A001580 -id:A001580 - cp's OEIS frontend

A061119 Primes in the sequence n^2 + 2^n (A001580).

3, 17, 593, 32993, 2097593, 8589935681

Offset: 1

Amarnath Murthy, Apr 21 2001

nonn

p and p^2 + 2^p are both prime only for p=3. All positive n satisfy the congruence n=3 (mod 6). - Lekraj Beedassy, Sep 07 2004
For values of n, see A064539. - Lekraj Beedassy, Jan 01 2007
The next term has 605 digits. - Harvey P. Dale, Jul 19 2017

			a(3) = 593 = 2^9 + 9^2.
a(4) = 32993 = 2^15 + 15^2.

J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 165 pp. 30; 160, Ellipses Paris 2004.

Subsequence of A094133.

Cf. A001580, A064539, A075896.

Select[Table[n^2+2^n,{n,1000}],PrimeQ] (* Harvey P. Dale, Jul 19 2017 *)

for(n=1,10^7, if(isprime(n^2+2^n),print(n^2+2^n)))

a(n) = A001580(A064539(n)). - Elmo R. Oliveira, Feb 18 2025

More terms from Jason Earls, Aug 09 2001. Next term too large to include.

A174121 Partial sums of A001580.

Original entry on oeis.org

1, 4, 12, 29, 61, 118, 218, 395, 715, 1308, 2432, 4601, 8841, 17202, 33782, 66775, 132567, 263928, 526396, 1051045, 2100021, 4197614, 8392402, 16781539, 33559331, 67114388, 134223928, 268442385, 536878625, 1073750378, 2147493102, 4294977711, 8589946031

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 08 2010

A001580 2^n+n^2 -> 1,3,8,17,32,57,100,177,320,593,1124,..

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-9,2).

Cf. A174120.

f[n_]:=Sum[2^i+i^2,{i,0,n}];Table[f[n],{n,0,5!}]
Accumulate[Table[2^n+n^2,{n,0,50}]] (* or *) LinearRecurrence[{6,-14,16,-9,2},{1,4,12,29,61},50] (* Harvey P. Dale, Sep 23 2019 *)

Vec(x*(1-2*x+2*x^2-3*x^3)/((1-x)^4*(1-2*x)) + O(x^40)) \\ Colin Barker, Feb 26 2016

From Colin Barker, Feb 26 2016: (Start)
a(n) = (n-2)*(2*n^2+n+3)/6+2^n.
a(n) = 6*a(n-1)-14*a(n-2)+16*a(n-3)-9*a(n-4)+2*a(n-5) for n>5.
G.f.: x*(1-2*x+2*x^2-3*x^3) / ((1-x)^4*(1-2*x)).
(End)

A007310 Numbers congruent to 1 or 5 mod 6.

Original entry on oeis.org

1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 125, 127, 131, 133, 137, 139, 143, 145, 149, 151, 155, 157, 161, 163, 167, 169, 173, 175

