)[,1]); for (k=1, #f~, if ((f[k]-1) % (f[1]-1), return(0))); return(1)); \\ Michel Marcus, Jan 03 2021