cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 11 results. Next

A194185 Primes of the form k^16 + (k+1)^16.

Original entry on oeis.org

65537, 4338014017, 2973697798081, 36054040477057, 314707907280257, 8746361693522261761, 4441930186581050471617, 1936348941361814438534657, 8260002645666200230661441, 157512780598351804823277697, 684655198104511486296198721, 21770695412796292350304592257
Offset: 1

Views

Author

Jonathan Vos Post, Aug 18 2011

Keywords

Comments

Prime 16-dimensional centered cube numbers. This is to dimension 16 as A194155 is to dimension 8 and as A152913 is to dimension 4.

Examples

			a(1) = 1^16 + (1+1)^16 = 65537 = A100266(2).
a(2) = 3^16 + (3+1)^16 = 4338014017 = A100266(3).
a(3) = 5^16 + (5+1)^16 = 2973697798081 = A100266(4).
a(4) = 6^16 + (6+1)^16 = 36054040477057 = A100266(5).
a(5) = 7^16 + (7+1)^16 = 314707907280257 = A100266(6).
a(6) = 14^16 + (14+1)^16 = 8746361693522261761 = A100266(11).
a(7) = 21^16 + (21+1)^16 = 4441930186581050471617 = A100266(22).

Links

Crossrefs

Cf. A000040, A154535, A100266, A152913, A194155.

Programs

  • Magma
    [ a: n in [1..100] | IsPrime(a) where a is n^16+(n+1)^16 ]; // Vincenzo Librandi, Dec 07 2011
  • Mathematica
    Select[Table[n^16+(n+1)^16,{n,0,800}],PrimeQ] (* Vincenzo Librandi, Dec 07 2011 *)
Select[Total/@Partition[Range[60]^16,2,1],PrimeQ] (* Harvey P. Dale, Dec 07 2017 *)

A194216 Primes of the form k^32 + (k+1)^32.

Original entry on oeis.org

3512911982806776822251393039617, 2211377674535255285545615254209921, 476961452964007550415682034114910337, 46677208572152524490331633250547044320123137
Offset: 1

Views

Author

Jonathan Vos Post, Aug 18 2011

Keywords

Comments

Prime 32-dimensional centered cube numbers. This is to dimension 32 as A194185 is to dimension 16; as A194155 is to dimension 8; and as A152913 is to dimension 4.

Examples

			a(1) = 8^32 + (8 + 1)^32 = A100267(2).
a(2) = 10^32 + (10 + 1)^32 = A100267(3) = A176935(2).
a(3) = 12^32 + (12 + 1)^32 = A100267(4).
a(4) = 22^32 + (22 + 1)^32.

Links

Crossrefs

Cf. A000040, A100267, A154535, A100266, A152913, A194155, A194185.

Programs

  • Magma
    [a: n in [1..200] | IsPrime(a) where a is n^32+(n+1)^32]; // Vincenzo Librandi, Dec 08 2011
  • Mathematica
    Select[Table[n^32+(n+1)^32,{n,1,3000}],PrimeQ] (* Vincenzo Librandi, Dec 08 2011 *)

A036085 Centered cube numbers: (n+1)^7 + n^7.

Original entry on oeis.org

1, 129, 2315, 18571, 94509, 358061, 1103479, 2920695, 6880121, 14782969, 29487171, 55318979, 98580325, 168162021, 276272879, 439294831, 678774129, 1022558705, 1506091771, 2173871739, 3081088541, 4295446429, 5899183335, 7991296871, 10689987049, 14135325801
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Never prime, as a(n) = (2n+1)*(n^6 + 3n^5 + 9n^4 + 13n^3 + 11n^2 + 5n + 1). Semiprimes in the sequence begin for n = 1, 2, 8, 9, 21, 30, 33, 53, 65, 81, 83. - Jonathan Vos Post, Aug 26 2011

References

  • B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.

Links

Crossrefs

Cf. A100267, A154535, A100266, A152913, A194155, A194185, A194216.

Programs

Formula

a(n) = A001015(n+1) + A001015(n).
G.f.: (1+x)*(x^6 + 120*x^5 + 1191*x^4 + 2416*x^3 + 1191*x^2 + 120*x + 1) / (x-1)^8. - R. J. Mathar, Aug 27 2011

A036087 Centered cube numbers: a(n) = (n+1)^9 + n^9.

Original entry on oeis.org

1, 513, 20195, 281827, 2215269, 12030821, 50431303, 174571335, 521638217, 1387420489, 3357947691, 7517728043, 15764279725, 31265546157, 59104406159, 107162836111, 187307353233, 316947166865
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Never prime nor semiprime, as a(n) = (2n+1) * (n^2 + n + 1) * (n^6 + 3n^5 + 12n^4 + 19n^3 + 15n^2 + 6n + 1). - Jonathan Vos Post, Aug 26 2011
Triprimes (A014612) if n = 2, 5, 6, 14, 21, 75, 90, ... - R. J. Mathar, Aug 27 2011

References

  • B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.