Offset: 1

C. Christofferson (Magpie56(AT)aol.com)

Numbers n such that phi(4n) = phi(3n). - Benoit Cloitre, Aug 06 2003
Or, numbers relatively prime to 2 and 3, or coprime to 6, or having only prime factors >= 5; also known as 5-rough numbers. (Edited by M. F. Hasler, Nov 01 2014: merged with comments from Zak Seidov, Apr 26 2007 and Michael B. Porter, Oct 09 2009)
Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 38 ).
Numbers k such that k mod 2 = 1 and (k+1) mod 3 <> 1. - Klaus Brockhaus, Jun 15 2004
Also numbers n such that the sum of the squares of the first n integers is divisible by n, or A000330(n) = n*(n+1)*(2*n+1)/6 is divisible by n. - Alexander Adamchuk, Jan 04 2007
Numbers n such that the sum of squares of n consecutive integers is divisible by n, because A000330(m+n) - A000330(m) = n*(n+1)*(2*n+1)/6 + n*(m^2+n*m+m) is divisible by n independent of m. - Kaupo Palo, Dec 10 2016
A126759(a(n)) = n + 1. - Reinhard Zumkeller, Jun 16 2008
Terms of this sequence (starting from the second term) are equal to the result of the expression sqrt(4!*(k+1) + 1) - but only when this expression yields integral values (that is when the parameter k takes values, which are terms of A144065). - Alexander R. Povolotsky, Sep 09 2008
For n > 1: a(n) is prime if and only if A075743(n-2) = 1; a(2*n-1) = A016969(n-1), a(2*n) = A016921(n-1). - Reinhard Zumkeller, Oct 02 2008
A156543 is a subsequence. - Reinhard Zumkeller, Feb 10 2009
Numbers n such that ChebyshevT(x, x/2) is not an integer (is integer/2). - Artur Jasinski, Feb 13 2010
If 12*k + 1 is a perfect square (k = 0, 2, 4, 10, 14, 24, 30, 44, ... = A152749) then the square root of 12*k + 1 = a(n). - Gary Detlefs, Feb 22 2010
A089128(a(n)) = 1. Complement of A047229(n+1) for n >= 1. See A164576 for corresponding values A175485(a(n)). - Jaroslav Krizek, May 28 2010
Cf. property described by Gary Detlefs in A113801 and in Comment: more generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (with h, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 6). Also a(n)^2 - 1 == 0 (mod 12). - Bruno Berselli, Nov 05 2010 - Nov 17 2010
Numbers n such that ( Sum_{k = 1..n} k^14 ) mod n = 0. (Conjectured) - Gary Detlefs, Dec 27 2011
From Peter Bala, May 02 2018: (Start)
The above conjecture is true. Apply Ireland and Rosen, Proposition 15.2.2. with m = 14 to obtain the congruence 6*( Sum_{k = 1..n} k^14 )/n = 7 (mod n), true for all n >= 1.  Suppose n is coprime to 6, then 6 is a unit in Z/nZ, and it follows from the congruence that ( Sum_{k = 1..n} k^14 )/n is an integer. On the other hand, if either 2 divides n or 3 divides n then the congruence shows that ( Sum_{k = 1..n} k^14 )/n cannot be integral. (End)
A126759(a(n)) = n and A126759(m) < n for m < a(n). - Reinhard Zumkeller, May 23 2013
(a(n-1)^2 - 1)/24 = A001318(n), the generalized pentagonal numbers. - Richard R. Forberg, May 30 2013
Numbers k for which A001580(k) is divisible by 3. - Bruno Berselli, Jun 18 2014
Numbers n such that sigma(n) + sigma(2n) = sigma(3n). - Jahangeer Kholdi and Farideh Firoozbakht, Aug 15 2014
a(n) are values of k such that Sum_{m = 1..k-1} m*(k-m)/k is an integer. Sums for those k are given by A062717. Also see Detlefs formula below based on A062717. - Richard R. Forberg, Feb 16 2015
a(n) are exactly those positive integers m such that the sequence b(n) = n*(n + m)*(n + 2*m)/6 is integral, and also such that the sequence c(n) = n*(n + m)*(n + 2*m)*(n + 3*m)/24 is integral. Cf. A007775. - Peter Bala, Nov 13 2015
Along with 2, these are the numbers k such that the k-th Fibonacci number is coprime to every Lucas number. - Clark Kimberling, Jun 21 2016
This sequence is the Engel expansion of 1F2(1; 5/6, 7/6; 1/36) + 1F2(1; 7/6, 11/6; 1/36)/5. - Benedict W. J. Irwin, Dec 16 2016
The sequence a(n), n >= 4 is generated by the successor of the pair of polygonal numbers {P_s(4) + 1, P_(2*s - 1)(3) + 1}, s >= 3. - Ralf Steiner, May 25 2018
The asymptotic density of this sequence is 1/3. - Amiram Eldar, Oct 18 2020
Also, the only vertices in the odd Collatz tree A088975 that are branch values to other odd nodes t == 1 (mod 2) of A005408. - Heinz Ebert, Apr 14 2021
From Flávio V. Fernandes, Aug 01 2021: (Start)
For any two terms j and k, the product j*k is also a term (the same property as p^n and smooth numbers).
From a(2) to a(phi(A033845(n))), or a((A033845(n))/3), the terms are the totatives of the A033845(n) itself. (End)
Also orders n for which cyclic and semicyclic diagonal Latin squares exist (see A123565 and A342990). - Eduard I. Vatutin, Jul 11 2023
If k is in the sequence, then k*2^m + 3 is also in the sequence, for all m > 0. - Jules Beauchamp, Aug 29 2024

			G.f. = x + 5*x^2 + 7*x^3 + 11*x^4 + 13*x^5 + 17*x^6 + 19*x^7 + 23*x^8 + ...

