A006552 -id:A006552 - cp's OEIS frontend

A004006 a(n) = C(n,1) + C(n,2) + C(n,3), or n*(n^2 + 5)/6.

0, 1, 3, 7, 14, 25, 41, 63, 92, 129, 175, 231, 298, 377, 469, 575, 696, 833, 987, 1159, 1350, 1561, 1793, 2047, 2324, 2625, 2951, 3303, 3682, 4089, 4525, 4991, 5488, 6017, 6579, 7175, 7806, 8473, 9177, 9919, 10700, 11521, 12383, 13287, 14234, 15225

Offset: 0

Albert D. Rich (Albert_Rich(AT)msn.com)

3-dimensional analog of centered polygonal numbers.
The Burnside group B(3,n) has order 3^a(n).
Answer to the question: if you have a tall building and 3 plates and you need to find the highest story, a plate thrown from which does not break, what is the number of stories you can handle given n tries? - Leonid Broukhis, Oct 24 2000
Equals row sums of triangle A144329 starting with "1". - Gary W. Adamson, Sep 18 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=A[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=4, a(n-1)=-coeff(charpoly(A,x),x^(n-3)). - Milan Janjic, Jan 24 2010
From J. M. Bergot, Aug 03 2011: (Start)
If one formed the 3 X 3 square
  |  n  |  n+1 | n+2 |
  | n+3 |  n+4 | n+5 |
  | n+6 |  n+7 | n+8 |
  and found the sum of the horizontal products n*(n + 1)*(n + 2) + (n + 3)*(n + 4)*(n + 5) + (n + 6)*(n + 7)*(n + 8) and added the sum of the vertical products n*(n + 3)*(n + 6) + (n + 1)*(n + 4)*(n + 7) + (n + 2)*(n + 5)(n + 8) one gets 6*n^3 + 72*n^2 + 318*n + 504. This will give 36 times the values of all the terms in this sequence. (End)
a(n) is divisible by n for n congruent to {1,5} mod 6. (see A007310). - Gary Detlefs, Dec 08 2011
From Beimar Naranjo, Feb 22 2024: (Start)
Number of compositions with at most three parts and sum at most n.
Also the number of compositions with at most one part distinct from 1 and with a sum at most n. (End)
a(n) is the number of strings of length n defined on {0, 1, 2, 3} that contain one 1 and any number of 0's, or two 2's and any number of 0's, or three 3's and any number of 0's. For example, a(6) = 41 since the strings are the 20 permutations of 333000, the 15 permutations of 220000 and the 6 permutations of 100000. - Enrique Navarrete, Jun 18 2025

			G.f. = x + 3*x^2 + 7*x^3 + 14*x^4 + 25*x^5 + 41*x^6 + 63*x^7 + 92*x^8 + ... - _Michael Somos_, Dec 29 2019

W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Wiley, 1966, see p. 380.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Michael Boardman, The Egg-Drop Numbers, Mathematics Magazine, 77 (2004), 368-372. [From Parthasarathy Nambi, Sep 30 2009]
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics, 2017, Vol. 41 Issue 6, pp. 849-853.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Laurent Saloff-Coste, Random walks on finite groups, in Probability on discrete structures, 263-346, Encyclopaedia Math. Sci., 110, Springer, 2004.
Bridget Eileen Tenner, Reduced word manipulation: patterns and enumeration, J. Algebr. Comb. 46, No. 1, 189-217 (2017), w in S_n(231) l(w)=3.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A051576, A055795, A006552. Differences give A000217 + 1 or A000124.
1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
Cf. A000292, A001477, A007310, A128834, A144329, A228074.

List([0..50], n-> n*(n^2+5)/6); # G. C. Greubel, Aug 27 2019

a004006 n = a000292 n + n + 1  -- Reinhard Zumkeller, Mar 31 2012

[n*(n^2+5)/6: n in [0..50]]; // Vincenzo Librandi, May 15 2011

A004006 := proc(n) n*(n^2+5)/6 ;end proc:
seq(A004006(n),n=0..10) ; # R. J. Mathar, Jun 05 2011

Table[Total[Table[Binomial[n,i], {i,3}]], {n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1}, {0,1,3,7}, 50] (* Harvey P. Dale, Aug 21 2011 *)

A004006(n):=n*(n^2+5)/6$ makelist(A004006(n),n,0,30); /* Martin Ettl, Jan 08 2013 */

