A005408 -id:A005408 - cp's OEIS frontend

A166101 Integers k such that A166100(k)/A005408(k) is not an integer.

1, 13, 37, 121, 181, 337, 433, 793, 937, 1093, 1261, 1633, 2521, 3037, 3313, 3901, 4213, 4537, 5221, 7141, 7561, 8437, 9841, 10333, 10837, 11353, 11881, 14701, 15301, 17173, 19153, 19837, 21961, 22693, 23437, 25741, 27337, 28153, 29821

Offset: 1

Antti Karttunen, Oct 13 2009

nonn

A. Karttunen, Table of n, a(n) for n = 1..92

A166102 gives the corresponding odd numbers. Cf. A166272.

A130845 a(4n) = a(4n+1) = a(4n+2) = A001477(n), a(4n+3) = A005408(n).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 3, 2, 2, 2, 5, 3, 3, 3, 7, 4, 4, 4, 9, 5, 5, 5, 11, 6, 6, 6, 13, 7, 7, 7, 15, 8, 8, 8, 17, 9, 9, 9, 19, 10, 10, 10, 21, 11, 11, 11, 23, 12, 12, 12, 25, 13, 13, 13, 27, 14, 14, 14, 29, 15, 15, 15, 31, 16, 16, 16, 33, 17, 17, 17

Offset: 0

Paul Curtz, Jul 20 2007

nonn

Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

CoefficientList[Series[x^3(1+x+x^2+x^3+x^4)/((1-x)^2(1+x)^2(1+x^2)^2),{x,0,80}],x] (* or *) LinearRecurrence[{0,0,0,2,0,0,0,-1},{0,0,0,1,1,1,1,3},80] (* Harvey P. Dale, Mar 04 2012 *)

O.g.f.: x^3*(1+x+x^2+x^3+x^4)/((1-x)^2*(1+x)^2*(1+x^2)^2). - R. J. Mathar, Aug 22 2008
a(0)=0, a(1)=0, a(2)=0, a(3)=1, a(4)=1, a(5)=1, a(6)=1, a(7)=3, a(n)=2*a(n-4)- a(n-8). - Harvey P. Dale, Mar 04 2012
a(n) = cos(n*Pi/2)/4-(n-1)*(2*sin(n*Pi/2)+(-1)^n-5)/16. - Wesley Ivan Hurt, May 05 2021

Edited by N. J. A. Sloane, Sep 28 2007

A131733 Primes (A000040) - odds (A005408).

Original entry on oeis.org

1, 0, 0, 0, 2, 2, 4, 4, 6, 10, 10, 14, 16, 16, 18, 22, 26, 26, 30, 32, 32, 36, 38, 42, 48, 50, 50, 52, 52, 54, 66, 68, 72, 72, 80, 80, 84, 88, 90, 94, 98, 98, 106, 106, 108, 108, 118, 128, 130, 130, 132, 136, 136, 144, 148, 152, 156, 156, 160, 162, 162, 170

Offset: 1

Paul Curtz, Sep 17 2007

A131733 := proc(n) ithprime(n)-2*n+1 ; end proc: # R. J. Mathar, Jul 10 2011

With[{nn=70},Prime[Range[nn]]-Range[1,2nn,2]] (* Harvey P. Dale, Jun 12 2013 *)

a(n) = A000040(n)-2n+1.

A166407 a(n) = floor(3*(A166406(n)/A005408(n))).

Original entry on oeis.org

-3, 1, 0, 3, -9, 3, 0, 6, 0, 3, 0, 9, -30, 1, 0, 9, 0, 6, 0, 12, 0, 3, 0, 15, -63, 6, 0, 12, 0, 9, 0, 6, 0, 3, 0, 21, 0, 2, 0, 15, -81, 9, 0, 18, 0, 6, 0, 24, 0, 0, 0, 15, 0, 9, 0, 24, 0, 6, 0, 30, -165, 6, 0, 15, 0, 15, 0, 6, 0, 9, 0, 30, 0, 0, 0, 21, 0, 12, 0, 30, 0, 3, 0, 33, -234, 6, 0, 6

Offset: 0

Antti Karttunen, Oct 21 2009

sign

Conjecture: the quotient A166406(i)/A005408(i) has denominator 3 when i is one of the terms of A166101, and it is integral in other cases. If true, then floor in the formula is unnecessary.

