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%I #21 Jan 14 2024 12:38:35
%S 0,0,0,0,0,1,0,0,0,1,1,1,0,2,0,0,0,1,1,2,1,3,1,2,0,2,2,2,0,3,0,0,0,1,
%T 1,3,1,4,2,4,1,5,3,5,1,5,2,3,0,3,2,4,2,5,2,4,0,3,3,3,0,4,0,0,0,1,1,4,
%U 1,5,3,5,1,6,4,8,2,7,4,5,1,6,5,8,3,10,5,7,1,7,5,8,2,7,3,4,0,4,3,6,2,8,4,7,2,8,5,9,2,8,4,5,0,5,3,6,3,7,3,6,0
%N a(n) = A001222(A283983(n)).
%C a(n) = Sum_{c} floor(c/2), where c ranges over each coefficient of terms c * x^k in the Stern polynomial B(n,x), thus sum of the halved terms (for odd terms floored down) on row n of table A125184.
%H Antti Karttunen, <a href="/A284264/b284264.txt">Table of n, a(n) for n = 0..8192</a>
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F a(n) = A001222(A283983(n)).
%F Other identities and observations. For all n >= 0:
%F a(2n) = a(n).
%F a(n) = (1/2) * (A002487(n) - A277700(n)).
%F 2*a(n) <= A284272(n).
%t A003961[p_?PrimeQ] := A003961[p] = Prime[ PrimePi[p] + 1]; A003961[1] = 1; A003961[n_] := A003961[n] = Times @@ ( A003961[First[#]] ^ Last[#] & ) /@ FactorInteger[n] (* after _Jean-François Alcover_, Dec 01 2011 *); A260443[n_]:= If[n<2, n + 1, If[EvenQ[n], A003961[A260443[n/2]], A260443[(n - 1)/2] * A260443[(n + 1)/2]]]; A000188[n_]:= Sum[Boole[Mod[i^2, n] == 0], {i, n}]; Table[PrimeOmega[A000188[A260443[n]]], {n, 0, 120}] (* _Indranil Ghosh_, Mar 28 2017 *)
%o (PARI)
%o A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From _Michel Marcus_
%o A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2)))); \\ Cf. _Charles R Greathouse IV_'s code for "ps" in A186891 and A277013.
%o A000188(n) = core(n, 1)[2]; \\ This function from _Michel Marcus_, Feb 27 2013
%o A283983(n) = A000188(A260443(n));
%o A284264(n) = bigomega(A283983(n));
%o (Scheme) (define (A284264 n) (/ (- (A002487 n) (A277700 n)) 2))
%Y Cf. A000188, A002487, A001222, A125184, A260443, A277700, A283983, A284265 (odd bisection), A284272.
%Y Cf. A023758 (gives the positions of zeros).
%K nonn
%O 0,14
%A _Antti Karttunen_, Mar 25 2017
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