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, 1980.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Andreas Enge, William Hart, and Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], 2016-2018.
B. W. J. Irwin, Constants Whose Engel Expansions are the k-rough Numbers.
L. B. W. Jolley, Summation of Series, Dover, 1961
Cedric A. B. Smith, Prime factors and recurring duodecimals, Math. Gaz. 59 (408) (1975) 106-109.
William A. Stein's The Modular Forms Database, PARI-readable dimension tables for Gamma_0(N).
Eric Weisstein's World of Mathematics, Rough Number.
Eric Weisstein's World of Mathematics, Pi Formulas. [_Jaume Oliver Lafont_, Oct 23 2009]
Index entries for sequences related to smooth numbers [_Michael B. Porter_, Oct 09 2009]
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

A005408 \ A016945. Union of A016921 and A016969; union of A038509 and A140475. Essentially the same as A038179. Complement of A047229. Subsequence of A186422.
Cf. A000330, A001580, A002194, A019670, A032528 (partial sums), A038509 (subsequence of composites), A047209, A047336, A047522, A056020, A084967, A090771, A091998, A144065, A175885-A175887.
For k-rough numbers with other values of k, see A000027, A005408, A007775, A008364-A008366, A166061, A166063.
Cf. A126760 (a left inverse).
Row 3 of A260717 (without the initial 1).
Cf. A105397 (first differences).

Filtered([1..150],n->n mod 6=1 or n mod 6=5); # Muniru A Asiru, Dec 19 2018

a007310 n = a007310_list !! (n-1)
a007310_list = 1 : 5 : map (+ 6) a007310_list
-- Reinhard Zumkeller, Jan 07 2012

[n: n in [1..250] | n mod 6 in [1, 5]]; // Vincenzo Librandi, Feb 12 2016

seq(seq(6*i+j,j=[1,5]),i=0..100); # Robert Israel, Sep 08 2014

Select[Range[200], MemberQ[{1, 5}, Mod[#, 6]] &] (* Harvey P. Dale, Aug 27 2013 *)
a[n_] := (6 n + (-1)^n - 3)/2; a[rem156, 60] (* Robert G. Wilson v, May 26 2014 from a suggestion by N. J. A. Sloane *)
Flatten[Table[6n + {1, 5}, {n, 0, 24}]] (* Alonso del Arte, Feb 06 2016 *)
Table[2*Floor[3*n/2] - 1, {n, 1000}] (* Mikk Heidemaa, Feb 11 2016 *)

isA007310(n) = gcd(n,6)==1 \\ Michael B. Porter, Oct 09 2009

A007310(n)=n\2*6-(-1)^n \\ M. F. Hasler, Oct 31 2014

\\ given an element from the sequence, find the next term in the sequence.
nxt(n) = n + 9/2 - (n%6)/2 \\ David A. Corneth, Nov 01 2016

