A001029 -id:A001029 - cp's OEIS frontend

A159653 Numerator of Hermite(n, 15/19).

1, 30, 178, -37980, -1524948, 63937800, 7423196280, -54282661200, -39145313835120, -860822763962400, 228541566381737760, 13071387347260660800, -1422935499785941465920, -155938564970244609148800, 8677515651883508324661120, 1836552484275737759015904000

Offset: 0

N. J. A. Sloane, Nov 12 2009

			Numerator of 1, 30/19, 178/361, -37980/6859, -1524948/130321, 63937800/2476099, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
DLMF Digital library of mathematical functions, Table 18.9.1 for H_n(x)

Cf. A001029 (denominators).

[Numerator((&+[(-1)^k*Factorial(n)*(30/19)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 11 2018

A159653 := proc(n)
        orthopoly[H](n,15/19) ;
        numer(%) ;
end proc: # R. J. Mathar, Feb 16 2014

Numerator[Table[HermiteH[n, 15/19], {n, 0, 30}]] (* Vladimir Joseph Stephan Orlovsky, Jun 16 2011 *)
Table[19^n*HermiteH[n, 15/19], {n,0,50}] (* G. C. Greubel, Jul 11 2018 *)

a(n)=numerator(polhermite(n,15/19)) \\ Charles R Greathouse IV, Jan 29 2016

D-finite with recurrence a(n) - 30*a(n-1) + 722*(n-1)*a(n-2) = 0. [DLMF] - R. J. Mathar, Feb 16 2014
From G. C. Greubel, Jul 11 2018: (Start)
a(n) = 19^n * Hermite(n, 15/19).
E.g.f.: exp(30*x - 361*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(30/19)^(n-2*k)/(k!*(n-2*k)!)). (End)

A159654 Numerator of Hermite(n, 16/19).

Original entry on oeis.org

1, 32, 302, -36544, -1823540, 47185792, 8092924744, 54564740864, -39155569948528, -1568144181583360, 204252279714867424, 17858073941907616768, -1050713239354433344832, -188345176292029458712576, 3834948823235768695790720, 2026511404303378366932021248

Offset: 0

N. J. A. Sloane, Nov 12 2009

			Numerator of 1, 32/19, 302/361, -36544/6859, -1823540/130321, 47185792/2476099, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
DLMF Digital library of mathematical functions, Table 18.9.1 for H_n(x)

Cf. A001029 (denominators).

[Numerator((&+[(-1)^k*Factorial(n)*(32/19)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 11 2018

A159654 := proc(n)
        orthopoly[H](n,16/19) ;
        numer(%) ;
end proc: # R. J. Mathar, Feb 16 2014

Numerator[Table[HermiteH[n, 16/19], {n, 0, 30}]] (* Vladimir Joseph Stephan Orlovsky, Jun 16 2011 *)
Table[19^n*HermiteH[n, 16/19], {n, 0, 50}] (* G. C. Greubel, Jul 11 2018 *)

a(n)=numerator(polhermite(n,16/19)) \\ Charles R Greathouse IV, Jan 29 2016

D-finite with recurrence a(n) - 32*a(n-1) + 722*(n-1)*a(n-2) = 0. [DLMF] - R. J. Mathar, Feb 16 2014
From G. C. Greubel, Jul 11 2018: (Start)
a(n) = 19^n * Hermite(n, 16/19).
E.g.f.: exp(32*x - 361*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(32/19)^(n-2*k)/(k!*(n-2*k)!)). (End)

A159655 Numerator of Hermite(n, 17/19).

Original entry on oeis.org

1, 34, 434, -34340, -2107604, 27515384, 8543973496, 171298455376, -37357094566000, -2259561093495776, 165921323311011616, 21955356087613897664, -571265042757181733696, -209644216596830988306560, -1766009672973345849952384, 2059039412479673870904327424

Offset: 0

N. J. A. Sloane, Nov 12 2009

			Numerator of 1, 34/19, 434/361, -34340/6859, -2107604/130321, 27515384/2476099,..

Vincenzo Librandi, Table of n, a(n) for n = 0..200
DLMF Digital library of mathematical functions, Table 18.9.1 for H_n(x)

Cf. A001029 (denominators).

