A152913 -id:A152913 - cp's OEIS frontend

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

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

Offset: 1

Jonathan Vos Post, Aug 18 2011

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

			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).

Vincenzo Librandi, Table of n, a(n) for n = 1..3000

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

[ a: n in [1..100] | IsPrime(a) where a is n^16+(n+1)^16 ]; // Vincenzo Librandi, Dec 07 2011

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 *)

A194155 Primes of the form k^8 + (k+1)^8.

Original entry on oeis.org

257, 2070241, 17995718017, 188386299457, 2505920246017, 3192202523137, 5072985298081, 11905609260481, 21370852274017, 766108283826337, 970961614082017, 2348771079002657, 2887223180589697, 9007197376151521, 55110306149736577, 77802445498340417

Offset: 1

Jonathan Vos Post, Aug 17 2011

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

			a(2) = 5^8 + (5+1)^8 = 2070241 is prime.
a(3) = 17^8 + (17+1)^8.
a(4) = 23^8 + (23+1)^8.
a(5) = 32^8 + (32+1)^8.
a(6) = 33^8 + (33+1)^8.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A000040, A153504, A152913.

[ a: n in [0..200] | IsPrime(a) where a is n^8+(n+1)^8 ];  // Vincenzo Librandi, Dec 07 2011

Select[Table[n^8+(n+1)^8,{n,0,900}],PrimeQ] (* Vincenzo Librandi, Dec 07 2011 *)

A008514 4-dimensional centered cube numbers.

Original entry on oeis.org

1, 17, 97, 337, 881, 1921, 3697, 6497, 10657, 16561, 24641, 35377, 49297, 66977, 89041, 116161, 149057, 188497, 235297, 290321, 354481, 428737, 514097, 611617, 722401, 847601, 988417, 1146097, 1321937, 1517281, 1733521, 1972097, 2234497, 2522257, 2836961, 3180241

Offset: 0

N. J. A. Sloane

Summation of n^4 taken two at a time. - Al Hakanson (hawkuu(AT)gmail.com), May 27 2009

The primes in this sequence are given by A152913. - Jonathan Vos Post, Aug 17 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002061, A047838, A152913.

List([0..30],n->n^4+(n+1)^4); # Muniru A Asiru, Aug 02 2018

[(n+1)^4+n^4: n in [0..30]]; // Vincenzo Librandi, Aug 27 2011

Maple
```
seq(n^4+(n+1)^4, n=0..40);
```

Total/@Partition[Range[0, 30]^4, 2, 1] (* or *) LinearRecurrence[{5,-10, 10,-5,1}, {1,17,97,337,881}, 30] (* Harvey P. Dale, Jan 28 2013 *)

a(n) = n^4 + (n+1)^4; \\ Altug Alkan, Aug 01 2018

[i^4+(i+1)^4 for i in range(0,36)] # Zerinvary Lajos, Jul 03 2008

a(n) = n^4 + (n+1)^4.
a(n) = 2*n^4 + 4*n^3 + 6*n^2 + 4*n + 1. - Al Hakanson (hawkuu(AT)gmail.com), May 27 2009, corrected R. J. Mathar, May 29 2009
G.f.: (1+10*x+x^2)*(1+x)^2/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 09 2009
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), with a(0) = 1, a(1) = 17, a(2) = 97, a(3) = 337, a(4) = 881. - Harvey P. Dale, Jan 28 2013
a(n) = 4*(n+n^2) + 2*(n+n^2)^2 + 1. - Avi Friedlich, Mar 31 2015
a(n) = 2*A002061(n+1)^2 - 1. - Bruce J. Nicholson, Apr 14 2017
a(n) = A047838(2*(n^2+n+1)). - David James Sycamore, Aug 01 2018
E.g.f.: (1 + 16*x + 32*x^2 + 16*x^3 + 2*x^4)*exp(x). - G. C. Greubel, Nov 09 2019
Sum_{n>=0} 1/a(n) = (tanh((sqrt(2)-1)*Pi/2)*Pi*(2+sqrt(2)) - tanh((sqrt(2)+1)*Pi/2)*Pi*(2-sqrt(2)))/4. - Amiram Eldar, Sep 20 2022

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

Original entry on oeis.org

3512911982806776822251393039617, 2211377674535255285545615254209921, 476961452964007550415682034114910337, 46677208572152524490331633250547044320123137

Offset: 1

Jonathan Vos Post, Aug 18 2011

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.

