A064487 -id:A064487 - cp's OEIS frontend

A037250 a(n) = n^2(n^2 + 1)(n-1).

0, 0, 20, 180, 816, 2600, 6660, 14700, 29120, 53136, 90900, 147620, 229680, 344760, 501956, 711900, 986880, 1340960, 1790100, 2352276, 3047600, 3898440, 4929540, 6168140, 7644096, 9390000, 11441300

Offset: 0

N. J. A. Sloane

Conjecture: satisfies a linear recurrence having signature (6, -15, 20, -15, 6, -1). - Harvey P. Dale, Jul 27 2019

This conjecture is true since for any series a(n) = P(n) (P polynomial in n of degree d) there is an o.g.f. Q(x)/(1-x)^(d+1). - Georg Fischer, Feb 17 2021

R. W. Carter, Simple Groups of Lie Type, Wiley 1972, Chap. 14.
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A064487, A064583.

[n^2*(n^2+1)*(n-1): n in [0..30]]; // Vincenzo Librandi, Sep 14 2011

seq(coeff(series(4*x^2*(x^3+9*x^2+15*x+5)/(x-1)^6, x, n+1),x,n), n = 0..30); # Georg Fischer, Feb 17 2021

Table[n^2 (n^2+1)(n-1),{n,0,30}] (* Harvey P. Dale, Jul 27 2019 *)

A064583 a(n) = n^4(n^4+1)(n^2-1).

Original entry on oeis.org

0, 0, 816, 53136, 986880, 9390000, 58831920, 276825696, 1057222656, 3444262560, 9900990000, 25724822640, 61490347776, 137047559376, 287786357040, 574098840000, 1095233372160, 2009042197056, 3559481173296, 6114129610320, 10214463840000, 16642143690480, 26505160063536

Offset: 0

N. J. A. Sloane, Oct 17 2001

nonn

R. W. Carter, Simple Groups of Lie Type, Wiley 1972, Chap. 14.
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.

Harry J. Smith, Table of n, a(n) for n=0,...,500
Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. A064487, A037250.

[n^4*(n^4+1)*(n^2-1): n in [0..25]]; // Vincenzo Librandi, Jun 20 2018

Table[n^4 (n^4+1)(n^2-1),{n,0,30}] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{0,0,816,53136,986880,9390000,58831920,276825696,1057222656,3444262560,9900990000},30] (* Harvey P. Dale, Aug 17 2015 *)
CoefficientList[Series[48 x^2 (17 + 920 x + 9318 x^2 + 27545 x^3 + 27545 x^4 + 9318 x^5 + 920 x^6 + 17 x^7)/(1 - x)^11, {x, 0, 33}], x] (* Vincenzo Librandi, Jun 20 2018 *)

a(n) = { n^4*(n^4 + 1)*(n^2 - 1) } \\ Harry J. Smith, Sep 18 2009

a(0)=0, a(1)=0, a(2)=816, a(3)=53136, a(4)=986880, a(5)=9390000, a(6)=58831920, a(7)=276825696, a(8)=1057222656, a(9)=3444262560, a(10)=9900990000, a(n)= 11*a(n-1)- 55*a(n-2)+ 165*a(n-3)- 330*a(n-4)+ 462*a(n-5)- 462*a(n-6)+ 330*a(n-7)- 165*a(n-8)+ 55*a(n-9)- 11*a(n-10)+ a(n-11). - Harvey P. Dale, Aug 17 2015

G.f.: 48*x^2*(17 + 920*x + 9318*x^2 + 27545*x^3 + 27545*x^4 + 9318*x^5 + 920*x^6 + 17*x^7)/(1-x)^11. - Vincenzo Librandi, Jun 20 2018

A257391 Numbers of the form 4^p(4^p+1)(2^p-1) with p an odd prime.

Original entry on oeis.org

29120, 32537600, 34093383680, 36011213418659840, 36888985097480437760, 38685331082014736871587840, 39614005699412557795646504960, 41538369916519054182462860998737920, 44601490313984496701256699111250939955118080, 45671926145323068271210017365594287580527984640

Offset: 1

Danny Rorabaugh, Apr 21 2015

5 divides (4^m+1) for odd m, so every term in this sequence is a multiple of 5 (A008587).
A064487(k) = 4^(2k+1)*(4^(2k+1)+1)*(2^(2k+1)-1), so this sequence is a subsequence of A064487.
Every non-solvable number (A056866) is divisible by 12 or 20. All non-solvable numbers not divisible by 12 (A008594) are divisible by a member of this sequence. In particular, every primitive non-solvable number (A257146) not divisible by 12 is in this sequence.
All terms are divisible by 320 and have at least 4 distinct prime factors. - Jianing Song, Apr 04 2022

See A056866 and A064487 for references.

Danny Rorabaugh, Table of n, a(n) for n = 1..100

Subsequence of A008587, A008602, A056866, and A064487.

Table[4^p (4^p+1)(2^p-1),{p,Prime[Range[2,20]]}] (* Harvey P. Dale, Jul 17 2024 *)

a(n)=my(p=prime(n+1)); 4^p*(4^p+1)*(2^p-1) \\ Charles R Greathouse IV, Apr 21 2015

[4^nth_prime(n)*(4^nth_prime(n)+1)*(2^nth_prime(n)-1) for n in range(2,12)]

a(n) = 4^p*(4^p+1)*(2^p-1) where p = A065091(n) = A000040(n+1).

cp's OEIS Frontend

A037250 a(n) = n^2(n^2 + 1)(n-1).

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A064583 a(n) = n^4(n^4+1)(n^2-1).

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A257391 Numbers of the form 4^p(4^p+1)(2^p-1) with p an odd prime.

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cp's OEIS Frontend

A037250 a(n) = n^2*(n^2 + 1)*(n-1).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

A064583 a(n) = n^4*(n^4+1)*(n^2-1).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A257391 Numbers of the form 4^p*(4^p+1)*(2^p-1) with p an odd prime.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A037250 a(n) = n^2(n^2 + 1)(n-1).

A064583 a(n) = n^4(n^4+1)(n^2-1).

A257391 Numbers of the form 4^p(4^p+1)(2^p-1) with p an odd prime.