A258997 -id:A258997 - cp's OEIS frontend

A258851 The pi-based arithmetic derivative of n: a(p) = pi(p) for p prime, a(uv) = a(u)v + u*a(v), where pi = A000720.

0, 0, 1, 2, 4, 3, 7, 4, 12, 12, 11, 5, 20, 6, 15, 19, 32, 7, 33, 8, 32, 26, 21, 9, 52, 30, 25, 54, 44, 10, 53, 11, 80, 37, 31, 41, 84, 12, 35, 44, 84, 13, 73, 14, 64, 87, 41, 15, 128, 56, 85, 55, 76, 16, 135, 58, 116, 62, 49, 17, 136, 18, 53, 120, 192, 69, 107

Offset: 0

Alois P. Heinz, Jun 12 2015

The pi-based variant of the arithmetic derivative of n (A003415).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Wikipedia, Arithmetic derivative
Index entries for sequences computed from indices in prime factorization

Column k=1 of A258850, A258997.
First differences give A258863.
Partial sums give A258864.
Cf. A000720 (pi), A003415, A020639, A032742, A055396, A056239, A258861, A258862, A258995, A278510.

with(numtheory):
a:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
seq(a(n), n=0..100);

a[n_] := n*Total[Last[#]*PrimePi[First[#]]/First[#]& /@ FactorInteger[n]]; a[0] = 0; Array[a, 100, 0] (* Jean-François Alcover, Apr 24 2016 *)

A258851(n)=n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i]) \\ M. F. Hasler, Jul 13 2015

(define (A258851 n) (if (<= n 1) 0 (+ (* (A055396 n) (A032742 n)) (* (A020639 n) (A258851 (A032742 n)))))) ;; Antti Karttunen, Mar 07 2017

a(n) = n * Sum e_j*pi(p_j)/p_j for n = Product p_j^e_j with pi = A000720.
a(A258861(n)) = n; A258861 = pi-based antiderivative of n.
a(a(A258862(n))) = n; A258862 = second pi-based antiderivative of n.
a(a(a(A258995(n)))) = n; A258995 = third pi-based antiderivative of n.
a(0) = a(0*p) = a(0)*p + 0*a(p) = a(0)*p for all p => a(0) = 0.
a(p) = a(1*p) = a(1)*p + 1*a(p) = a(1)*p + a(p) for all p => a(1) = 0.
a(u^v) = v * u^(v-1) * a(u).

A018215 a(n) = n*4^n.

Original entry on oeis.org

0, 4, 32, 192, 1024, 5120, 24576, 114688, 524288, 2359296, 10485760, 46137344, 201326592, 872415232, 3758096384, 16106127360, 68719476736, 292057776128, 1236950581248, 5222680231936, 21990232555520, 92358976733184, 387028092977152, 1618481116086272

Offset: 0

N. J. A. Sloane, Peter Winkler (pw(AT)bell-labs.com)

Bisection of A001787. That is, a(n) = A001787(2*n). - Graeme McRae, Jul 12 2006

All numbers of the form n*4^n+(4^n-1)/3 have the property that they are sums of two squares and also their indices are the sum of two squares. This follows from the identity n*4^n+(4^n-1)/3 = 4*(4*(..(4*(4*n+1)+1)..)+1)+1. - Artur Jasinski, Nov 12 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (8,-16).

Cf. A000302 (4^n), A001787, A002450, A083679.

Row n=4 of A258997.

[n*4^n: n in [0..25]]; // Vincenzo Librandi, Jun 01 2011

Table[n 4^n,{n,0,20}] (* or *) LinearRecurrence[{8,-16},{0,4},30] (* Harvey P. Dale, Apr 22 2018 *)

a(n) = n<<(2*n) \\ David A. Corneth, Apr 22 2018

G.f.: 4*x/(1-4*x)^2.
E.g.f.: 4*x*exp(4*x).
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = log(4/3) = A083679.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(5/4). (End)

A212697 a(n) = 2n3^(n-1).

Original entry on oeis.org

2, 12, 54, 216, 810, 2916, 10206, 34992, 118098, 393660, 1299078, 4251528, 13817466, 44641044, 143489070, 459165024, 1463588514, 4649045868, 14721978582, 46490458680, 146444944842, 460255540932, 1443528742014, 4518872583696, 14121476824050, 44059007691036

Offset: 1

Stanislav Sykora, May 24 2012

Main transitions in systems of n particles with spin 1.
Consider the set S of all b^n numbers which have n digits in base b. Define as "main transition" a pair (x,y) of elements of S such that x and y differ in base b in only one digit which in y exceeds that in x by 1. This particular sequence a(n) gives the number of such transitions for the case b=3.
The terminology originates from quantum theory of coupled spin systems (such as in magnetic resonance) with n particles, each with spin S = (b-1)/2. Then the i-th digit's value in base b can be intended as a label for the b = 2S+1 quantum states of the i-th particle. The most intense main quantum transitions then correspond to the above definition. Due to continuity, the correspondence holds regardless of how strongly coupled are the particles among themselves.
a(n) is the number of functions from {1,2,...,n} into {1,2,3} with a specially designated element of the domain that is restricted to be mapped into {1,2}. Hence the e.g.f. is 2*x*exp(x)^3. - Geoffrey Critzer, Mar 01 2015
a(n) is the distance spectral radius of the dimension-regular generalized recursive circulant graph (commonly known as multiplicative circulant graph) of order 3^n. - John Rafael M. Antalan, Sep 25 2020

			n=2, b=3, S={00, 01, 02, 10, 11, 12, 20, 21, 22}, main transitions = {(00,01), (00,10), (01,02), (01,12), (02,12), (10,11), (10,20), (11,12), (11,21), (12,22), (20,21), (21,22)}, main transitions count = 12.

