A009975 -id:A009975 - cp's OEIS frontend

A160317 Numerator of Hermite(n, 19/31).

1, 38, -478, -164236, -3484820, 1130223208, 76437602104, -10129105154704, -1413297494585968, 102039816064461920, 28324733071797627424, -884865408030648260288, -632466392109110072889152, -3625187129327311294505344, 15665048162323786452017148800

Offset: 0

N. J. A. Sloane, Nov 12 2009

			Numerators of 1, 38/31, -478/961, -164236/29791, -3484820/923521, ...

G. C. Greubel, Table of n, a(n) for n = 0..368

Cf. A009975 (denominators).

[Numerator((&+[(-1)^k*Factorial(n)*(38/31)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Oct 04 2018

Numerator[HermiteH[Range[0,20],19/31]] (* Harvey P. Dale, Dec 26 2017 *)
Table[31^n*HermiteH[n, 19/31], {n, 0, 30}] (* G. C. Greubel, Oct 04 2018 *)

a(n)=numerator(polhermite(n, 19/31)) \\ Charles R Greathouse IV, Jan 29 2016

x='x+O('x^30); Vec(serlaplace(exp(38*x - 961*x^2))) \\ G. C. Greubel, Oct 04 2018

From G. C. Greubel, Oct 04 2018: (Start)
a(n) = 31^n * Hermite(n, 19/31).
a(n+2) = 38*a(n+1) - 1922*(n+1)*a(n)
E.g.f.: exp(38*x - 961*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(38/31)^(n-2*k)/(k!*(n-2*k)!)). (End)

A160328 Numerator of Hermite(n, 20/31).

Original entry on oeis.org

1, 40, -322, -166640, -4808948, 1088770400, 89764806280, -8965108001600, -1566300023755120, 75195499682396800, 30101677798211937760, -241190391967188985600, -646057287688484347545920, -20279476307208127137958400, 15331208337896144822264021120

Offset: 0

N. J. A. Sloane, Nov 12 2009

			Numerators of 1, 40/31, -322/961, -166640/29791, -4808948/923521, ...

G. C. Greubel, Table of n, a(n) for n = 0..368

Cf. A009975 (denominators).

[Numerator((&+[(-1)^k*Factorial(n)*(40/31)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Oct 04 2018

Table[31^n*HermiteH[n, 20/31], {n, 0, 30}] (* G. C. Greubel, Oct 04 2018 *)

a(n)=numerator(polhermite(n, 20/31)) \\ Charles R Greathouse IV, Jan 29 2016

x='x+O('x^30); Vec(serlaplace(exp(40*x - 961*x^2))) \\ G. C. Greubel, Oct 04 2018

From G. C. Greubel, Oct 04 2018: (Start)
a(n) = 31^n * Hermite(n, 20/31).
a(n+2) = 40*a(n+1) - 1922*(n+1)*a(n)
E.g.f.: exp(40*x - 961*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(40/31)^(n-2*k)/(k!*(n-2*k)!)). (End)

A160334 Numerator of Hermite(n, 23/31).

Original entry on oeis.org

1, 46, 194, -167900, -8842004, 884083016, 125639477176, -4415829390416, -1893481677885040, -19202364475675424, 31870137298174352416, 1835095760938501860416, -589384037754831073199936, -69436314367007836275831680, 11532279106459848726285343616

Offset: 0

N. J. A. Sloane, Nov 12 2009

			Numerators of 1, 46/31, 194/961, -167900/29791, -8842004/923521, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A009975 (denominators).

[Numerator((&+[(-1)^k*Factorial(n)*(46/31)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 12 2018

Numerator[HermiteH[Range[0,20],23/31]] (* Harvey P. Dale, Aug 10 2014 *)
Table[31^n*HermiteH[n, 23/31], {n,0,30}] (* G. C. Greubel, Jul 12 2018 *)

a(n)=numerator(polhermite(n,23/31)) \\ Charles R Greathouse IV, Jan 29 2016

From G. C. Greubel, Jul 12 2018: (Start)
a(n) = 31^n * Hermite(n, 23/31).
E.g.f.: exp(46*x - 961*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(46/31)^(n-2*k)/(k!*(n-2*k)!)). (End)

A160344 Numerator of Hermite(n, 26/31).

Original entry on oeis.org

1, 52, 782, -159224, -12788660, 559103792, 151972419784, 1454980899424, -1968977929003888, -124758638617745600, 27571931007786483424, 3831601446637967570048, -383682490141447518907712, -108323545252613355018788096, 3953866345538313246451111040

Offset: 0

N. J. A. Sloane, Nov 12 2009

			Numerators of 1, 52/31, 782/961, -159224/29791, -12788660/923521, ...

Harvey P. Dale, Table of n, a(n) for n = 0..369

Cf. A009975 (denominators).