Links

Crossrefs

Cf. A036085, A100267, A154535, A100266, A152913, A194155, A194185, A194216.

Programs

  • Magma
    [(n+1)^9+n^9: n in [0..20]]; // Vincenzo Librandi, Aug 27 2011
  • Mathematica
    Total/@Partition[Range[0,20]^9,2,1] (* Harvey P. Dale, Jan 31 2015 *)
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,513,20195,281827,2215269,12030821,50431303,174571335,521638217,1387420489},20] (* Harvey P. Dale, Jan 21 2023 *)
  • PARI
    a(n)=(n+1)^9+n^9 \\ Charles R Greathouse IV, Jan 31 2017

Formula

a(n) = A001017(n+1) + A001017(n).
G.f.: (1+x)*(x^8 + 502*x^7 + 14608*x^6 + 88234*x^5 + 156190*x^4 + 88234*x^3 + 14608*x^2 + 502*x + 1) / (x-1)^10. - R. J. Mathar, Aug 27 2011

A036088 Centered cube numbers: (n+1)^10 + n^10.

Original entry on oeis.org

1, 1025, 60073, 1107625, 10814201, 70231801, 342941425, 1356217073, 4560526225, 13486784401, 35937424601, 87854788825, 199775856073, 427113146825, 865905045601, 1676162018401, 3115505528225
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Never prime, as a(n) = (2n^2 + 2n + 1) * (n^8 + 4n^7 + 18n^6 + 40n^5 + 56n^4 + 50n^3 + 27n^2 + 8n + 1), multiple of A001844(n). Semiprime for n in {2, 4, 7, 14, 19, 22, 32, 60, 65, 70, 87, 99, 102, 135, 137, ...}. - Jonathan Vos Post, Aug 26 2011

References

  • B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.

Links

Crossrefs

Cf. A036085, A036087, A100267, A154535, A100266, A152913, A194155, A194185, A194216.

Programs

  • Magma
    [(n+1)^10+n^10: n in [0..20]]; // Vincenzo Librandi, Aug 27 2011
  • Mathematica
    Total/@Partition[Range[0,20]^10,2,1] (* Harvey P. Dale, Aug 04 2019 *)

Formula

G.f.: -(x^8 + 1012*x^7 + 46828*x^6 + 408364*x^5 + 901990*x^4 + 408364*x^3 + 46828*x^2 + 1012*x + 1)*(1+x)^2 / (x-1)^11. - R. J. Mathar, Aug 27 2011

A036089 Centered cube numbers: (n+1)^11 + n^11.

Original entry on oeis.org

1, 2049, 179195, 4371451, 53022429, 411625181, 2340123799, 10567261335, 39970994201, 131381059609, 385311670611, 1028320041299, 2535168764725, 5841725563701, 12699321029039, 26241941903791, 51864082352049
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Never prime, as a(n) = (2*n+1) * (n^10 + 5*n^9 + 25*n^8 + 70*n^7 + 130*n^6 + 166*n^5 + 148*n^4 + 91*n^3 + 37*n^2 + 9*n + 1). - Jonathan Vos Post, Aug 26 2011

Links

Crossrefs

Cf. A008455, A036087, A036088, A100267, A154535, A100266, A152913, A194155, A194185, A194216.

Programs

  • Magma
    [(n+1)^11+n^11: n in [0..20]]; // Vincenzo Librandi, Aug 27 2011
  • PARI
    Vec((1 + x)*(1 + 2036*x + 152637*x^2 + 2203488*x^3 + 9738114*x^4 + 15724248*x^5 + 9738114*x^6 + 2203488*x^7 + 152637*x^8 + 2036*x^9 + x^10) / (1 - x)^12 + O(x^40)) \\ Colin Barker, Feb 06 2020

Formula

From Colin Barker, Feb 06 2020: (Start)
G.f.: (1 + x)*(1 + 2036*x + 152637*x^2 + 2203488*x^3 + 9738114*x^4 + 15724248*x^5 + 9738114*x^6 + 2203488*x^7 + 152637*x^8 + 2036*x^9 + x^10) / (1 - x)^12.
a(n) = 12*a(n-1) - 66*a(n-2) + 220*a(n-3) - 495*a(n-4) + 792*a(n-5) - 924*a(n-6) + 792*a(n-7) - 495*a(n-8) + 220*a(n-9) - 66*a(n-10) + 12*a(n-11) - a(n-12) for n>11.
(End)

A036090 Centered cube numbers: (n+1)^12 + n^12.

Original entry on oeis.org

1, 4097, 535537, 17308657, 260917841, 2420922961, 16018069537, 82560763937, 351149013217, 1282429536481, 4138428376721, 12054528824977, 32214185570737, 79991997497777, 186440250265921, 411221314601281
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Never prime, as a(n) = (2n^4 + 4n^3 + 6n^2 + 4n + 1) * (n^8 + 4n^7 + 22n^6 + 52n^5 + 69n^4 + 56n^3 + 28n^2 + 8n + 1) Semiprime for n in {1, 2, 3, 6, 14, 16, 36, 87, 97, 109, 110, 119, 121, 163, 195, ...}. - Jonathan Vos Post, Aug 26 2011

References

  • B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.

