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Square root of the largest square dividing prime factorization representation of the n-th Stern polynomial: a(n) = A000188(A260443(n)).
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%I #14 Mar 28 2017 14:46:00

%S 1,1,1,1,1,3,1,1,1,3,5,3,1,15,1,1,1,3,5,15,7,45,5,15,1,15,35,15,1,105,

%T 1,1,1,3,5,105,7,225,35,525,11,1575,175,1125,7,1575,35,105,1,105,35,

%U 525,77,1575,35,525,1,105,385,105,1,1155,1,1,1,3,5,1155,7,1575,385,3675,11,7875,1225,275625,77,55125,2695,5775,13,17325,13475,275625,539

%N Square root of the largest square dividing prime factorization representation of the n-th Stern polynomial: a(n) = A000188(A260443(n)).

%H Antti Karttunen, <a href="/A283983/b283983.txt">Table of n, a(n) for n = 0..4096</a>

%F a(n) = A000188(A260443(n)).

%F a(n) = A000196(A283989(n)).

%F Other identities. For all n >= 0:

%F a(2n) = A003961(a(n)).

%F A001222(a(n)) = A284264(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[A000188[A260443[n]], {n, 0, 50}] (* _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 (Scheme)

%o (define (A283983 n) (A000188 (A260443 n)))

%o (define (A283983 n) (A000196 (A283989 n)))

%Y Cf. A000188, A000196, A001222, A003961, A260443, A283989, A284264, A283484 (odd bisection).

%Y Cf. A023758 (positions of ones).

%K nonn

%O 0,6

%A _Antti Karttunen_, Mar 25 2017