[Numerator((&+[(-1)^k*Factorial(n)*(34/19)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 11 2018

A159655 := proc(n)
        orthopoly[H](n,17/19) ;
        numer(%) ;
end proc: # R. J. Mathar, Feb 16 2014

Numerator[Table[HermiteH[n, 17/19], {n, 0, 30}]] (* Vladimir Joseph Stephan Orlovsky, Jun 16 2011 *)
Table[19^n*HermiteH[n, 17/19], {n,0,50}] (* G. C. Greubel, Jul 11 2018 *)

a(n)=numerator(polhermite(n,17/19)) \\ Charles R Greathouse IV, Jan 29 2016

D-finite with recurrence a(n) -34*a(n-1) +722*(n-1)*a(n-2)=0. [DLMF] - R. J. Mathar, Feb 16 2014
From G. C. Greubel, Jul 11 2018: (Start)
a(n) = 19^n * Hermite(n, 17/19).
E.g.f.: exp(34*x - 361*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(34/19)^(n-2*k)/(k!*(n-2*k)!)). (End)

A159656 Numerator of Hermite(n, 18/19).

Original entry on oeis.org

1, 36, 574, -31320, -2370804, 5103216, 8742318216, 292616324064, -33649488597360, -2901533477298624, 114199171722894816, 25060241888120278656, -4801113850900597056, -217294775817306515769600, -7777548674818481563737984, 1916423841667868925104549376

Offset: 0

N. J. A. Sloane, Nov 12 2009

			Numerator of 1, 36/19, 574/361, -31320/6859, -2370804/130321, 5103216/2476099,...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
DLMF Digital library of mathematical functions, Table 18.9.1 for H_n(x)

Cf. A001029 (denominators).

[Numerator((&+[(-1)^k*Factorial(n)*(36/19)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 11 2018

A159656 := proc(n)
        orthopoly[H](n,18/19) ;
        numer(%) ;
end proc: # R. J. Mathar, Feb 16 2014

Numerator[Table[HermiteH[n, 18/19], {n, 0, 30}]] (* Vladimir Joseph Stephan Orlovsky, Jun 16 2011 *)
Table[19^n*HermiteH[n,18/19], {n,0,50}] (* G. C. Greubel, Jul 11 2018 *)

a(n)=numerator(polhermite(n,18/19)) \\ Charles R Greathouse IV, Jan 29 2016

D-finite with recurrence a(n) -36*a(n-1) +722*(n-1)*a(n-2)=0. [DLMF] - R. J. Mathar, Feb 16 2014
From G. C. Greubel, Jul 11 2018: (Start)
a(n) = 19^n * Hermite(n, 18/19).
E.g.f.: exp(36*x - 361*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(36/19)^(n-2*k)/(k!*(n-2*k)!)). (End)

A165840 Totally multiplicative sequence with a(p) = 19.

Original entry on oeis.org

1, 19, 19, 361, 19, 361, 19, 6859, 361, 361, 19, 6859, 19, 361, 361, 130321, 19, 6859, 19, 6859, 361, 361, 19, 130321, 361, 361, 6859, 6859, 19, 6859, 19, 2476099, 361, 361, 361, 130321, 19, 361, 361, 130321, 19, 6859, 19, 6859, 6859, 361, 19, 2476099, 361

Offset: 1

Jaroslav Krizek, Sep 28 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000

19^PrimeOmega[Range[100]] (* G. C. Greubel, Apr 09 2016 *)

a(n) = A001029(A001222(n)) = 19^bigomega(n) = 19^A001222(n).

A197353 a(0)=0, a(1)=1, a(2n)=19*a(n), a(2n+1)=a(2n)+1.

Original entry on oeis.org

0, 1, 19, 20, 361, 362, 380, 381, 6859, 6860, 6878, 6879, 7220, 7221, 7239, 7240, 130321, 130322, 130340, 130341, 130682, 130683, 130701, 130702, 137180, 137181, 137199, 137200, 137541, 137542, 137560, 137561, 2476099, 2476100, 2476118, 2476119

Offset: 0

Philippe Deléham, Oct 14 2011

Numbers whose set of base 19 digits is {0,1}.
Sums of distinct powers of 19.
a(n) modulo 2 is the Prouhet-Thue-Morse sequence A010060. - Philippe Deléham, Oct 17 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 45.

Cf. A001029, A010060, A104257, A030308, A197352.

[n: n in [0..2500000] | Set(IntegerToSequence(n, 19)) subset {0, 1}]; // Vincenzo Librandi, Jun 05 2012

FromDigits[#,19]&/@Tuples[{0,1},5] (* Vincenzo Librandi, Jun 05 2012 *)

a(n) = Sum_{k>=0} A030308(n,k)*19^k.