Links

Crossrefs

Cf. A008456, A036088, A036089, A100267, A154535, A100266, A152913, A194155, A194185, A194216.

Programs

  • Magma
    [(n+1)^12+n^12: n in [0..20]]; // Vincenzo Librandi, Aug 27 2011
  • Mathematica
    Total/@Partition[Range[0,20]^12,2,1] (* Harvey P. Dale, May 09 2018 *)

Formula

G.f.: -(x^10 + 4082*x^9 + 474189*x^8 + 9713496*x^7 + 56604978*x^6 + 105907308*x^5 + 56604978*x^4 + 9713496*x^3 + 474189*x^2 + 4082*x + 1)*(1+x)^2 / (x-1)^13. - R. J. Mathar, Aug 27 2011

A036092 Centered cube numbers: a(n) = (n+1)^14 + n^14.

Original entry on oeis.org

1, 16385, 4799353, 273218425, 6371951081, 84467679721, 756587236945, 5076269583953, 27274838966065, 122876792454961, 479749833583241, 1663668298132105, 5221294850248153, 15049383211257305, 40304932850948641, 101250520063318561, 240435420597328865
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Never prime, as a(n) = (2n^2 + 2n + 1) * (n^12 + 6n^11 + 39n^10 + 140n^9 + 341n^8 + 590n^7 + 741n^6 + 680n^5 + 451n^4 + 210n^3 + 65n^2 + 12n + 1). Semiprime for n in {2, 5, 22, 24, 34, 35, 39, 84, 217, 220, 285, ...}. - Jonathan Vos Post, Aug 26 2011

Links

Crossrefs

Cf. A000040, A036085, A036087, A036088, A036089, A036090, A036091, A100267, A154535, A100266, A152913, A194155, A194185, A194216.

Programs

Formula

G.f.: -(x +1)^2*(x^12 +16368*x^11 +4520946*x^10 +193889840*x^9 +2377852335*x^8 +10465410528*x^7 +17505765564*x^6 +10465410528*x^5 +2377852335*x^4 +193889840*x^3 +4520946*x^2 +16368*x +1) / (x -1)^15. - Colin Barker, Feb 16 2015

A194553 Centered cube numbers: (n+1)^25 + n^25.

Original entry on oeis.org

1, 33554433, 847322163875, 1126747195452067, 299149123783795749, 28728311253806654501, 1369498907693894602183, 39120000482621126610375, 755676919554809750479817, 10717897987691852588770249, 118347059433883722041830251
Offset: 0

Views

Author

Jonathan Vos Post, Aug 28 2011

Keywords

Comments

Can never be prime as a(n) = (2*n+1) * (n^4 + 2*n^3 + 4*n^2 + 3*n+1) * (n^20 + 10*n^19 + 120*n^18 + 795*n^17 + 3685*n^16 + 12752*n^15 + 33965*n^14 + 71205*n^13 + 119580*n^12 + 162965*n^11 + 181754*n^10 + 166595*n^9 + 125515*n^8 +77415*n^7 + 38745*n^6 + 15503*n^5 + 4845*n^4 + 1140*n^3 + 190*n^2 + 20*n + 1).

Links

Crossrefs

Cf. A000040, A036085-A036102, A100267, A154535, A100266, A152913, A194155, A194185, A194216.

Programs

  • Magma
    [(n+1)^25+n^25: n in [0..10]]; // Vincenzo Librandi, Sep 21 2011
  • Mathematica
    Total/@Partition[Range[0,20]^25,2,1] (* Harvey P. Dale, Dec 03 2015 *)

A036095 Centered cube numbers: a(n) = (n+1)^17 + n^17.

Original entry on oeis.org

1, 131073, 129271235, 17309009347, 780119322309, 17689598897861, 249557173431943, 2484430327672455, 18928981513351817, 116677181699666569, 605447028499293771, 2724058135239730763, 10869027026121774925
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Never prime, as a(n) = (2n + 1) * (n^16 + 8n^15 + 64n^14 + 308n^13 + 1036n^12 + 2576n^11 + 4900n^10 + 7274n^9 + 8518n^8 + 7896n^7 + 5776n^6 + 3300n^5 + 1444n^4 + 468n^3 + 106n^2 + 15n + 1). Semiprime for n in {1, 5, 21, 29, 33, ...}. - Jonathan Vos Post, Aug 27 2011

References

  • B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.

Links

Crossrefs

Cf. A000040, A036085, A036087, A036088, A036089, A036090, A036091, A036092, A036093, A100267, A154535, A100266, A152913, A194155, A194185, A194216.

Programs

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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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