A303281 -id:A303281 - cp's OEIS frontend

A002064 Cullen numbers: a(n) = n*2^n + 1.

1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, 229377, 491521, 1048577, 2228225, 4718593, 9961473, 20971521, 44040193, 92274689, 192937985, 402653185, 838860801, 1744830465, 3623878657, 7516192769, 15569256449, 32212254721, 66571993089

Offset: 0

N. J. A. Sloane

Binomial transform is A084859. Inverse binomial transform is A004277. - Paul Barry, Jun 12 2003
Let A be the Hessenberg matrix of order n defined by: A[1,j]=1, A[i,i]:=2,(i>1), A[i,i-1] =-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= (-1)^(n-1)*coeff(charpoly(A,x),x). - Milan Janjic, Jan 26 2010
Indices of primes are listed in A005849. - M. F. Hasler, Jan 18 2015
Add the list of fractions beginning with 1/2 + 3/4 + 7/8 + ... + (2^n - 1)/2^n and take the sums pairwise from left to right. For 1/2 + 3/4 = 5/4, 5 + 4 = 9 = a(2); for 5/4 + 7/8 = 17/8, 17 + 8 = 25 = a(3); for 17/8 + 15/16 = 49/16, 49 + 16 = 65 = a(4); for 49/16 + 31/32 = 129/32, 129 + 32 = 161 = a(5). For each pairwise sum a/b, a + b = n*2^(n+1). - J. M. Bergot, May 06 2015
Number of divisors of (2^n)^(2^n). - Gus Wiseman, May 03 2021
Named after the Irish Jesuit priest James Cullen (1867-1933), who checked the primality of the terms up to n=100. - Amiram Eldar, Jun 05 2021

			G.f. = 1 + 3*x + 9*x^2 + 25*x^3 + 65*x^4 + 161*x^5 + 385*x^6 + 897*x^7 + ... - _Michael Somos_, Jul 18 2018

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
R. K. Guy, Unsolved Problems in Number Theory, B20.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 240-242.
W. Sierpiński, Elementary Theory of Numbers. Państ. Wydaw. Nauk., Warsaw, 1964, p. 346.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..300
Ray Ballinger, Cullen Primes: Definition and Status.
Attila Bérczes, István Pink, and Paul Thomas Young, Cullen numbers and Woodall numbers in generalized Fibonacci sequences, J. Num. Theor. (2024) Vol. 262, 86-102.
Yuri Bilu, Diego Marques, and Alain Togbé, Generalized Cullen numbers in linear recurrence sequences, Journal of Number Theory, Vol. 202 (2019), pp. 412-425; arXiv preprint, arXiv:1806.09441 [math.NT], 2018.
Daniel Birmajer, Juan B. Gil, David S. Kenepp, and Michael D. Weiner, Restricted generating trees for weak orderings, arXiv:2108.04302 [math.CO], 2021.
C. K. Caldwell, The Top Twenty: Cullen Primes.
James Cullen, Question 15897, Educational Times, Vol. 58 (December 1905), p. 534.
Orhan Eren and Yüksel Soykan, Gaussian Generalized Woodall Numbers, Arch. Current Res. Int'l (2023) Vol. 23, Iss. 8, Art. No. ACRI.108618, 48-68. See p. 50.
Orhan Eren and Yüksel Soykan, On Dual Hyperbolic Generalized Woodall Numbers, Archives Current Res. Int'l (2024) Vol. 24, Iss. 11, Art. No. ACRI.126420, 398-423. See p. 401.
Jon Grantham and Hester Graves, The abc Conjecture Implies That Only Finitely Many Cullen Numbers Are Repunits, arXiv:2009.04052 [math.NT], 2020.
José María Grau and Florian Luca, Cullen numbers with the Lehmer property, Proceedings of the American Mathematical Society, Vol. 140, No. 1 (2012), pp. 129-134; arXiv preprint, arXiv:1103.3578 [math.NT], Mar 18 2011.
Paul Leyland, Factors of Cullen and Woodall numbers.
Paul Leyland, Generalized Cullen and Woodall numbers.
Diego Marques, On Generalized Cullen and Woodall Numbers That are Also Fibonacci Numbers, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.4.
Hisanori Mishima, Factorizations of many number sequences, Cullen numbers (n = 1 to 100), (n = 101 to 200), (n = 201 to 300), (n = 301 to 323).
Simon Plouffe, Approximations de Séries Génératrices et Quelques Conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Wacław Sierpiński, Elementary Theory of Numbers, Warszawa 1964.
Amelia Carolina Sparavigna, On the generalized sums of Mersenne, Fermat, Cullen and Woodall Numbers, Politecnico di Torino (Italy, 2019).
Amelia Carolina Sparavigna, Composition Operations of Generalized Entropies Applied to the Study of Numbers, International Journal of Sciences, Vol. 8, No. 4 (2019), pp. 87-92.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences, Vol. 8, No. 10 (2019).
Eric Weisstein's World of Mathematics, Cullen Number.
Wikipedia, Cullen number.
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Cf. A005849, A003261, A050914, A130197, A134081, A001787, A143038, A156708, A181527.
Cf. A000325, A186947.
Diagonal k = n + 1 of A046688.
A000005 counts divisors of n.
A000312 = n^n.
A002109 gives hyperfactorials (sigma: A260146, omega: A303281).
A057156 = (2^n)^(2^n).
A062319 counts divisors of n^n.
A173339 lists positions of squares in A062319.
A188385 gives the highest prime exponent in n^n.
A249784 counts divisors of n^n^n.
Cf. A000169, A000272, A036289, A066959, A176029, A340841, A343656.