Antti Karttunen, Table of n, a(n) for n = 0..65535

Cf. A166408.

from sympy import floor, jacobi_symbol as J
def a(n):
    l=0
    m=0
    for i in range(1, 2*n + 2):
        if J(i, 2*n + 1)==-1: l+=i
        elif J(i, 2*n + 1)==1: m+=i
    return floor(3*((l - m)/(2*n + 1)))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 12 2017

A289786 p-INVERT of the odd positive integers (A005408), where p(S) = 1 - S - S^2.

Original entry on oeis.org

1, 5, 20, 77, 291, 1098, 4149, 15689, 59332, 224369, 848447, 3208370, 12132345, 45878109, 173486772, 656035301, 2480778763, 9380993978, 35473960589, 134143768193, 507260826084, 1918192318185, 7253589435975, 27429241169378, 103722891648049, 392225150722037

Offset: 0

Clark Kimberling, Aug 10 2017

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the INVERT transform of s, so that p-INVERT is a generalization of the INVERT transform (e.g., A033453).

See A289780 for a guide to related sequences.

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -6, 5, 1)

Cf. A005408, A289780.

z = 60; s = x (1 + x)/(1 - x)^2; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A005408 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289786 *)
LinearRecurrence[{5,-6,5,1},{1,5,20,77},30] (* Harvey P. Dale, May 06 2018 *)

G.f.: (-1 - x^2 - 2 x^3)/(-1 + 5 x - 6 x^2 + 5 x^3 + x^4).

a(n) = 5*a(n-1) - 6*a(n-2) + 5*a(n-3) + a(n-4).

A113170 Ascending descending base exponent transform of odd numbers A005408.

Original entry on oeis.org

1, 4, 33, 376, 5665, 115356, 3014209, 95722288, 3619661121, 161338248820, 8349617508961, 493959321484584, 33041900704133473, 2479933070973253516, 207343189445230918785, 19175058576632809926496, 1949302342535131018462849, 216707770770991401785821668

Offset: 1

Jonathan Vos Post, Jan 06 2006

A003101 is the ascending descending base exponent transform of natural numbers A000027. The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154. The parity of this sequence cycles odd, even, odd, even, ... There is no nontrivial integer fixed point of the transform.

			a(2) = 4 because 1^3 + 3^1 = 1 + 3 = 4.
a(3) = 33 because 1^5 + 3^3 + 5^1 = 1 + 27 + 5 = 33.
a(4) = 406 because 1^7 + 3^5 + 5^3 + 7^1 = 1 + 243 + 125 + 7 = 376.
a(5) = 5665 because 1^9 + 3^7 + 5^5 + 7^3 + 9^1 = 5665.
a(6) = 115356 = 1^11 + 3^9 + 5^7 + 7^5 + 9^3 + 11^1.
a(7) = 3014209 = 1^13 + 3^11 + 5^9 + 7^7 + 9^5 + 11^3 + 13^1.
a(8) = 95722288 = 1^15 + 3^13 + 5^11 + 7^9 + 9^7 + 11^5 + 13^3 + 15^1.
a(9) = 3619661121 = 1^17 + 3^15 + 5^13 + 7^11 + 9^9 + 11^7 + 13^5 + 15^3 + 17^1.
a(10) = 161338248820 = 1^19 + 3^17 + 5^15 + 7^13 + 9^11 + 11^9 + 13^7 + 15^5 + 17^3 + 19^1.

G. C. Greubel, Table of n, a(n) for n = 1..295

Cf. A005408, A113122, A113153, A113154.

Table[Sum[(2 k + 1)^(2 n - 2 k + 1), {k, 1, n}], {n, 0, 10}] + 1 (* G. C. Greubel, May 18 2017 *)

for(n=0,25, print1(1 + sum(k=1,n, (2*k+1)^(2*n-2*k+1)), ", ")) \\ G. C. Greubel, May 18 2017

a(1) = 1. For n>1: a(n) = Sum_{i=1..n} (2n+1)^(2n-i).

A169642 a(n) = A005408(n) * A022998(n).

