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A284272 Sum of coefficients > 1 in the Stern polynomial B(n,x): a(n) = A275812(A260443(n)).

%I A284272 #12 Mar 28 2017 14:53:32
%S A284272 0,0,0,0,0,2,0,0,0,2,2,3,0,4,0,0,0,2,2,6,2,7,3,5,0,5,4,6,0,6,0,0,0,2,
%T A284272 2,8,2,9,6,9,2,10,7,11,3,11,5,7,0,7,5,11,4,12,6,9,0,8,6,9,0,8,0,0,0,2,
%U A284272 2,10,2,12,8,11,2,13,9,17,6,16,9,12,2,13,10,18,7,20,11,16,3,15,11,17,5,15,7,9,0,9,7,15,5,17,11,16,4,17,12,19,6,18,9
%N A284272 Sum of coefficients > 1 in the Stern polynomial B(n,x): a(n) = A275812(A260443(n)).
%C A284272 Sum of terms larger than one on row n of table A125184.
%H A284272 Antti Karttunen, <a href="/A284272/b284272.txt">Table of n, a(n) for n = 0..8192</a>
%F A284272 a(n) = A275812(A260443(n)).
%F A284272 Other identities and observations. For all n >= 0:
%F A284272 A002487(n) = A284271(n) + a(n).
%F A284272 a(n) >= 2*A284264(n).
%t A284272 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]]]; A275812[n_]:= PrimeOmega[n] - If[n<2, 0,Count[Transpose[FactorInteger[n]][[2]], 1]]; Table[A275812[A260443[n]], {n, 0, 150}] (* _Indranil Ghosh_, Mar 28 2017 *)
%o A284272 (PARI)
%o A284272 A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From _Michel Marcus_
%o A284272 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 A284272 A056169(n) = { my(f=factor(n)[, 2]); sum(i=1, #f, f[i]==1); }; \\ This function from _Charles R Greathouse IV_, Apr 29 2015
%o A284272 A275812(n) = bigomega(n) - A056169(n);
%o A284272 A284272(n) = A275812(A260443(n));
%o A284272 (Scheme) (define (A284272 n) (A275812 (A260443 n)))
%Y A284272 Cf. A002487, A125184, A260443, A275812, A284264, A284271, A284268 (odd bisection).
%K A284272 nonn
%O A284272 0,6
%A A284272 _Antti Karttunen_, Mar 25 2017