def A007310(n): return (n+(n>>1)<<1)-1 # Chai Wah Wu, Oct 10 2023

[i for i in range(150) if gcd(6,i) == 1] # Zerinvary Lajos, Apr 21 2009

a(n) = (6*n + (-1)^n - 3)/2. - Antonio Esposito, Jan 18 2002
a(n) = a(n-1) + a(n-2) - a(n-3), n >= 4. - Roger L. Bagula
a(n) = 3*n - 1 - (n mod 2). - Zak Seidov, Jan 18 2006
a(1) = 1 then alternatively add 4 and 2. a(1) = 1, a(n) = a(n-1) + 3 + (-1)^n. - Zak Seidov, Mar 25 2006
1 + 1/5^2 + 1/7^2 + 1/11^2 + ... = Pi^2/9 [Jolley]. - Gary W. Adamson, Dec 20 2006
For n >= 3 a(n) = a(n-2) + 6. - Zak Seidov, Apr 18 2007
From R. J. Mathar, May 23 2008: (Start)
Expand (x+x^5)/(1-x^6) = x + x^5 + x^7 + x^11 + x^13 + ...
O.g.f.: x*(1+4*x+x^2)/((1+x)*(1-x)^2). (End)
a(n) = 6*floor(n/2) - 1 + 2*(n mod 2). - Reinhard Zumkeller, Oct 02 2008
1 + 1/5 - 1/7 - 1/11 + + - - ... = Pi/3 = A019670 [Jolley eq (315)]. - Jaume Oliver Lafont, Oct 23 2009
a(n) = ( 6*A062717(n)+1 )^(1/2). - Gary Detlefs, Feb 22 2010
a(n) = 6*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i), with n > 1. - Bruno Berselli, Nov 05 2010
a(n) = 6*n - a(n-1) - 6 for n>1, a(1) = 1. - Vincenzo Librandi, Nov 18 2010
Sum_{n >= 1} (-1)^(n+1)/a(n) = A093766 [Jolley eq (84)]. - R. J. Mathar, Mar 24 2011
a(n) = 6*floor(n/2) + (-1)^(n+1). - Gary Detlefs, Dec 29 2011
a(n) = 3*n + ((n+1) mod 2) - 2. - Gary Detlefs, Jan 08 2012
a(n) = 2*n + 1 + 2*floor((n-2)/2) = 2*n - 1 + 2*floor(n/2), leading to the o.g.f. given by R. J. Mathar above. - Wolfdieter Lang, Jan 20 2012
1 - 1/5 + 1/7 - 1/11 + - ... = Pi*sqrt(3)/6 = A093766 (L. Euler). - Philippe Deléham, Mar 09 2013
1 - 1/5^3 + 1/7^3 - 1/11^3 + - ... = Pi^3*sqrt(3)/54 (L. Euler). - Philippe Deléham, Mar 09 2013
gcd(a(n), 6) = 1. - Reinhard Zumkeller, Nov 14 2013
a(n) = sqrt(6*n*(3*n + (-1)^n - 3)-3*(-1)^n + 5)/sqrt(2). - Alexander R. Povolotsky, May 16 2014
a(n) = 3*n + 6/(9*n mod 6 - 6). - Mikk Heidemaa, Feb 05 2016
From Mikk Heidemaa, Feb 11 2016: (Start)
a(n) = 2*floor(3*n/2) - 1.
a(n) = A047238(n+1) - 1. (suggested by Michel Marcus) (End)
E.g.f.: (2 + (6*x - 3)*exp(x) + exp(-x))/2. - Ilya Gutkovskiy, Jun 18 2016
From Bruno Berselli, Apr 27 2017: (Start)
a(k*n) = k*a(n) + (4*k + (-1)^k - 3)/2 for k>0 and odd n, a(k*n) = k*a(n) + k - 1 for even n. Some special cases:
k=2: a(2*n) = 2*a(n) + 3 for odd n, a(2*n) = 2*a(n) + 1 for even n;
k=3: a(3*n) = 3*a(n) + 4 for odd n, a(3*n) = 3*a(n) + 2 for even n;
k=4: a(4*n) = 4*a(n) + 7 for odd n, a(4*n) = 4*a(n) + 3 for even n;
k=5: a(5*n) = 5*a(n) + 8 for odd n, a(5*n) = 5*a(n) + 4 for even n, etc. (End)
From Antti Karttunen, May 20 2017: (Start)
a(A273669(n)) = 5*a(n) = A084967(n).
a((5*n)-3) = A255413(n).
A126760(a(n)) = n. (End)
a(2*m) = 6*m - 1, m >= 1; a(2*m + 1) = 6*m + 1, m >= 0. - Ralf Steiner, May 17 2018
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(3) (A002194).
Product_{n>=2} (1 + (-1)^n/a(n)) = Pi/3 (A019670). (End)

A100357 Numbers k such that 2^k + k^2 + 1 is prime.

Original entry on oeis.org

0, 6, 12, 18, 162, 192, 216, 420, 1524, 5112, 7404, 24216, 25944, 101832, 346854

Offset: 1

Labos Elemer, Nov 19 2004

nonn

a(15) > 200000. - Giovanni Resta, Mar 23 2014
All terms are multiples of 6. Corresponding primes of the form 2^n+n^2+1 are in A035325. - Zak Seidov, Apr 05 2014
a(16) > 5*10^5. - Robert Price, Jun 15 2014

Cf. A001580, A063997, A035325.

[n: n in [0..800] | IsPrime(2^n + n^2 + 1) ]; // Vincenzo Librandi, Sep 03 2012

{ta={{0}}, tb={{0}}}; Do[g=n;s=2^n+n^2+1;If[PrimeQ[s], Print[n];ta=Append[ta, n];tb=Append[tb, s]], {n, 0, 10000, 6}];{ta, tb, g}
Select[Range[0, 10000, 6], PrimeQ[2^# + #^2 + 1] &] (* Vincenzo Librandi, Sep 03 2012 *)

is(n)=isprime(2^n+n^2+1) \\ Charles R Greathouse IV, Jul 01 2013

Added a(1) from Vincenzo Librandi, Sep 03 2012
a(12)-a(14) from Giovanni Resta, Mar 23 2014
Mathematica codes edited by Zak Seidov, Apr 05 2014
a(15) from Robert Price, Jun 15 2014

A001585 a(n) = 3^n + n^3.