M. H. Levitt, Spin Dynamics, Basics of Nuclear Magnetic Resonance, 2nd Edition, John Wiley & Sons, 2007, Section 6 (Mathematical techniques).
J. A. Pople, W. G. Schneider, H. J. Bernstein, High-Resolution Nuclear Magnetic Resonance, McGraw-Hill, 1959, Chapter 6.

Stanislav Sykora, Table of n, a(n) for n = 1..100
John Rafael M. Antalan and Francis Joseph H. Campeña, Distance eigenvalues and forwarding indices of dimension-regular generalized recursive circulant graph of order power of two and three, arXiv:2009.11608[math.CO], 2020.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (6,-9).

Cf. A001787 (b = 2).
Cf. A212698, A212699, A212700, A212701, A212702, A212703, A212704 (b = 4, 5, 6, 7, 8, 9, 10).
Row n=3 of A258997.

List([1..30], n-> 2*3^(n-1)*n) # G. C. Greubel, Jun 08 2019

[2*3^(n-1)*n: n in [1..30]]; // G. C. Greubel, Jun 08 2019

A212697:=n->2*n*3^(n-1): seq(A212697(n), n=1..30); # Wesley Ivan Hurt, Mar 01 2015

Table[Sum[Binomial[n, j] j 2^j, {j, n}], {n, 30}] (* Geoffrey Critzer, Mar 01 2015 *)
Table[2*3^(n-1)*n, {n,30}] (* G. C. Greubel, Jun 08 2019 *)

mtrans(n,b) = n*(b-1)*b^(n-1);
for (n=1,100,write("b212697.txt",n," ",mtrans(n,3)))

[2*3^(n-1)*n for n in (1..30)] # G. C. Greubel, Jun 08 2019

a(n) = n*(b-1)*b^(n-1). For this sequence, set b=3.
From R. J. Mathar, Oct 15 2013: (Start)
G.f.: 2*x/(1-3*x)^2.
a(n) = 2*A027471(n+1). (End)
a(n) = A005843(n)*A000244(n-1). - Omar E. Pol, Jan 21 2014
a(n) = Sum_{j=1..n} binomial(n,j)*j*2^j. - Geoffrey Critzer, Mar 01 2015
E.g.f.: 2*x*exp(3*x). - G. C. Greubel, Jun 08 2019

A230539 a(n) = 3n2^(3*n-1).

Original entry on oeis.org

0, 12, 192, 2304, 24576, 245760, 2359296, 22020096, 201326592, 1811939328, 16106127360, 141733920768, 1236950581248, 10720238370816, 92358976733184, 791648371998720, 6755399441055744, 57420895248973824, 486388759756013568, 4107282860161892352

Offset: 0

Bruno Berselli, Oct 23 2013

Arithmetic derivative of 8^n: a(n) = A003415(8^n).

Sum of reciprocals of a(n), for n>0: (2/3)*log(8/7).

Bruno Berselli, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (16,-64).

Cf. A001018, A003415.
Cf. arithmetic derivative of k^n: A001787 (k=2), A027471 (k=3), A018215 (k=4), A053464 (k=5), A212700 (k=6), A027473 (k=7), this sequence, A230540 (k=9), A085708 (k=10), A081127 (k=11).
Row n=8 of A258997.

Magma
```
[3*n*2^(3*n-1): n in [0..20]];
```

A230539:=n->3*n*2^(3*n-1): seq(A230539(n), n=0..30); # Wesley Ivan Hurt, May 03 2017

Table[3 n 2^(3 n - 1), {n,0,20}]
LinearRecurrence[{16,-64},{0,12},20] (* Harvey P. Dale, Dec 25 2022 *)

a(n) = 3*n*2^(3*n-1); \\ Michel Marcus, Oct 23 2013

G.f.: 12*x/(1-8*x)^2.

a(n) = 12*A053539(n).

A258846 The pi-based arithmetic derivative of n^n.

Original entry on oeis.org

0, 0, 4, 54, 1024, 9375, 326592, 3294172, 201326592, 4649045868, 110000000000, 1426558353055, 178322008965120, 1817250639553518, 166680102383370240, 8319983917236328125, 590295810358705651712, 5790681833204357349239, 1298431466484785739988992

Offset: 0

Alois P. Heinz, Jun 12 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..380

Cf. A000312, A000720, A258851.

Main diagonal of A258997.

with(numtheory):
a:= n-> n^(n+1)*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
seq(a(n), n=0..20);

a[n_] := n^(n+1)*Sum[i[[2]]*PrimePi[i[[1]]]/i[[1]], {i, FactorInteger[n]}];
a[0] = 0; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 24 2017, translated from Maple *)

a(n) = A258851(A000312(n)).
a(n) = n^n * A258851(n).
a(n) = A258997(n,n).

cp's OEIS Frontend

A258851 The pi-based arithmetic derivative of n: a(p) = pi(p) for p prime, a(u*v) = a(u)*v + u*a(v), where pi = A000720.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Scheme

Formula

A018215 a(n) = n*4^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A212697 a(n) = 2*n*3^(n-1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A230539 a(n) = 3*n*2^(3*n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A258846 The pi-based arithmetic derivative of n^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A258851 The pi-based arithmetic derivative of n: a(p) = pi(p) for p prime, a(uv) = a(u)v + u*a(v), where pi = A000720.

A212697 a(n) = 2n3^(n-1).

A230539 a(n) = 3n2^(3*n-1).