a002064 n = n * 2 ^ n + 1
a002064_list = 1 : 3 : (map (+ 1) $ zipWith (-) (tail xs) xs)
   where xs = map (* 4) a002064_list
-- Reinhard Zumkeller, Mar 16 2013

[n*2^n + 1: n in [0..40]]; // Vincenzo Librandi, May 07 2015

A002064:=-(1-2*z+2*z**2)/((z-1)*(-1+2*z)**2); # conjectured by Simon Plouffe in his 1992 dissertation

Table[n*2^n+1,{n,0,2*4!}] (* Vladimir Joseph Stephan Orlovsky, Apr 25 2010 *)
LinearRecurrence[{5,-8,4},{1,3,9},51] (* Harvey P. Dale, Oct 13 2011 *)
CoefficientList[Series[(1 - 2 x + 2 x^2)/((1 - x) (2 x - 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, May 07 2015 *)

A002064(n)=n*2^n+1  \\ M. F. Hasler, Oct 31 2012

a(n) = 4a(n-1) - 4a(n-2) + 1. - Paul Barry, Jun 12 2003
a(n) = sum of row (n+1) of triangle A130197. Example: a(3) = 25 = (12 + 8 + 4 + 1), row 4 of A130197. - Gary W. Adamson, May 16 2007
Row sums of triangle A134081. - Gary W. Adamson, Oct 07 2007
Equals row sums of triangle A143038. - Gary W. Adamson, Jul 18 2008
Equals row sums of triangle A156708. - Gary W. Adamson, Feb 13 2009
G.f.: -(1-2*x+2*x^2)/((-1+x)*(2*x-1)^2). a(n) = A001787(n+1)+1-A000079(n). - R. J. Mathar, Nov 16 2007
a(n) = 1 + 2^(n + log_2(n)) ~ 1 + A000079(n+A004257(n)). a(n) ~ A000051(n+A004257(n)). - Jonathan Vos Post, Jul 20 2008
a(0)=1, a(1)=3, a(2)=9, a(n) = 5*a(n-1)-8*a(n-2)+4*a(n-3). - Harvey P. Dale, Oct 13 2011
a(n) = A036289(n) + 1 = A003261(n) + 2. - Reinhard Zumkeller, Mar 16 2013
E.g.f.: 2*x*exp(2*x) + exp(x). - Robert Israel, Dec 12 2014
a(n) = 2^n * A000325(n) = 4^n * A186947(-n) for all n in Z. - Michael Somos, Jul 18 2018
a(n) = Sum_{i=0..n-1} a(i) + A000325(n+1). - Ivan N. Ianakiev, Aug 07 2019
a(n) = sigma((2^n)^(2^n)) = A000005(A057156(n)) = A062319(2^n). - Gus Wiseman, May 03 2021
Sum_{n>=0} 1/a(n) = A340841. - Amiram Eldar, Jun 05 2021

Edited by M. F. Hasler, Oct 31 2012

A303279 Expansion of (1/(1 - x)^2) * Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k)).