Original entry on oeis.org

1, 4, 17, 54, 145, 368, 945, 2530, 7073, 20412, 60049, 178478, 533169, 1596520, 4785713, 14352282, 43050817, 129145076, 387426321, 1162268326, 3486792401, 10460362464, 31381070257, 94143190994, 282429550305, 847288625068

Offset: 0

N. J. A. Sloane

In this sequence if we do a forward difference, then the 4th forward difference when considered as a sequence will be a geometric progression with common ratio 3. - Gopalakrishnan (gopala498(AT)yahoo.co.in), May 26 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. K. Hoeflin, Titan Test
Index entries for linear recurrences with constant coefficients, signature (7, -18, 22, -13, 3).

Cf. A001580.

[3^n+n^3: n in [0..30]]; // Vincenzo Librandi, Oct 27 2011

seq(seq(k^n+n^k, k=3..3), n=0..23); # Zerinvary Lajos, Jun 29 2007

Table[3^n+n^3,{n,0,3*4!}] (* Vladimir Joseph Stephan Orlovsky, May 07 2010 *)

G.f.: (-1+2*x^4+15*x^3-7*x^2+3*x)/((3*x-1)*(x-1)^4). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009

More terms from Vladimir Joseph Stephan Orlovsky, May 07 2010

A001589 a(n) = 4^n + n^4.

Original entry on oeis.org

1, 5, 32, 145, 512, 1649, 5392, 18785, 69632, 268705, 1058576, 4208945, 16797952, 67137425, 268473872, 1073792449, 4295032832, 17179952705, 68719581712, 274878037265, 1099511787776, 4398046705585, 17592186278672

Offset: 0

N. J. A. Sloane

a(n) is prime if and only if n = 1. - Reinhard Zumkeller, May 24 2009

The statement above (and the corollary that 5 is the only prime term in this sequence) can be proved using Sophie Germain's identity x^4 + 4y^4 = (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2). - Alonso del Arte, Oct 31 2013 [exponents corrected by Thomas Ordowski, Nov 29 2017]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Notes, Mathematical Intelligencer 2(2) (1980), p. 66. - _Reinhard Zumkeller_, May 24 2009
Index entries for linear recurrences with constant coefficients, signature (9,-30,50,-45,21,-4).

Cf. A001580, A001585.

[4^n+n^4: n in [0..30]]; // Vincenzo Librandi, Oct 27 2011

Table[4^n + n^4, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)
LinearRecurrence[{9,-30,50,-45,21,-4},{1,5,32,145,512,1649},30] (* Harvey P. Dale, Mar 06 2023 *)

PARI
```
a(n)=1<<(n+n)+n^4
```

G.f.: -(5*x^5 + 38*x^4 + 43*x^3 - 17*x^2 + 4*x - 1) / ((x - 1)^5*(4*x - 1)). - Colin Barker, Jan 01 2013

A001593 a(n) = 5^n + n^5.

Original entry on oeis.org

1, 6, 57, 368, 1649, 6250, 23401, 94932, 423393, 2012174, 9865625, 48989176, 244389457, 1221074418, 6104053449, 30518337500, 152588939201, 762940872982, 3814699155193, 19073488804224, 95367434840625, 476837162287226, 2384185796169257, 11920928961514468

Offset: 0

N. J. A. Sloane

a(24) is prime; a(1036) and a(104824) are probable primes (3-PRP). - David Radcliffe, Dec 23 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-45,95,-115,81,-31,5).

Cf. A001580, A001585, A001589.

[5^n+n^5: n in [0..30]]; // Vincenzo Librandi, Oct 27 2011

seq(seq(k^n+n^k, k=5..5), n=0..20); # Zerinvary Lajos, Jun 29 2007

f[n_]:=5^n+n^5;f[Range[0,40]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)
LinearRecurrence[{11,-45,95,-115,81,-31,5},{1,6,57,368,1649,6250,23401},30] (* Harvey P. Dale, Jan 15 2018 *)

a(n)=5^n+n^5 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (4*x^6+135*x^5+289*x^4+84*x^3-36*x^2+5*x-1) / ((x-1)^6*(5*x-1)). - Colin Barker, May 07 2013

More terms from Colin Barker, May 07 2013

A001594 a(n) = 6^n + n^6.