[Numerator((&+[(-1)^k*Factorial(n)*(52/31)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jul 12 2018

Numerator[HermiteH[Range[0,20],26/31]] (* Harvey P. Dale, Jan 26 2016 *)
Table[31^n*HermiteH[n, 26/31], {n,0,30}] (* G. C. Greubel, Jul 12 2018 *)

a(n)=numerator(polhermite(n,26/31)) \\ Charles R Greathouse IV, Jan 29 2016

From G. C. Greubel, Jul 12 2018: (Start)
a(n) = 31^n * Hermite(n, 26/31).
E.g.f.: exp(52*x - 961*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(52/31)^(n-2*k)/(k!*(n-2*k)!)). (End)

A165852 Totally multiplicative sequence with a(p) = 31.

Original entry on oeis.org

1, 31, 31, 961, 31, 961, 31, 29791, 961, 961, 31, 29791, 31, 961, 961, 923521, 31, 29791, 31, 29791, 961, 961, 31, 923521, 961, 961, 29791, 29791, 31, 29791, 31, 28629151, 961, 961, 961, 923521, 31, 961, 961, 923521, 31, 29791, 31, 29791, 29791

Offset: 1

Jaroslav Krizek, Sep 28 2009

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

Cf. A001222, A009975.

[n eq 1 select 1 else 31^(&+[p[2]: p in Factorization(n)]): n in [1..100]]; // Vincenzo Librandi, Apr 14 2016

31^PrimeOmega[Range[100]] (* G. C. Greubel, Apr 13 2016 *)

a(n) = 31^bigomega(n); \\ Michel Marcus, Apr 14 2016

a(n) = A009975(A001222(n)) = 31^bigomega(n) = 31^A001222(n).

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 = ", ")

A160329 Numerator of Hermite(n, 21/31).

Original entry on oeis.org

1, 42, -158, -168084, -6148500, 1033992792, 102514782264, -7618384022256, -1699206009514608, 45773620326594720, 31315357606300667424, 435476036787477513408, -643779296967334655115072, -37082549785094436884075136, 14528002423051857343574033280

Offset: 0

N. J. A. Sloane, Nov 12 2009

			Numerators of 1, 42/31, -158/961, -168084/29791, -6148500/923521,...

Cf. A009975 (denominators).

Numerator[HermiteH[Range[0,30],21/31]] (* Harvey P. Dale, May 07 2013 *)

a(n)=numerator(polhermite(n, 21/31)) \\ Charles R Greathouse IV, Jan 29 2016

A160330 Numerator of Hermite(n, 22/31).

Original entry on oeis.org

1, 44, 14, -168520, -7495604, 965775184, 114526862536, -6098137470304, -1809162457252720, 14161813624274624, 31918011985025634016, 1132202469482569623424, -624993700730178890935616, -53612840588273856995818240, 13257127620560200061101298816

Offset: 0

N. J. A. Sloane, Nov 12 2009

			Numerators of 1, 44/31, 14/961, -168520/29791, -7495604/923521,...

Cf. A009975 (denominators).

HermiteH[Range[0,20],22/31]//Numerator (* Harvey P. Dale, Oct 02 2022 *)

a(n)=numerator(polhermite(n, 22/31)) \\ Charles R Greathouse IV, Jan 29 2016

A160335 Numerator of Hermite(n, 24/31).

Original entry on oeis.org

1, 48, 382, -166176, -10179060, 788966208, 135691144584, -2585183370624, -1949677461023088, -53834738622393600, 31141453266902483424, 2529493433133724196352, -536976920178433543125312, -84115128710361024934677504, 9379379149481011311664525440

Offset: 0

N. J. A. Sloane, Nov 12 2009

			Numerators of 1, 48/31, 382/961, -166176/29791, -10179060/923521,...

Cf. A009975 (denominators).

a(n)=numerator(polhermite(n, 24/31)) \\ Charles R Greathouse IV, Jan 29 2016

A160336 Numerator of Hermite(n, 25/31).

Original entry on oeis.org

1, 50, 578, -163300, -11497748, 680563000, 144521508280, -622177102000, -1975501227499120, -89208466254604000, 29711796920549577760, 3200176567440967768000, -468157982122210784601920, -97216771457569019831248000, 6836556768427107672501173120

Offset: 0

N. J. A. Sloane, Nov 12 2009

			Numerators of 1, 50/31, 578/961, -163300/29791, -11497748/923521,...

Cf. A009975 (denominators).

a(n)=numerator(polhermite(n, 25/31)) \\ Charles R Greathouse IV, Jan 29 2016

cp's OEIS Frontend

A160317 Numerator of Hermite(n, 19/31).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A160328 Numerator of Hermite(n, 20/31).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A160334 Numerator of Hermite(n, 23/31).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A160344 Numerator of Hermite(n, 26/31).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A165852 Totally multiplicative sequence with a(p) = 31.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Python

A160329 Numerator of Hermite(n, 21/31).

Views

Author

Keywords

Examples

Crossrefs

Programs