			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.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

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

[a: n in [1..200] | IsPrime(a) where a is n^32+(n+1)^32]; // Vincenzo Librandi, Dec 08 2011

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

N. J. A. Sloane

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

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

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

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

[(n+1)^7+n^7: n in [0..30]]; // Vincenzo Librandi, Aug 27 2011

a(n)=(n+1)^7+n^7 \\ Charles R Greathouse IV, Oct 07 2015

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

A078902 Generalized Fermat primes of the form (k+1)^2^m + k^2^m, with m>1.

Original entry on oeis.org

17, 97, 257, 337, 881, 3697, 10657, 16561, 49297, 65537, 66977, 89041, 149057, 847601, 988417, 1146097, 1972097, 2070241, 2522257, 2836961, 3553777, 3959297, 4398577, 5385761, 7166897, 11073217, 17653681, 32530177, 41532497, 44048497

Offset: 1

T. D. Noe, Dec 12 2002

nonn

For k=1, these are the Fermat primes A019434. Is the set of generalized Fermat primes infinite? Conjecture that there are only a finite number of generalized Fermat primes for each value of k. See A077659, which shows that in cases such as k=11, there appear to be no primes. See A078901 for generalized Fermat numbers.

See A080131 for the conjectured number of primes for each k. See A080208 for the least k such that (k+1)^2^n + k^2^n is prime. The largest probable prime of this form discovered to date is the 10217-digit 312^2^12 + 311^2^12.

T. D. Noe, Table of n, a(n) for n = 1..525 (terms < 10^14)
T. D. Noe, Table of generalized Fermat primes of the form (k+1)^2^m + k^2^m
Eric Weisstein's World of Mathematics, Generalized Fermat Number

Cf. A019434, A077659, A078900, A078901.

Cf. A080131, A080208, A019434, A078902, A080134, A153504, A152913, A194185.

lst3=Select[lst2, PrimeQ[ # ]&] (* lst2 is from A078901 *)

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

N. J. A. Sloane

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

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

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).

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

[(n+1)^9+n^9: n in [0..20]]; // Vincenzo Librandi, Aug 27 2011

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 *)

a(n)=(n+1)^9+n^9 \\ Charles R Greathouse IV, Jan 31 2017

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

N. J. A. Sloane

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

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

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

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

[(n+1)^10+n^10: n in [0..20]]; // Vincenzo Librandi, Aug 27 2011

Total/@Partition[Range[0,20]^10,2,1] (* Harvey P. Dale, Aug 04 2019 *)

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

N. J. A. Sloane

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

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

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

[(n+1)^11+n^11: n in [0..20]]; // Vincenzo Librandi, Aug 27 2011

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

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)

A080208 a(n) is the least k such that the generalized Fermat number (k+1)^(2^n) + k^(2^n) is prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 8, 95, 31, 85, 59, 1078, 754, 311, 3508, 1828, 49957, 22844

Offset: 0

T. D. Noe, Feb 10 2003

The first five terms correspond to the five known Fermat primes. The sequence A078902 lists some of the generalized Fermat primes. Bjorn and Riesel examined generalized Fermat numbers for k <= 11 and n <= 999. The sequence A080134 lists the conjectured number of primes for each k.

For n >= 10, a(n) yields a probable prime. a(13) was found by Henri Lifchitz. It is known that a(14) > 1000.

			a(5) = 8 because (k+1)^32 + k^32 is prime for k = 8 and composite for k < 8.

T. D. Noe, Table of generalized Fermat primes of the form (k+1)^2^m + k^2^m
Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Eric Weisstein's World of Mathematics, Generalized Fermat Number

Cf. A019434, A078902, A080134, A153504, A152913, A194185, A253633.

a(n) = A253633(n) - 1.

a(14)-a(15) from Jeppe Stig Nielsen, Nov 27 2020

a(16) by Kellen Shenton communicated by Jeppe Stig Nielsen, May 19 2023

cp's OEIS Frontend

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

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A194155 Primes of the form k^8 + (k+1)^8.

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A008514 4-dimensional centered cube numbers.

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

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

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A078902 Generalized Fermat primes of the form (k+1)^2^m + k^2^m, with m>1.

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

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