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

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, 189, 7, 441, 21, 21, 13, 63, 21, 1323, 49, 567, 1029, 1323, 7, 3969, 1029, 1701, 343, 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

Offset: 0

Author

Antti Karttunen, Mar 29 2017

Keywords

• nonn

Comments

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.

Examples

			n A260443(n)                      Stern            With even powers
             prime factorization  polynomial       of x cleared  -> a(n)
------------------------------------------------------------------------
0       1    (empty)              B_0(x) = 0                    0  |  1
1       2    p_1                  B_1(x) = 1                    0  |  1
2       3    p_2                  B_2(x) = x                    x  |  3
3       6    p_2 * p_1            B_3(x) = x + 1                x  |  3
4       5    p_3                  B_4(x) = x^2                  0  |  1
5      18    p_2^2 * p_1          B_5(x) = 2x + 1              2x  |  9
6      15    p_3 * p_2            B_6(x) = x^2 + x              x  |  3
7      30    p_3 * p_2 * p_1      B_7(x) = x^2 + x + 1          x  |  3
8       7    p_4                  B_8(x) = x^3                x^3  |  7
9      90    p_3 * p_2^2 * p_1    B_9(x) = x^2 + 2x + 1        2x  |  9
10     75    p_3^2 * p_2          B_10(x) = 2x^2 + x            x  |  3
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..8192

Crossrefs

Cf. A001222, A003961, A064989, A248101, A260443, A284553, A284556, A284564 (odd bisection).

Programs

Mathematica

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 *)

PARI

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From Michel Marcus
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.
A248101(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 2] *= (primepi(f[i, 1])+1) % 2; ); factorback(f); } \\ After Michel Marcus
A284554(n) = A248101(A260443(n));

Scheme

(define (A284554 n) (A248101 (A260443 n)))

Formula

a(0) = a(1) = 1, a(2n) = A003961(A284553(n)), a(2n+1) = a(n)*a(n+1).

Other identities. For all n >= 0:

a(n) = A248101(A260443(n)).

a(n) = A260443(n) / A284553(n).

a(n) = A064989(A284553(2n)).

A001222(a(n)) = A284556(n).