Original entry on oeis.org

0, 3, 20, 21, 72, 55, 156, 105, 272, 171, 420, 253, 600, 351, 812, 465, 1056, 595, 1332, 741, 1640, 903, 1980, 1081, 2352, 1275, 2756, 1485, 3192, 1711, 3660, 1953, 4160, 2211, 4692, 2485, 5256, 2775, 5852, 3081, 6480, 3403, 7140, 3741, 7832, 4095, 8556

Offset: 0

Paul Curtz, Apr 04 2010

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A005408, A022998.

LinearRecurrence[{0, 3, 0, -3 , 0, 1}, {0 , 3, 20, 21, 72, 55}, 47] (* Georg Fischer, Feb 22 2019 *)

concat(0, Vec(-x*(3+20*x+12*x^2+12*x^3+x^4)/ ((x-1)^3*(1+x)^3) + O(x^50))) \\ Colin Barker, Dec 29 2016

From R. J. Mathar, Oct 09 2010: (Start)
a(n)= +3*a(n-2) -3*a(n-4) +a(n-6).
G.f.: -x*(3+20*x+12*x^2+12*x^3+x^4)/ ( (x-1)^3*(1+x)^3 ). (End)
From Colin Barker, Dec 29 2016: (Start)
a(n) = 4*n^2 + 2*n for n>0 and even.
a(n) = 2*n^2 + n for n odd. (End)
Sum_{n>=1} 1/a(n) = 1 + Pi/8 - 5*log(2)/4. - Amiram Eldar, Aug 12 2022

Edited by N. J. A. Sloane, Apr 05 2010

More terms from R. J. Mathar, Oct 09 2010

A173283 A(x) satisfies A005408(x) = A(x)/A(x^2), A005408 = odd numbers.

Original entry on oeis.org

1, 3, 8, 16, 32, 56, 96, 152, 240, 360, 536, 768, 1096, 1520, 2096, 2824, 3792, 5000, 6568, 8496, 10960, 13960, 17728, 22264, 27896, 34624, 42872, 52640, 64504, 78464, 95248, 114856, 138256, 165448, 197640, 234832, 278592, 328920, 387744, 455064

Offset: 0

Gary W. Adamson, Feb 14 2010

nonn

(1 + 3x + 5x^2 + 7x^3 + ...) = (1 + 3x + 8x^2 + 16x^3 + ...) / (1 + 3x^2 + 8x^4 + 16x^6 + ...).

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A005408, A152204.

A173283 := proc(n) option remember; if n = 0 then 1; else add(procname(l)*(2*n-4*l+1),l=0..n/2) ; end if; end proc: seq(A173283(n),n=0..60) ; # R. J. Mathar, Apr 01 2010

m = 40;
A[_] = 1;
Do[A[x_] = A[x^2] (1 + x)/(1 - x)^2 + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Feb 06 2020 *)

Given M = triangle A152204, odd numbers shifted down twice in every column > 0.
A173283 = lim_{n->inf} M^n, the left-shifted vector considered as a sequence.
a(n) = Sum_{t=0..n/2} (2*n - 4*t + 1)*a(t). - R. J. Mathar, Apr 01 2010

More terms from R. J. Mathar, Apr 01 2010

A227569 Decimal expansion of maximal value of function F[a(n); b(n)] for pairs of complements a(n) and b(n) of natural numbers A000027, where a(n) = odd numbers (A005408) and b(n) = even numbers (A005843); see Comments for the definition of function F[a(n); b(n)].

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Jul 16 2013

Apart from the first digit, the same as A143280. The sum of the reciprocals of the double factorial numbers, Sum_{n>=1} 1/n!! = Sum_{n>=2} n!!/n!. - Robert G. Wilson v, Jun 27 2015
Definition of function F[a(n); b(n)]: Let a(n) and b(n) is pair of complements of natural numbers (A000027) with a(1) < a(2) < a(3) < ... and b(1) < b(2) < b(3) < ..., then F[a(n); b(n)] = F[a(n)] + F[b(n)]; where F[a(n)] = 1/a(1) + 1/a(1)a(2) + 1/a(1)a(2)a(3) + ... and F[b(n)] = 1/b(1) + 1/b(1)b(2) + 1/b(1)b(2)b(3) + ...
Value of function F[a(n); b(n)] is real number c = a + b, where a = real number whose Engel expansion is sequence a(n) and b = real number whose Engel expansion is sequence b(n). See A006784 for definition of Engel expansion.
Example for a(n) = odd numbers (A005408) and b(n) = even numbers (A005843): c = 2.059407... = a + b, where a = 1.410686... (A060196) and b = 0.648721... (A019774 - 1).
Example for a(n) = nonprime numbers (A018252) and b(n) = primes (A000040): c = 2.002747... = a + b, where a = 1.297516... and b = 0.705230... (A064648).
Conjecture: there are no pairs of complements a(n) and b(n) such that F[a(n); b(n)] = 2.
e - 1 <= F[a(n); b(n)] <= sqrt(e) + sqrt((e*Pi)/2)*erf(1/sqrt(2)) - 1.
1.71828182... (A091131) <= F[a(n); b(n)] <= 2.05940740....

