A284554 -id:A284554 - cp's OEIS frontend

A284556 Sequence c of the mutual recurrence pair: b(0) = 0, b(1) = 1, b(2n) = c(n), b(2n+1) = b(n) + b(n+1), c(0) = c(1) = 0, c(2n) = b(n), c(2n+1) = c(n) + c(n+1).

0, 0, 1, 1, 0, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 2, 0, 3, 2, 3, 2, 4, 2, 4, 1, 3, 3, 4, 1, 4, 2, 2, 1, 3, 2, 5, 2, 5, 4, 5, 1, 6, 4, 6, 3, 6, 3, 5, 1, 4, 4, 6, 2, 7, 4, 5, 2, 5, 3, 6, 2, 4, 3, 3, 0, 4, 3, 5, 3, 7, 4, 7, 2, 7, 6, 9, 3, 9, 5, 6, 2, 7, 5, 10, 4, 10, 7, 9, 2, 9, 6, 9, 4, 8, 4, 6, 1, 5, 5, 8, 3, 10, 6, 8, 3, 9, 6, 11, 4, 9, 6, 7, 1, 7, 5, 8, 4, 9, 5

Offset: 0

Antti Karttunen, Apr 05 2017

nonn

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

Cf. A000360, A001222, A002487, A102283, A284554, A284566 (odd bisection).

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[PrimeOmega[Times @@ (FactorInteger[#] /. {p_, e_} /; e > 0 :> (p^Mod[PrimePi@ p + 1, 2])^e) &@ a@ n], {n, 0, 118}] (* or *)
a[n_] := Which[n < 2, n, EvenQ@ n, a[n/2], True, a[(n - 1)/2] + a[(n + 1)/2]]; Table[(a[n] - JacobiSymbol[n, 3])/2, {n, 0, 118}] (* Michael De Vlieger, Apr 05 2017, after Alonso del Arte at A102283 *)

a(n) = A001222(A284554(n)).
Other identities. For all n >= 1:
a(n) = (A002487(n) - A102283(n))/2.
a(n) = A002487(n) - A000360(n-1).
A000360(n-1) - a(n) = A102283(n) ≡ n (mod 3).

A284564 a(n) = A248101(A277324(n)).

Original entry on oeis.org

1, 3, 9, 3, 9, 27, 9, 21, 63, 27, 81, 189, 63, 189, 441, 21, 63, 1323, 567, 1323, 3969, 1701, 3969, 1323, 441, 9261, 27783, 1323, 3087, 9261, 441, 273, 819, 1323, 27783, 64827, 27783, 583443, 1361367, 9261, 27783, 4084101, 1750329, 583443, 1361367, 583443, 194481, 17199, 5733, 453789, 9529569, 453789, 1361367, 28588707, 1361367, 120393, 280917, 453789, 1361367

Offset: 0

Antti Karttunen, Mar 29 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..4096

Cf. A001222, A248101, A277324, A284554, A284563, A284566.

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[2 n + 1], {n, 0, 58}] (* Michael De Vlieger, Apr 05 2017 *)

A284564(n) = A284554(n+n+1); \\ Other code as in A284554.

(define (A284564 n) (A248101 (A277324 n)))
(define (A284564 n) (A284554 (+ n n 1)))

a(n) = A248101(A277324(n)).
a(n) = A284554((2*n)+1).
Other identities. For all n >= 0:
A001222(a(n)) = A284566(n).

A284553 Prime factorization representation of Stern polynomials B(n,x) with only the even powers of x present: a(n) = A247503(A260443(n)).

Original entry on oeis.org

1, 2, 1, 2, 5, 2, 5, 10, 1, 10, 25, 10, 5, 50, 5, 10, 11, 10, 25, 250, 5, 250, 125, 50, 11, 250, 25, 250, 55, 50, 55, 110, 1, 110, 275, 250, 55, 6250, 125, 1250, 121, 1250, 625, 31250, 55, 6250, 1375, 550, 11, 2750, 275, 6250, 605, 6250, 1375, 13750, 11, 2750, 3025, 2750, 55, 6050, 55, 110, 17, 110, 275, 30250, 55, 68750, 15125, 13750, 121

Offset: 0

Antti Karttunen, Mar 29 2017

nonn

a(n) = Prime factorization representation of Stern polynomials B(n,x) where the coefficients of odd powers of x are replaced by zeros. In other words, only the constant term and other terms with even powers of x are present. See the examples.