Original entry on oeis.org

0, 1, 3, 7, 12, 19, 27, 38, 51, 66, 82, 101, 121, 143, 167, 195, 224, 256, 289, 325, 363, 403, 444, 489, 536, 585, 637, 692, 748, 807, 867, 932, 999, 1068, 1139, 1214, 1290, 1368, 1448, 1532, 1617, 1705, 1794, 1886, 1981, 2078, 2176, 2279, 2384, 2492, 2602, 2715, 2829, 2947, 3067

Offset: 1

Ilya Gutkovskiy, Apr 20 2018

nonn

Sum of exponents in prime-power factorization of product of first n factorials (A000178).

Partial sums of A022559.

			a(5) = 12 because 2!*3!*4!*5! = 2^8*3^3*5 and 8 + 3 + 1 = 12.

Daniel Suteu, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and William Verreault, Arithmetic functions at factorial arguments, Publications de l'Institut Mathématique, Vol. 115, No. 129 (2024), pp. 45-76.
Eric Weisstein's World of Mathematics, Barnes G-Function.
Eric Weisstein's World of Mathematics, Superfactorial.
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences related to factorial numbers.

Cf. A000178, A001222, A022559, A083342, A303281.

Half of row sums of triangle A356718.

b:= proc(n) option remember; `if`(n<1, [0$2],
      (p-> p+[numtheory[bigomega](n), p[1]])(b(n-1)))
    end:
a:= n-> b(n+1)[2]:
seq(a(n), n=1..55);  # Alois P. Heinz, Oct 07 2021

nmax = 55; Rest[CoefficientList[Series[1/(1 - x)^2 Sum[Boole[PrimePowerQ[k]] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[PrimeOmega[BarnesG[n + 2]], {n, 55}]
Table[ Sum[ PrimeOmega@ j, {k, n}, {j, k}], {n, 55}]

a(n) = my(t=0); sum(k=1, n, t+=bigomega(k)); \\ Daniel Suteu, Jan 17 2019

a(n) = (n^2/2) * (log(log(n)) + c + O(1/log(n))), where c = A083342 (De Koninck and Verreault, 2024, p. 49, eq. (3.1)). - Amiram Eldar, Dec 10 2024

a(n) = (1/2) * Sum_{k=0..n} A356718(n,k). - Dario T. de Castro, Feb 19 2025

A046688 Antidiagonals of square array in which k-th row (k>0) is an arithmetic progression of difference 2^(k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 5, 1, 1, 5, 7, 9, 9, 1, 1, 6, 9, 13, 17, 17, 1, 1, 7, 11, 17, 25, 33, 33, 1, 1, 8, 13, 21, 33, 49, 65, 65, 1, 1, 9, 15, 25, 41, 65, 97, 129, 129, 1, 1, 10, 17, 29, 49, 81, 129, 193, 257, 257, 1, 1, 11, 19, 33, 57, 97, 161, 257, 385, 513, 513, 1

Offset: 0

N. J. A. Sloane

			From _Gus Wiseman_, May 08 2021: (Start):
Array A(m,n) = 1 + n*2^(m-1) begins:
       n=0: n=1: n=2: n=3: n=4: n=5: n=6: n=7: n=8: n=9:
  m=0:   1    1    1    1    1    1    1    1    1    1
  m=1:   1    2    3    5    9   17   33   65  129  257
  m=2:   1    3    5    9   17   33   65  129  257  513
  m=3:   1    4    7   13   25   49   97  193  385  769
  m=4:   1    5    9   17   33   65  129  257  513 1025
  m=5:   1    6   11   21   41   81  161  321  641 1281
  m=6:   1    7   13   25   49   97  193  385  769 1537
  m=7:   1    8   15   29   57  113  225  449  897 1793
  m=8:   1    9   17   33   65  129  257  513 1025 2049
  m=9:   1   10   19   37   73  145  289  577 1153 2305
Triangle T(n,k) = 1 + (n-k)*2^(k-1) begins:
   1
   1   1
   1   2   1
   1   3   3   1
   1   4   5   5   1
   1   5   7   9   9   1
   1   6   9  13  17  17   1
   1   7  11  17  25  33  33   1
   1   8  13  21  33  49  65  65   1
   1   9  15  25  41  65  97 129 129   1
   1  10  17  29  49  81 129 193 257 257   1
   1  11  19  33  57  97 161 257 385 513 513   1
(End)

G. H. Hardy, A Theorem Concerning the Infinite Cardinal Numbers, Quart. J. Math., 35 (1904), p. 90 = Collected Papers, Vol. VII, p. 430.