			2.05940740534257614453947549923327861297725472633534020929971877980544281968...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Engel Expansion
Googology Wiki, Double factorial

Cf. A000027, A005408, A005843, A091131 (e-1), A006882 (n!!), A143280 (m(2)).

SetDefaultRealField(RealField(112)); R:= RealField(); -1 + Exp(1/2)*(1 + Sqrt(Pi(R)/2)*Erf(1/Sqrt(2)) ); // G. C. Greubel, Apr 01 2019

RealDigits[Sqrt[E] -1 + Sqrt[E*Pi/2]*Erf[1/Sqrt[2]], 10, 105][[1]] (* or *)
RealDigits[Sum[1/n!!, {n, 125}], 10, 105][[1]] (* Robert G. Wilson v, Apr 09 2014 *)

default(realprecision, 100); exp(1/2) - 1 + sqrt(exp(1)*Pi/2)*(1-erfc(1/sqrt(2))) \\ G. C. Greubel, Apr 01 2019

numerical_approx(-1 + exp(1/2)*(1 + sqrt(pi/2)*erf(1/sqrt(2))), digits=112) # G. C. Greubel, Apr 01 2019

A025115 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A005408 (odd natural numbers), t = A023533.

Original entry on oeis.org

0, 0, 1, 3, 5, 0, 0, 0, 1, 3, 5, 7, 9, 11, 13, 15, 17, 0, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 32, 36, 40, 44, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A005408, A023533.

A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
A025115:= func< n | (&+[(2*k-1)*A023533(n+2-k): k in [1..Floor((n+1)/2)]]) >;
[A025115(n): n in [1..100]]; // G. C. Greubel, Sep 13 2022

b[j_]:= b[j]= Sum[KroneckerDelta[j, Binomial[m+2,3]], {m,0,15}];
A025115[n_]:= A025115[n]= Sum[(2*(n-j+2)-1)*b[j], {j, Floor[(n+4)/2], n+1}];
Table[A025115[n], {n,100}] (* G. C. Greubel, Sep 13 2022 *)

@CachedFunction
def b(j): return sum(bool(j==binomial(m+2,3)) for m in (0..10))
@CachedFunction
def A025115(n): return sum((2*(n-j+2)-1)*b(j) for j in (((n+4)//2)..n+1))
[A025115(n) for n in (1..100)] # G. C. Greubel, Sep 13 2022

cp's OEIS Frontend

A166101 Integers k such that A166100(k)/A005408(k) is not an integer.

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A130845 a(4n) = a(4n+1) = a(4n+2) = A001477(n), a(4n+3) = A005408(n).

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A131733 Primes (A000040) - odds (A005408).

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A166407 a(n) = floor(3*(A166406(n)/A005408(n))).

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A289786 p-INVERT of the odd positive integers (A005408), where p(S) = 1 - S - S^2.

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A113170 Ascending descending base exponent transform of odd numbers A005408.

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A169642 a(n) = A005408(n) * A022998(n).

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A173283 A(x) satisfies A005408(x) = A(x)/A(x^2), A005408 = odd numbers.

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Maple

Mathematica

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A227569 Decimal expansion of maximal value of function F[a(n); b(n)] for pairs of complements a(n) and b(n) of natural numbers A000027, where a(n) = odd numbers (A005408) and b(n) = even numbers (A005843); see Comments for the definition of function F[a(n); b(n)].

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A025115 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)*t(n-k+1), where k = floor(n/2), s = A005408 (odd natural numbers), t = A023533.