A325147 Reduced Clausen numbers.

10, 546, 2, 46, 6630, 76670, 211659630, 6, 261870, 111418, 46, 13589784390, 524588442, 114, 1138240087314330, 2, 276742830, 26805565070, 1909802752494, 3210, 15370, 177430547680928732190, 358, 5760551069383110, 76004922, 1126, 4347631610092420338, 81366

Offset: 1

Author

Peter Luschny, May 21 2019

Keywords

• nonn

Comments

Let P(m) denote the prime factors of m and C(m) = Clausen(m-1, 1) (cf. A160014) then Product_{p in P(C(m)) setminus P(m)} p is in this sequence provided P(m) is a subset of P(C(m)).

Examples

			Let n = 561 then P(561) = {3, 11, 17} and P(Clausen(560,1)) = {2, 3, 5, 11, 17, 29, 41, 71, 113, 281}. Since P(561) is a subset of P(Clausen(560, 1)), a(18) = 2*5*29*41*71*113*281 = 26805565070.
		

Crossrefs

Weak Carmichael numbers are A225498. Clausen numbers are in A160014.

A324977 is a subsequence.

Programs

Maple

with(numtheory): a := proc(n) if isweakCarmichael(n) then # cf. A225498 and A160014
mul(m, m in factorset(Clausen(n-1, 1)) minus factorset(n)) else NULL fi end:
seq(a(n), n=2..1350);

Mathematica

pf[n_] := FactorInteger[n][[All, 1]];
Clausen[0, ] = 1; Clausen[n, k_] := Times @@ (Select[Divisors[n], PrimeQ[# + k]&] + k);
weakCarmQ[n_] := If[EvenQ[n] || PrimeQ[n], Return[False], pf[n] == (pf[n] ~Intersection~ pf[Clausen[n - 1, 1]])];
f[n_] := Times @@ Complement[pf[Clausen[n - 1, 1]], pf[n]];
f /@ Select[Range[2, 2000], weakCarmQ] (* Jean-François Alcover, Jul 21 2019 *)