Original entry on oeis.org

1, 7, 100, 945, 5392, 23401, 93312, 397585, 1941760, 10609137, 61466176, 364568617, 2179768320, 13065520825, 78371693632, 470196375201, 2821126684672, 16926683582305, 101559990680640, 609359787056377

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-63,161,-245,231,-133,43,-6).

Cf. sequences of the form k^n+n^k: A001580 (k=2), A001585 (k=3), A001589 (k=4), A001593 (k=5), this sequence (k=6), A001596 (k=7), A198401 (k=8), A185277 (k=9), A177068 (k=10), A177069 (k=11).

[6^n+n^6: n in [0..30]]; // Vincenzo Librandi, Oct 27 2011

seq(seq(k^n+n^k, k=6..6), n=0..19); # Zerinvary Lajos, Jun 29 2007

Table[6^n + n^6, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - 6 x + 72 x^2 - 75 x^3 - 1475 x^4 - 1776 x^5 - 334 x^6 - 7 x^7)/((1-x)^7 (1-6 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 28 2014 *)
LinearRecurrence[{13,-63,161,-245,231,-133,43,-6},{1,7,100,945,5392,23401,93312,397585},20] (* Harvey P. Dale, Jan 07 2023 *)

a(n)=6^n+n^6 \\ Charles R Greathouse IV, Feb 14 2011

[6^n+n^6 for n in (0..30)] # Bruno Berselli, Aug 28 2014

G.f.: (1 - 6*x + 72*x^2 - 75*x^3 - 1475*x^4 - 1776*x^5 - 334*x^6 - 7*x^7)/((1-x)^7*(1-6*x)). - Vincenzo Librandi, Aug 28 2014

A061421 Primes of the form 2^n+n+1.

Original entry on oeis.org

2, 7, 71, 110427941548649020598956093796432407239217743554726184882600387580788973

Offset: 1

Jason Earls, May 02 2001

nonn

Next term is 2^1884+1884+1, with 568 digits and is too large to include. - Emeric Deutsch, May 13 2006

The Wikipedia article "Zeisel number" gives a historical connection to A051015. - Jonathan Sondow, Oct 17 2017

K. S. Brown, The Sum of the Prime Factors of N (Part 2: Primes of the Form 2k-1 + k), Math Pages
Wikipedia, Zeisel number

Cf. A001580, A069539, A052007, A048744, A100357, A100358, A100359.

Cf. A061422.

a:=proc(n) if isprime(2^n+n+1)=true then 2^n+n+1 else fi end: seq(a(n),n=0..1000); # Emeric Deutsch, May 13 2006

{ta={{0}}, tb={{0}}};Do[g=n;s=2^n+n+1; If[PrimeQ[s], Print[n];ta=Append[ta, n]; tb=Append[tb, s]], {n, 1, 10000}];{ta, tb, g} (* Labos Elemer, Nov 19 2004 *)

Edited by N. J. A. Sloane, May 04 2007

A001596 a(n) = 7^n + n^7.

Original entry on oeis.org

1, 8, 177, 2530, 18785, 94932, 397585, 1647086, 7861953, 45136576, 292475249, 1996813914, 13877119009, 96951758924, 678328486353, 4747732369318, 33233199005057, 232630924325880, 1628414210130481

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (15,-84,252,-462,546,-420,204,-57,7).

Cf. A001580, A001585, A001589, A001593, A001594.

[7^n+n^7: n in [0..30]]; // Vincenzo Librandi, Oct 27 2011

seq(seq(k^n+n^k, k=7..7), n=0..18); # Zerinvary Lajos, Jun 29 2007

f[n_]:=7^n+n^7;f[Range[0,40]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)
LinearRecurrence[{15,-84,252,-462,546,-420,204,-57,7},{1,8,177,2530,18785,94932,397585,1647086,7861953},20] (* Harvey P. Dale, Sep 16 2018 *)

PARI
```
a(n)=7^n+n^7
```

cp's OEIS Frontend

A061119 Primes in the sequence n^2 + 2^n (A001580).

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A174121 Partial sums of A001580.

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A007310 Numbers congruent to 1 or 5 mod 6.

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A100357 Numbers k such that 2^k + k^2 + 1 is prime.

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A001585 a(n) = 3^n + n^3.

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A001589 a(n) = 4^n + n^4.

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