{a(n) = n*(n^2 + 5)/6}; /* Michael Somos, May 04 2007 */

[n*(n^2+5)/6 for n in (0..50)] # G. C. Greubel, Aug 27 2019

G.f.: x*(1-x+x^2)/(1-x)^4.
E.g.f.: x*(1 + x/2 + x^2/6) * exp(x).
a(-n) = -a(n).
a(n) = binomial(n+2,n-1) - binomial(n,n-2). - Zerinvary Lajos, May 11 2006
Euler transform of length 6 sequence [3, 1, 1, 0, 0, -1]. - Michael Somos, May 04 2007
Starting (1, 3, 7, 14, ...) = binomial transform of [1, 2, 2, 1, 0, 0, 0, ...]. - Gary W. Adamson, Apr 24 2008
a(0)=0, a(1)=1, a(2)=3, a(3)=7, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Aug 21 2011
a(n+1) = A000292(n) + n + 1. - Reinhard Zumkeller, Mar 31 2012
a(n) = 2*a(n-1) + (n-1) - a(n-2) with a(0) = 0, a(1) = 1. - Richard R. Forberg, Jan 23 2014
a(n) = Sum_{i=1..n} binomial(n-2i,2). - Wesley Ivan Hurt, Nov 18 2017
a(n) = n + Sum_{k=0..n} k*(n-k). - Gionata Neri, May 19 2018
a(n) = Sum_{k=0..n-1} A000124(k). - Torlach Rush, Aug 05 2018
G.f.: ((1 - x^5)/(1 - x)^5 - 1)/5. - Michael Somos, Dec 29 2019
G.f.: g(f(x)), where g is g.f. of A001477 and f is g.f. of A128834. - Oboifeng Dira, Jun 21 2020
Sum_{n>0} 1/a(n) = 3*(2*gamma + polygamma(0, 1-i*sqrt(5)) + polygamma(0, 1+i*sqrt(5)))/5 = 1.6787729555834452106286261834348972248... where i denotes the imaginary unit. - Stefano Spezia, Aug 31 2023

A242176 Numbers k such that k*6^k + 1 is prime.

Original entry on oeis.org

1, 2, 91, 185, 387, 488, 747, 800, 9901, 10115, 12043, 13118, 30981, 51496

Offset: 1

Vincenzo Librandi, May 08 2014

G. Loeh, Generalized Cullen primes

Cf. numbers n such that n*k^n + 1 is prime: A005849 (k=2), A006552 (k=3), A007646 (k=4), this sequence (k=6), A242177 (k=7), A242178 (k=8), A007647 (k=10), A242196 (k=12), A242197 (k=14), A242198 (k=15), A242199 (k=16), A007648 (k=18).

[n: n in [0..1500] | IsPrime(n*6^n+1)];

Select[Range[1500], PrimeQ[# 6^# + 1] &]

is(n)=ispseudoprime(n*6^n+1) \\ Charles R Greathouse IV, Feb 17 2017

a(9)-a(14) from Loeh's list (see Links) - Bruno Berselli, May 08 2014

A242203 Numbers n such that n*3^n + 1 is semiprime.

Original entry on oeis.org

1, 3, 10, 16, 20, 22, 24, 34, 39, 56, 63, 108, 128, 194, 202, 212, 214, 218, 314, 364, 662, 722

Offset: 1

Vincenzo Librandi, May 10 2014

The semiprimes of this form are 4, 82, 590491, 688747537, 69735688021, 690383311399, 6778308875545, 567024177788663347, 158049650967740074414, 29307467449532190083956645177, ...

a(23) >= 894. - Hugo Pfoertner, Aug 03 2019

Status of 894*3^894+1 in factordb.com.

Cf. numbers n such that n*k^n + 1 is semiprime: A242175 (k=2), this sequence (k=3), A242204 (k=4), A242205 (k=5), A242269 (k=6), A242270 (k=7), A242271 (k=8), A242272 (k=9), A216378 (k=10).

Cf. A006552, A050914.

IsSemiprime:=func; [n: n in [1..130] | IsSemiprime(s) where s is n*3^n+1];

Mathematica

Select[Range[130], PrimeOmega[# 3^# + 1] == 2 &]

PARI

isok(n) = bigomega(n*3^n + 1)==2; \\ Michel Marcus, Mar 30 2019

Extensions

a(14)-a(20) from Luke March, Jul 30 2015

a(21)-a(22) from Daniel Suteu, Mar 30 2019

A210339 Generalized Cullen primes: any primes that can be written in the form n*b^n + 1 with n+2 > b > 2.

Original entry on oeis.org

19, 193, 52489, 114689, 9000000001, 259374246011, 38280596832649217, 59296646043258913, 408700964355468751, 2434970217729660813313, 13576803638250229989377, 21000000000000000000001, 3140085798164163223281069127, 4818833289797717549937328129

Offset: 1

Arkadiusz Wesolowski, Mar 20 2012

nonn

			81*2^324 + 1 is a prime number and 81*2^324 + 1 = 81*16^81 + 1, so this number is in the sequence.

Harvey Dubner, Generalized Cullen numbers, J. Recreational Math. 21 (1989), pp. 190-194.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..92
Chris Caldwell, The Top 20 Generalized Cullen Primes
Chris Caldwell, The Prime Glossary, Cullen prime
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 57
G. L. Honaker, Jr. and Chris Caldwell, 15545...00001 (1008-digits)
G. L. Honaker, Jr. and Chris Caldwell, 36869...81251 (10002-digits)
PrimeGrid, Home Page
Wikipedia, Cullen number

Cf. A050920, A005849, A006552, A007646.

lst = {}; Do[p = n*b^n + 1; If[p < 10^200 && PrimeQ[p], AppendTo[lst, p]], {b, 3, 100}, {n, b - 1, 413}]; Sort@lst

A265121 Integers k such that k*3^k + 2 is prime.

