cp's OEIS Frontend

A024321 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = (composite numbers).

%I A024321 #11 Jan 20 2022 10:33:41
%S A024321 0,0,6,8,9,10,12,14,25,28,32,35,37,40,44,46,64,69,73,77,81,85,89,93,
%T A024321 96,100,128,133,139,144,148,154,162,166,170,176,181,187,223,229,236,
%U A024321 242,248,255,262,268,275,281,287,294,301,308,354,361,370,380,386,394,401,408,418,425
%N A024321 a(n) = s(1)*t(n) + s(2)*t(n-1) + ... + s(k)*t(n+1-k), where k = floor((n+1)/2), s = A023531, t = (composite numbers).
%H A024321 G. C. Greubel, <a href="/A024321/b024321.txt">Table of n, a(n) for n = 1..1000</a>
%F A024321 a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*A002808(n-j+1). - _G. C. Greubel_, Jan 19 2022
%t A024321 A023531[n_]:= SquaresR[1, 8n+9]/2;
%t A024321 Composite[n_]:= FixedPoint[n +PrimePi[#] +1 &, n];
%t A024321 a[n_]:= Sum[A023531[j]*Composite[n-j+1], {j, Floor[(n+1)/2]}];
%t A024321 Table[a[n], {n, 70}] (* _G. C. Greubel_, Jan 19 2022 *)
%o A024321 (Magma)
%o A024321 A002808:= [n : n in [2..100] | not IsPrime(n) ];
%o A024321 A023531:= func< n | IsIntegral( (Sqrt(8*n+9) -3)/2 ) select 1 else 0 >;
%o A024321 [ (&+[A023531(j)*A002808[n-j+1]: j in [1..Floor((n+1)/2)]]) : n in [1..70]]; // _G. C. Greubel_, Jan 19 2022
%o A024321 (Sage)
%o A024321 A002808 = [n for n in (1..250) if sloane.A001222(n) > 1]
%o A024321 def A023531(n):
%o A024321     if ((sqrt(8*n+9) -3)/2).is_integer(): return 1
%o A024321     else: return 0
%o A024321 [sum( A023531(j)*A002808[n-j] for j in (1..floor((n+1)/2)) ) for n in (1..70)] # _G. C. Greubel_, Jan 19 2022
%Y A024321 Cf. A024312, A024313, A024314, A024315, A024316, A024317, A024318, A024319, A024320, A024322, A024323, A024324, A024325, A024326, A024327.
%Y A024321 Cf. A002808, A023531.
%K A024321 nonn
%O A024321 1,3
%A A024321 _Clark Kimberling_