Proof that A001222(a(1+n)) matches Ralf Stephan's formula for A000360(n): Consider functions A001222(a(n)) and A001222(A284554(n)) (= A284556(n)). They can be reduced to the following mutual recurrence pair: b(0) = 0, b(1) = 1, b(2n) = c(n), b(2n+1) = b(n) + b(n+1) and c(0) = c(1) = 0, c(2n) = b(n), c(2n+1) = c(n) + c(n+1). From the definitions it follows that the difference b(n) - c(n) for even n is b(2n) - c(2n) = -(b(n) - c(n)), and for odd n, b(2n+1) - c(2n+1) = (b(n)+b(n+1))-(c(n)+c(n+1)) = (b(n)-c(n)) + (b(n+1)-c(n+1)). Then by induction, if we assume that for 3n, 3n+1, 3n+2, ..., 6n, the value of difference b(n)-c(n) is always [0, +1, -1; repeated], it follows that from 6n to 12n the differences are [0, +1, -1; 0, +1, -1; repeated], which proves that b(n) - c(n) = A102283(n). As an implication, recurrence b can be defined without referring to c as: b(0) = 0, b(1) = 1, b(2n) = b(n) - A102283(n), b(2n+1) = b(n)+b(n+1), and this is equal to Ralf Stephan's Oct 05 2003 formula for A000360, but shifted once right, with prepended zero.

			n A260443(n)                      Stern            With odd 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                    1  |  2
2       3    p_2                  B_2(x) = x                    0  |  1
3       6    p_2 * p_1            B_3(x) = x + 1                1  |  2
4       5    p_3                  B_4(x) = x^2                x^2  |  5
5      18    p_2^2 * p_1          B_5(x) = 2x + 1               1  |  2
6      15    p_3 * p_2            B_6(x) = x^2 + x            x^2  |  5
7      30    p_3 * p_2 * p_1      B_7(x) = x^2 + x + 1    x^2 + 1  | 10
8       7    p_4                  B_8(x) = x^3                  0  |  1
9      90    p_3 * p_2^2 * p_1    B_9(x) = x^2 + 2x + 1   x^2 + 1  | 10
10     75    p_3^2 * p_2          B_10(x) = 2x^2 + x         2x^2  | 25

Antti Karttunen, Table of n, a(n) for n = 0..8192
S. Klavzar, U. Milutinovic and C. Petr, Stern polynomials, Adv. Appl. Math. 39 (2007) 86-95.

Cf. A000360, A001222, A003961, A064989, A102283, A247503, A260443, A284554, A284556, A284563 (odd bisection).

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, 2])^e) &@ a@ n, {n, 0, 72}] (* Michael De Vlieger, Apr 05 2017 *)

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.
A247503(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 2] *= (primepi(f[i, 1]) % 2); ); factorback(f); } \\ After Michel Marcus
A284553(n) = A247503(A260443(n));

(define (A284553 n) (A247503 (A260443 n)))

a(0) = 1, a(1) = 2, a(2n) = A003961(A284554(n)), a(2n+1) = a(n)*a(n+1).
Other identities. For all n >= 0:
a(n) = A247503(A260443(n)).
a(n) = A260443(n) / A284554(n).
a(n) = A064989(A284554(2n)).
A001222(a(1+n)) = A000360(n). [Proof in Comments section.]

cp's OEIS Frontend

A284556 Sequence c of the mutual recurrence pair: b(0) = 0, b(1) = 1, b(2n) = c(n), b(2n+1) = b(n) + b(n+1), c(0) = c(1) = 0, c(2n) = b(n), c(2n+1) = c(n) + c(n+1).

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A284564 a(n) = A248101(A277324(n)).

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A284553 Prime factorization representation of Stern polynomials B(n,x) with only the even powers of x present: a(n) = A247503(A260443(n)).

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