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Row sums are A000079.
Diagonal n = m + 1 of the array is A002064.
Diagonal n = m of the array is A005183.
Column m = 1 of the array is A094373.
Diagonal n = m - 1 of the array is A131056.
A002109 gives hyperfactorials (sigma: A260146, omega: A303281).
A009998(k,n) = n^k.
A009999(n,k) = n^k.
A057156 = (2^n)^(2^n).
A062319 counts divisors of n^n.
Cf. A000169, A000272, A000312, A036289, A343656, A343658.

Table[If[k==0,1,n*2^(k-1)+1],{n,0,9},{k,0,9}] (* ARRAY, Gus Wiseman, May 08 2021 *)
Table[If[k==0,1,1+(n-k)*2^(k-1)],{n,0,10},{k,0,n}] (* TRIANGLE, Gus Wiseman, May 08 2021 *)

A(m,n)={if(m>0, 1+n*2^(m-1), 1)}
{ for(m=0, 10, for(n=0, 10, print1(A(m,n), ", ")); print) } \\ Andrew Howroyd, Mar 07 2020

A(m,n) = 1 + n*2^(m-1) for m > 1. - Andrew Howroyd, Mar 07 2020

As a triangle, T(n,k) = A(k,n-k) = 1 + (n-k)*2^(k-1). - Gus Wiseman, May 08 2021

More terms from Larry Reeves (larryr(AT)acm.org), Apr 06 2000

A323444 Sum of exponents in prime-power factorization of Product_{k=0..n} binomial(n,k) (A001142).

Original entry on oeis.org

0, 0, 1, 2, 6, 6, 11, 10, 23, 28, 33, 28, 45, 38, 44, 50, 86, 74, 96, 82, 106, 110, 114, 96, 147, 150, 153, 182, 211, 184, 215, 186, 281, 280, 279, 278, 347, 308, 306, 304, 380, 336, 374, 328, 368, 408, 403, 352, 489, 482, 524, 516, 559, 498, 596, 586, 686, 674

Offset: 0

Daniel Suteu, Jan 15 2019

nonn

Also sum of exponents in prime-power factorization of hyperfactorial(n) / superfactorial(n).

			a(4) = 6 because C(4,0)*C(4,1)*C(4,2)*C(4,3)*C(4,4) = 2^5 * 3^1 and 5 + 1 = 6, where C(n,k) is the binomial coefficient.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, International Journal of Number Theory, Vol. 12, No. 1 (2016), pp. 57-91; arXiv preprint, arXiv:1409.4145 [math.NT], 2014-2015.
Eric Weisstein's World of Mathematics, Hyperfactorial.
Eric Weisstein's World of Mathematics, Superfactorial.
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences related to factorial numbers.

Cf. A000178, A001142, A001222, A002109, A004788, A007318, A022559, A303279, A303281.

Array[Sum[PrimeOmega@ Binomial[#, k], {k, 0, #}] &, 57] (* Michael De Vlieger, Jan 19 2019 *)

a(n) = sum(k=0, n, bigomega(binomial(n, k)));

a(n) = my(t=0); sum(k=1, n, my(b=bigomega(k)); t+=b; k*b-t);

first(n) = my(res = List([0]), r=0, t=0, b=0); for(k=1, n, b=bigomega(k); t += b; r += k*b-t; listput(res, r)); res \\ adapted from Daniel Suteu \\ David A. Corneth, Jan 16 2019

a(n) = A303281(n) - A303279(n), for n > 0.

a(n) = A001222(A001142(n)).

cp's OEIS Frontend

A002064 Cullen numbers: a(n) = n*2^n + 1.

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A303279 Expansion of (1/(1 - x)^2) * Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k)).

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A046688 Antidiagonals of square array in which k-th row (k>0) is an arithmetic progression of difference 2^(k-1).

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A323444 Sum of exponents in prime-power factorization of Product_{k=0..n} binomial(n,k) (A001142).

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