A284272 -id:A284272 - cp's OEIS frontend

A284264 a(n) = A001222(A283983(n)).

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

Offset: 0

Antti Karttunen, Mar 25 2017

nonn

a(n) = Sum_{c} floor(c/2), where c ranges over each coefficient of terms c * x^k in the Stern polynomial B(n,x), thus sum of the halved terms (for odd terms floored down) on row n of table A125184.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A000188, A002487, A001222, A125184, A260443, A277700, A283983, A284265 (odd bisection), A284272.

Cf. A023758 (gives the positions of zeros).

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[PrimeOmega[A000188[A260443[n]]], {n, 0, 120}] (* 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.
A000188(n) = core(n, 1)[2]; \\ This function from Michel Marcus, Feb 27 2013
A283983(n) = A000188(A260443(n));
A284264(n) = bigomega(A283983(n));

(define (A284264 n) (/ (- (A002487 n) (A277700 n)) 2))

a(n) = A001222(A283983(n)).
Other identities and observations. For all n >= 0:
a(2n) = a(n).
a(n) = (1/2) * (A002487(n) - A277700(n)).
2*a(n) <= A284272(n).

A284268 Sum of coefficients > 1 in the Stern polynomial B(2n+1,x): a(n) = A275812(A277324(n)).

Original entry on oeis.org

0, 0, 2, 0, 2, 3, 4, 0, 2, 6, 7, 5, 5, 6, 6, 0, 2, 8, 9, 9, 10, 11, 11, 7, 7, 11, 12, 9, 8, 9, 8, 0, 2, 10, 12, 11, 13, 17, 16, 12, 13, 18, 20, 16, 15, 17, 15, 9, 9, 15, 17, 16, 17, 19, 18, 12, 11, 16, 17, 13, 11, 12, 10, 0, 2, 12, 15, 14, 17, 22, 21, 15, 17, 25, 27, 24, 23, 26, 22, 15, 16, 24, 29, 26, 28, 32, 30, 21, 20, 28, 30, 24, 21, 23, 19, 11, 11, 19

Offset: 0

Antti Karttunen, Mar 25 2017

nonn

Sum of terms larger than one on row 2n+1 of table A125184.

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

Cf. A007306, A125184, A260443, A275812, A277324, A284267.

Odd bisection of A284272.

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]]]; A275812[n_]:= PrimeOmega[n] - If[n<2, 0,Count[Transpose[FactorInteger[n]][[2]], 1]]; Table[A275812[A260443[2n + 1]], {n, 0, 150}] (* Indranil Ghosh, Mar 28 2017 *)

A284268(n) = A284272(n+n+1); \\ Other code as in A284272.

(define (A284268 n) (A284272 (+ n n 1)))
(define (A284268 n) (A275812 (A277324 n)))

a(n) = A284272((2*n)+1).
a(n) = A275812(A277324(n)).
Other identities. For all n >= 0:
A007306(1+n) = A284267(n) + 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).

cp's OEIS Frontend

A284264 a(n) = A001222(A283983(n)).

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A284268 Sum of coefficients > 1 in the Stern polynomial B(2n+1,x): a(n) = A275812(A277324(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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