Original entry on oeis.org

0, 1, 3, 5, 11, 15, 17, 153, 169, 273, 317, 373, 923, 1403, 1969, 2349, 7809, 10313, 12291, 24865, 41289

Offset: 1

Altug Alkan, Dec 01 2015

Initial corresponding primes are 2, 5, 83 and 1217.
How do this sequence and A006552 compare asymptotically?
a(22) > 10^5. - Michael S. Branicky, Oct 08 2024

			a(3) = 3 because 3^3 * 3 + 2 = 83 is prime.

Cf. A006552, A036290, A182373.

[n: n in [0..400] | IsPrime(n*3^n+2)]; // Vincenzo Librandi, Dec 02 2015

Select[Range[0, 10000], PrimeQ[# 3^# + 2] &] (* Vincenzo Librandi, Dec 02 2015 *)

for(n=0, 1e6, if(ispseudoprime(3^n*n + 2), print1(n, ", ")));

a(1) = 0 added by Vincenzo Librandi, Dec 02 2015

a(21) from Michael S. Branicky, May 16 2023

A265013 Numbers n such that n*9^n + 1 is prime.

Original entry on oeis.org

2, 12382, 27608, 31330, 117852

Offset: 1

Tim Johannes Ohrtmann, Nov 30 2015

All terms are even. - Robert Israel, Jan 18 2016

Cf. A005849, A006552, A007646, A242176, A242177, A242178, A007647, A242196, A242197, A242198, A242199, A007648.

[n: n in [0..100000] | IsPrime(n*9^n+1)];

Select[Range[100000], PrimeQ[# 9^# + 1] &]

for(n=1,100000, if(isprime(n*9^n+1), print1(n,", ")))

A266694 Numbers n such that 2n3^n + 1 is prime.

Original entry on oeis.org

1, 2, 3, 14, 19, 21, 27, 33, 46, 70, 80, 441, 540, 567, 625, 1057, 1119, 1213, 1542, 4263, 4419, 4507, 5186, 7345, 8626, 8853, 11256, 12885, 14688, 15236, 17697, 26770, 57801, 71665

Offset: 1

Juri-Stepan Gerasimov, Jan 02 2016

Primes: 2, 3, 19, 1213, 4507, ...

No further terms < 10^5. - Ray Chandler, Apr 13 2016

			1 is in this sequence because 2*1*3^1 + 1 = 7 is prime.

Cf. A006552.

[n: n in [1..2000]| IsPrime(2*n*3^n+1)]; // Vincenzo Librandi, Jan 03 2016

Select[Range[1600], PrimeQ[2 # 3^# + 1] &] (* Vincenzo Librandi, Jan 03 2016 *)

is(n)=ispseudoprime(2*n*3^n+1) \\ Charles R Greathouse IV, Feb 08 2016

a(24) and a(26) inserted by Charles R Greathouse IV, Feb 08 2016
a(28)-a(30) from Charles R Greathouse IV, Feb 08 2016
a(31)-a(32) from Ray Chandler, Apr 05 2016
a(33) from Ray Chandler, Apr 08 2016
a(34) from Ray Chandler, Apr 09 2016

A338412 Numbers k such that k * 20^k + 1 is prime.

Original entry on oeis.org

3, 6207, 8076, 22356, 151456

Offset: 1

Tim Johannes Ohrtmann, Oct 25 2020

a(6) > 219976.

Steven Harvey, Generalized Cullen Search
G. Loeh, Generalized Cullen primes
Wikipedia, Cullen number

Numbers k such that k * b^k + 1 is prime: A006093 (b=1), A005849 (b=2), A006552 (b=3), A007646 (b=4), A242176 (b=6), A242177 (b=7), A242178 (b=8), A265013 (b=9), A007647(b=10), A242196(b=12), A242197 (b=14), A242198 (b=15), A242199 (b=16), A007648 (b=18), this sequence (b=20).

Magma
```
[n: n in [1..10000] |IsPrime(n*20^n+1)]
```

Select[Range[1, 10000], PrimeQ[n*20^n+1] &]

for(n=1, 10000, if(isprime(n*20^n+1), print1(n, ", ")))

cp's OEIS Frontend

A004006 a(n) = C(n,1) + C(n,2) + C(n,3), or n*(n^2 + 5)/6.

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A242176 Numbers k such that k*6^k + 1 is prime.

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Extensions

A242203 Numbers n such that n*3^n + 1 is semiprime.

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Magma

Mathematica

PARI

Extensions

A210339 Generalized Cullen primes: any primes that can be written in the form n*b^n + 1 with n+2 > b > 2.

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Links

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Mathematica

A265121 Integers k such that k*3^k + 2 is prime.

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Programs

Magma

Mathematica

PARI

Extensions

A265013 Numbers n such that n*9^n + 1 is prime.

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Comments

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Programs

Magma

Mathematica

PARI

A266694 Numbers n such that 2*n*3^n + 1 is prime.

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A266694 Numbers n such that 2n3^n + 1 is prime.