cp's OEIS Frontend

A284554 Prime factorization representation of Stern polynomials B(n,x) with only the odd powers of x present: a(n) = A248101(A260443(n)).

%I A284554 #15 Apr 06 2017 21:21:37
%S A284554 1,1,3,3,1,9,3,3,7,9,3,27,7,9,21,21,1,63,21,27,49,81,21,189,7,63,147,
%T A284554 189,7,441,21,21,13,63,21,1323,49,567,1029,1323,7,3969,1029,1701,343,
%U A284554 3969,147,1323,13,441,1029,9261,49,27783,1029,1323,91,3087,147,9261,91,441,273,273,1,819,273,1323,637,27783,1029,64827,91,27783,50421,583443,343
%N A284554 Prime factorization representation of Stern polynomials B(n,x) with only the odd powers of x present: a(n) = A248101(A260443(n)).
%C A284554 a(n) = Prime factorization representation of Stern polynomials B(n,x) where the coefficients of even powers of x (including the constant term) are replaced by zeros. In other words, only the terms with odd powers of x are present. See the examples.
%H A284554 Antti Karttunen, <a href="/A284554/b284554.txt">Table of n, a(n) for n = 0..8192</a>
%F A284554 a(0) = a(1) = 1, a(2n) = A003961(A284553(n)), a(2n+1) = a(n)*a(n+1).
%F A284554 Other identities. For all n >= 0:
%F A284554 a(n) = A248101(A260443(n)).
%F A284554 a(n) = A260443(n) / A284553(n).
%F A284554 a(n) = A064989(A284553(2n)).
%F A284554 A001222(a(n)) = A284556(n).
%e A284554 n A260443(n)                      Stern            With even powers
%e A284554              prime factorization  polynomial       of x cleared  -> a(n)
%e A284554 ------------------------------------------------------------------------
%e A284554 0       1    (empty)              B_0(x) = 0                    0  |  1
%e A284554 1       2    p_1                  B_1(x) = 1                    0  |  1
%e A284554 2       3    p_2                  B_2(x) = x                    x  |  3
%e A284554 3       6    p_2 * p_1            B_3(x) = x + 1                x  |  3
%e A284554 4       5    p_3                  B_4(x) = x^2                  0  |  1
%e A284554 5      18    p_2^2 * p_1          B_5(x) = 2x + 1              2x  |  9
%e A284554 6      15    p_3 * p_2            B_6(x) = x^2 + x              x  |  3
%e A284554 7      30    p_3 * p_2 * p_1      B_7(x) = x^2 + x + 1          x  |  3
%e A284554 8       7    p_4                  B_8(x) = x^3                x^3  |  7
%e A284554 9      90    p_3 * p_2^2 * p_1    B_9(x) = x^2 + 2x + 1        2x  |  9
%e A284554 10     75    p_3^2 * p_2          B_10(x) = 2x^2 + x            x  |  3
%t A284554 a[n_] := a[n] = Which[n < 2, n + 1, EvenQ@ n, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &@ a[n/2], True, a[#] a[# + 1] &[(n - 1)/2]]; Table[Times @@ (FactorInteger[#] /. {p_, e_} /; e > 0 :> (p^Mod[PrimePi@ p + 1, 2])^e) &@ a@ n, {n, 0, 76}] (* _Michael De Vlieger_, Apr 05 2017 *)
%o A284554 (PARI)
%o A284554 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From _Michel Marcus_
%o A284554 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 A284554 A248101(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 2] *= (primepi(f[i, 1])+1) % 2; ); factorback(f); } \\ After _Michel Marcus_
%o A284554 A284554(n) = A248101(A260443(n));
%o A284554 (Scheme) (define (A284554 n) (A248101 (A260443 n)))
%Y A284554 Cf. A001222, A003961, A064989, A248101, A260443, A284553, A284556, A284564 (odd bisection).
%K A284554 nonn
%O A284554 0,3
%A A284554 _Antti Karttunen_, Mar 29 2017