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A284554
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Prime factorization representation of Stern polynomials B(n,x) with only the odd powers of x present: a(n) = A248101(A260443(n)).
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4
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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Other identities. For all n >= 0:
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EXAMPLE
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n A260443(n) Stern With even powers
prime factorization polynomial of x cleared -> a(n)
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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
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MATHEMATICA
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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 *)
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PROG
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(PARI)
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From Michel Marcus
A248101(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 2] *= (primepi(f[i, 1])+1) % 2; ); factorback(f); } \\ After Michel Marcus
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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