G.f.: (1/(1 - x))*Sum_{k>=0} 19^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

A359059 Numbers k such that phi(k) + rad(k) + psi(k) is a multiple of 3.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 11, 13, 17, 18, 19, 20, 23, 27, 29, 31, 32, 36, 37, 41, 42, 43, 44, 45, 47, 49, 50, 53, 54, 59, 61, 63, 67, 68, 71, 72, 73, 78, 79, 80, 81, 83, 84, 89, 90, 92, 97, 99, 101, 103, 105, 107, 108, 109, 110, 113, 114, 116, 117, 125, 126, 127, 128, 131, 135, 137, 139

Offset: 1

Torlach Rush, Dec 14 2022

nonn

When k is prime (denote as p), phi(p) = p - 1, rad(p) = p, and psi(p) = p + 1, so phi(p) + rad(p) + psi(p) = 3*p. Therefore, A000040 is a subsequence.

When k = p^m (m>=1) with p prime, phi(p^m) = (p-1)*p^(m-1), rad(p^m) = p, and psi(p^m) = (p+1)*p^(m-1), so phi(p^m) + rad(p^m) + psi(p^m) = 2*p^m + p = p * (1+2*p^(m-1)). Then, this expression is a multiple of 3 iff p == 0 or 1 (mod 3), equivalently iff p is a generalized cuban prime of A007645. Therefore, as 1 is also a term, every sequence {p^m, p in A007645, m>=0} is a subsequence. See crossrefs section. - Bernard Schott, Jan 25 2023 after an observation of Alois P. Heinz

			8 is a term because 4+2+12 is divisible by 3.

Cf. A000010 (phi), A000040, A001615 (psi), A007645, A007947 (rad), A001748 (3*p), A000244.

Subsequences of the form {p^n, n>=0}: A000244 (p=3), A000420 (p=7), A001022 (p=13), A001029 (p=19), A009975 (p=31), A009981 (p=37), A009987 (p=43).

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Divisible[Times @@ ((p - 1)*p^(e - 1)) + Times @@ p + Times @@ ((p + 1)*p^(e - 1)), 3]]; Select[Range[170], q] (* Amiram Eldar, Dec 15 2022 *)

isok(m) = ((eulerphi(m) + factorback(factorint(m)[, 1]) + m*sumdiv(m, d, moebius(d)^2/d)) % 3) == 0; \\ Michel Marcus, Dec 27 2022

from sympy.ntheory.factor_ import totient
from sympy import primefactors, prod
def rad(n): return 1 if n < 2 else prod(primefactors(n))
def psi(n):
    plist = primefactors(n)
    return n*prod(p+1 for p in plist)//prod(plist)
# Output display terms.
for n in range(1,170):
    if(0 == (totient(n) + rad(n) + psi(n)) % 3):
        print(n, end = ", ")

A231290 Numbers k dividing u(k), where the Lucas sequence is defined u(i) = u(i-1) - 5*u(i-2) with initial conditions u(0)=0, u(1)=1.

Original entry on oeis.org

1, 12, 19, 24, 36, 48, 72, 84, 96, 108, 144, 168, 192, 216, 228, 252, 288, 324, 336, 361, 384, 432, 456, 504, 576, 588, 648, 672, 684, 744, 756, 768, 816, 864, 912, 972, 1008, 1092, 1152, 1176, 1296, 1344, 1368, 1488, 1512, 1536, 1596, 1632, 1728, 1764

Offset: 0

Thomas M. Bridge, Nov 06 2013

nonn

Contains every nonnegative power of 19. All terms that are not a power of 19 are multiples of 12.

C. Smyth, The terms in Lucas sequences divisible by their indices, Journal of Integer Sequences, Vol.13 (2010), Article 10.2.4.

Cf. A001029 (powers of 19 (subsequence)).

nn = 3000; s = LinearRecurrence[{1, -5}, {1, 1}, nn]; t = {}; Do[If[Mod[s[[n]], n] == 0, AppendTo[t, n]], {n, nn}]; t (* T. D. Noe, Nov 08 2013 *)

cp's OEIS Frontend

A159653 Numerator of Hermite(n, 15/19).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A159654 Numerator of Hermite(n, 16/19).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A159655 Numerator of Hermite(n, 17/19).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A159656 Numerator of Hermite(n, 18/19).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A165840 Totally multiplicative sequence with a(p) = 19.

Views

Author

Keywords

Links

Programs

Mathematica

Formula

A197353 a(0)=0, a(1)=1, a(2n)=19*a(n), a(2n+1)=a(2n)+1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A359059 Numbers k such that phi(k) + rad(k) + psi(k) is a multiple of 3.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs