OFFSET
0,3
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.
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..8192
FORMULA
EXAMPLE
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Mar 29 2017
STATUS
approved