A284564 -id:A284564 - cp's OEIS frontend

A277324 Odd bisection of A260443 (the even terms): a(n) = A260443((2*n)+1).

2, 6, 18, 30, 90, 270, 450, 210, 630, 6750, 20250, 9450, 15750, 47250, 22050, 2310, 6930, 330750, 3543750, 1653750, 4961250, 53156250, 24806250, 727650, 1212750, 57881250, 173643750, 18191250, 8489250, 25467750, 2668050, 30030, 90090, 40020750, 1910081250, 891371250, 9550406250, 455814843750, 212713593750

Offset: 0

Antti Karttunen, Oct 10 2016

nonn

From David A. Corneth, Oct 22 2016: (Start)
The exponents of the prime factorization of a(n) are first nondecreasing, then nonincreasing.
The exponent of 2 in the prime factorization of a(n) is 1. (End)

			A method to find terms of this sequence, explained by an example to find a(7). To find k = a(7), we find k such that A048675(k) = 2*7+1 = 15. 7 has the binary partitions: {[7, 0, 0], [5, 1, 0], [3, 2, 0], [1, 3, 0], [3, 0, 1], [1, 1, 1]}. To each of those, we prepend a 1. This gives the binary partitions of 15 starting with a 1. For example, for the first we get [1, 7, 0, 0]. We see that only [1, 5, 1, 0], [1, 3, 2, 0] and [1, 1, 1, 1] start nondecreasing, then nonincreasing, so we only check those. These numbers will be the exponents in a prime factorization. [1, 5, 1, 0] corresponds to prime(1)^1 * prime(2)^5 * prime(3)^1 * prime(4)^0 = 2430. We find that [1, 1, 1, 1] gives k = 210 for which A048675(k) = 15 so a(7) = 210. - _David A. Corneth_, Oct 22 2016

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A005811, A005940, A007306, A007949, A156552, A260443, A277189, A277323, A283484, A283975, A284267, A284268, A284563, A284564, A284573.

Cf. A277200 (same sequence sorted into ascending order).

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[a[2 n + 1], {n, 0, 38}] (* Michael De Vlieger, Apr 05 2017 *)

from sympy import factorint, prime, primepi
from operator import mul
def a003961(n):
    F=factorint(n)
    return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**F[i] for i in F])
def a260443(n): return n + 1 if n<2 else a003961(a260443(n//2)) if n%2==0 else a260443((n - 1)//2)*a260443((n + 1)//2)
def a(n): return a260443(2*n + 1)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 21 2017

a(n) = A260443((2*n)+1).
a(0) = 2; for n >= 1, a(n) = A260443(n) * A260443(n+1).
Other identities. For all n >= 0:
A007949(a(n)) = A005811(n). [See comments in A125184.]
A156552(a(n)) = A277189(n), a(n) = A005940(1+A277189(n)).
A048675(a(n)) = 2n + 1. - David A. Corneth, Oct 22 2016
A001222(a(n)) = A007306(1+n).
A056169(a(n)) = A284267(n).
A275812(a(n)) = A284268(n).
A248663(a(n)) = A283975(n).
A000188(a(n)) = A283484(n).
A247503(a(n)) = A284563(n).
A248101(a(n)) = A284564(n).
A046523(a(n)) = A284573(n).
a(n) = A277198(n) * A284008(n).
a(n) = A284576(n) * A284578(n) = A284577(n) * A000290(A284578(n)).

More linking formulas added by Antti Karttunen, Apr 16 2017

A284563 a(n) = A247503(A277324(n)).

Original entry on oeis.org

2, 2, 2, 10, 10, 10, 50, 10, 10, 250, 250, 50, 250, 250, 50, 110, 110, 250, 6250, 1250, 1250, 31250, 6250, 550, 2750, 6250, 6250, 13750, 2750, 2750, 6050, 110, 110, 30250, 68750, 13750, 343750, 781250, 156250, 151250, 151250, 781250, 19531250, 1718750, 343750, 8593750, 756250, 6050, 30250, 756250, 1718750, 3781250, 3781250, 8593750, 18906250, 151250

Offset: 0

Antti Karttunen, Mar 29 2017

nonn

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

Odd bisection of A284553.

Cf. A001222, A247503, A277324, A284564, A284565.

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[2 n + 1], {n, 0, 55}] (* Michael De Vlieger, Apr 05 2017 *)

A284563(n) = A284553(n+n+1); \\ Other code as in A284553.

(define (A284563 n) (A247503 (A277324 n)))
(define (A284563 n) (A284553 (+ n n 1)))

a(n) = A247503(A277324(n)).
a(n) = A284553((2*n)+1).
Other identities. For all n >= 0:
A001222(a(n)) = A284565(n).

A284566 Odd bisection of A284556.

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 2, 2, 3, 3, 4, 4, 3, 4, 4, 2, 3, 5, 5, 5, 6, 6, 6, 5, 4, 6, 7, 5, 5, 6, 4, 3, 4, 5, 7, 7, 7, 9, 9, 6, 7, 10, 10, 9, 9, 9, 8, 6, 5, 8, 10, 8, 9, 11, 9, 7, 7, 8, 9, 8, 6, 7, 6, 3, 4, 7, 8, 8, 10, 11, 11, 9, 9, 13, 15, 12, 12, 14, 11, 8, 9, 12, 15, 14, 14, 17, 16, 11, 11, 15, 15, 13, 12, 12, 10, 7, 6

Offset: 0

Antti Karttunen, Apr 05 2017

nonn

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

Cf. A001222, A007306, A284556, A284564, A284565.

a[n_] := Which[n < 2, n, EvenQ@ n, a[n/2], True, a[(n - 1)/2] + a[(n + 1)/2]]; Table[(a[#] - JacobiSymbol[#, 3])/2 &[2 n + 1], {n, 0, 96}] (* Michael De Vlieger, Apr 05 2017 *)

(define (A284566 n) (A284556 (+ n n 1)))
(define (A284566 n) (A001222 (A284564 n)))

a(n) = A284556((2*n)+1).
a(n) = A001222(A284564(n)).
Other identities. For all n >= 1:
A007306(n) = a(n-1) + A284565(n-1).

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

Original entry on oeis.org

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

Antti Karttunen, Mar 29 2017

nonn

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.

			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

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

Cf. A001222, A003961, A064989, A248101, A260443, A284553, A284556, A284564 (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 + 1, 2])^e) &@ a@ n, {n, 0, 76}] (* 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.
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));

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

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).

cp's OEIS Frontend

A277324 Odd bisection of A260443 (the even terms): a(n) = A260443((2*n)+1).

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A284563 a(n) = A247503(A277324(n)).

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A284566 Odd bisection of A284556.

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A284554 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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