A277013 -id:A277013 - cp's OEIS frontend

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

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.]

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).

A283989 Largest square dividing prime factorization representation of the n-th Stern polynomial: a(n) = A008833(A260443(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 9, 1, 1, 1, 9, 25, 9, 1, 225, 1, 1, 1, 9, 25, 225, 49, 2025, 25, 225, 1, 225, 1225, 225, 1, 11025, 1, 1, 1, 9, 25, 11025, 49, 50625, 1225, 275625, 121, 2480625, 30625, 1265625, 49, 2480625, 1225, 11025, 1, 11025, 1225, 275625, 5929, 2480625, 1225, 275625

Offset: 0

Antti Karttunen, Mar 25 2017

nonn

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

Cf. A003961, A008833, A260443, A277330, A283983.

Cf. A023758 (positions of ones).

A003961[p_?PrimeQ] := A003961[p] = Prime[ PrimePi[p] + 1]; A003961[1] = 1; A003961[n_] := A003961[n] = Times @@ ( A003961[First[#]] ^ Last[#] & ) /@ FactorInteger[n]  (* after Jean-François Alcover, Dec 01 2011 *); A260443[n_]:= If[n<2, n + 1, If[EvenQ[n], A003961[A260443[n/2]], A260443[(n - 1)/2] * A260443[(n + 1)/2]]]; A000188[n_]:= Sum[Boole[Mod[i^2, n] == 0], {i, n}]; Table[A000188[A260443[n]]^2, {n, 0, 20}] (* Indranil Ghosh, Mar 28 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.
A008833(n) = n/core(n); \\ This function from Michael B. Porter, Oct 17 2009
A283989(n) = A008833(A260443(n));

(define (A283989 n) (A008833 (A260443 n)))
(define (A283989 n) (/ (A260443 n) (A277330 n)))

a(n) = A008833(A260443(n)).
a(n) = A260443(n) / A277330(n).
a(n) = A283983(n)^2.
a(2n) = A003961(a(n)).

A284271 Number of terms with coefficient 1 in the Stern polynomial B(n,x): a(n) = A056169(A260443(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 2, 1, 3, 4, 1, 3, 2, 1, 1, 1, 2, 2, 2, 2, 1, 2, 3, 1, 4, 5, 1, 4, 3, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 3, 2, 1, 2, 4, 1, 5, 6, 1, 5, 4, 1, 3, 2, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 3, 2, 2, 1, 1, 1, 2, 2, 4

Offset: 0

Antti Karttunen, Mar 25 2017

nonn

Number of 1's on row n of table A125184.

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

Cf. A002487, A056169, A125184, A260443, A277700, A284272, A284267 (odd bisection).

A003961[p_?PrimeQ] := A003961[p] = Prime[ PrimePi[p] + 1]; A003961[1] = 1; A003961[n_] := A003961[n] = Times @@ ( A003961[First[#]] ^ Last[#] & ) /@ FactorInteger[n](* Jean-François Alcover, Dec 01 2011 *); A260443[n_]:= If[n<2, n + 1, If[EvenQ[n], A003961[A260443[n/2]], A260443[(n - 1)/2] * A260443[(n + 1)/2]]]; a[n_]:= If[n<2, 0, Count[Transpose[FactorInteger[n]][[2]], 1]]; Table[a[A260443[n]], {n, 0, 150}] (* Indranil Ghosh, Mar 28 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.
A056169(n) = { my(f=factor(n)[, 2]); sum(i=1, #f, f[i]==1); }; \\ This function from Charles R Greathouse IV, Apr 29 2015
A284271(n) = A056169(A260443(n));

(define (A284271 n) (A056169 (A260443 n)))

a(n) = A056169(A260443(n)).
Other identities and observations. For all n >= 0:
A002487(n) = a(n) + A284272(n).
a(n) <= A277700(n).

A284573 Odd bisection of A278243: a(n) = A046523(A277324(n)).

Original entry on oeis.org

2, 6, 12, 30, 60, 120, 180, 210, 420, 1080, 2160, 2520, 2520, 7560, 6300, 2310, 4620, 37800, 90720, 75600, 226800, 544320, 453600, 138600, 138600, 756000, 2268000, 831600, 415800, 2079000, 485100, 30030, 60060, 2910600, 24948000, 12474000, 49896000, 272160000, 136080000, 29106000, 87318000, 1360800000, 3265920000, 1496880000, 748440000

Offset: 0

Antti Karttunen, Apr 11 2017

nonn

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

Cf. A007306, A046523, A277324, A278243.

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.
A277324(n) = A260443((2*n)+1);
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From Charles R Greathouse IV, Aug 17 2011
A284573(n) = A046523(A277324(n));

a(n) = A046523(A277324(n)).

a(n) = A278243((2*n)+1).

A305822 Number of irreducible factors (counted with multiplicity) of the (0,1)-polynomial encoded in the binary expansion of n has when it is factored over Q.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 3, 3, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 2, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 4, 1, 2, 3, 6, 2, 3, 1, 3, 1, 3, 1, 5, 1, 2, 3, 3, 1, 3, 1, 5, 1, 2, 1, 4, 2, 2, 1, 4, 1, 4, 1, 3, 2, 2, 2, 6, 1, 3, 3, 3, 1, 3, 1, 4, 3

Offset: 1

Antti Karttunen, Jun 10 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A206074 (gives the positions of 1's), A206719, A257000, A304751.
Cf. also A001222, A091222.
Differs from A277013 for the first time at n=65, where a(65) = 2, while A277013(65) = 1.

A305822(n) = vecsum(factor(Pol(binary(n)))[, 2]);

For all n >= 1, a(n) >= A206719(n).

cp's OEIS Frontend

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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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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A283989 Largest square dividing prime factorization representation of the n-th Stern polynomial: a(n) = A008833(A260443(n)).

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A284271 Number of terms with coefficient 1 in the Stern polynomial B(n,x): a(n) = A056169(A260443(n)).

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A284573 Odd bisection of A278243: a(n) = A046523(A277324(n)).

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A305822 Number of irreducible factors (counted with multiplicity) of the (0,1)-polynomial encoded in the binary expansion of n has when it is factored over Q.

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