A002064 -id:A002064 - cp's OEIS frontend

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A128001 Numbers k such that (k-1)*2^k + 1 is prime.

2, 3, 7, 27, 51, 55, 81, 1471, 1483, 8668, 10885, 20803, 32605, 36391, 57004, 61627, 88651, 89731, 133928, 153428

Offset: 1

N. J. A. Sloane, Jan 25 2008

Cf. A029544, A002064, A196303.

Select[Range[200], PrimeQ[(# - 1)*2^# + 1] &]  (* G. C. Greubel, May 08 2018 *)

is(n)=ispseudoprime((n-1)<Charles R Greathouse IV, Jun 06 2017

a(8)-a(14) from Jason Earls, Jan 29 2008
a(15)-a(18) from Charles R Greathouse IV, Oct 09 2011
a(19)-a(20) from Michael S. Branicky, May 14 2025

A129765 Triangle, (1, 1, 2, 2, 2, ...) in every column.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1

Offset: 1

Gary W. Adamson, May 16 2007

Row sums = A004277, (1, 2, 4, 6, 8, 10, ...). Binomial transform of (1, 1, 2, 2, 2, ...) = A000325, starting (1, 2, 5, 12, 27, 58, ...). Binomial transform of A130196 = A130197, a triangle with row sums = the Cullen numbers, A002064.

			First few rows of the triangle:
  1;
  1, 1;
  2, 1, 1;
  2, 2, 1, 1;
  2, 2, 2, 1, 1;
  ...

Cf. A004277, A002064, A000325, A130197.

A129765 := proc(n,m) if abs(n-m)<2 then 1 ; else 2 ; end if ; end proc:
for n from 1 to 18 do for m from 1 to n do printf("%d,", A129765(n,m)) ; od ; od ; # R. J. Mathar, Jun 08 2007

Table[PadLeft[{1,1},n,2],{n,20}]//Flatten (* Harvey P. Dale, May 20 2019 *)

Triangle, (1, 1, 2, 2, 2, ...) in every column. By rows, (1; 1, 1; 2, 1, 1; ...), continuing with (n-2) 2's followed by two 1's. Inverse of A000012 as an infinite lower triangular matrix (all 1's and the rest zeros), signed by columns: (+ - - + + - -, ...).

More terms from R. J. Mathar, Jun 08 2007

A130197 Binomial transform of A130196.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 12, 8, 4, 1, 27, 20, 12, 5, 1, 58, 47, 32, 17, 6, 1, 121, 105, 79, 49, 23, 7, 1, 248, 226, 184, 128, 72, 30, 8, 1, 503, 474, 410, 312, 200, 102, 38, 9, 1, 1014, 977, 884, 722, 512, 302, 140, 47, 10, 1, 2037, 1991, 1861, 1606, 1234, 814, 442, 187, 57, 11, 1

Offset: 1

Gary W. Adamson, May 16 2007

Row sums = A002064, the Cullen numbers: (1, 3, 9, 25, 65, 161, 385, ...).
Left border = A000325, (2^n - n) starting (1, 2, 5, 12, 27, 58, ...).
Equals A129689 with first column removed. - Georg Fischer, Jul 25 2023

			First few rows of the triangle:
    1;
    2,   1;
    5,   3,  1;
   12,   8,  4,  1;
   27,  20, 12,  5,  1;
   58,  47, 32, 17,  6, 1;
  121, 105, 79, 49, 23, 7, 1;
  ...

Cf. A000325, A002064, A007318, A129689, A130196.

A007318 * A130196 as infinite lower triangular matrices.

a(32) corrected and more terms from Georg Fischer, Jul 25 2023

A173474 Numbers n such that n*2^n + 1 is not prime.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72

Offset: 1

Juri-Stepan Gerasimov, Feb 19 2010

nonn

Complement of "prime Cullen numbers" A005849.
Where a(n)=n for n <= 140, and a(141)=142,..., a(4711)=4712, a(4712)=4714,..., a(5792)=5794, a(5793)=5796,..., a(6607)=6610, a(6608)=6612,..., a(18491)=18495, a(18492)=18497,..., a(32286)=32291, a(32287)=32293,..., a(32462)=32468, a(32463)=32470,..., a(59648)=59655, a(59649)=59657,..., a(90816)=90824, a(90817)=90826,..., a(262403)=262418, a(262404)=262420,..., a(361264)=361274, a(361265)=361276,..., a(481887)=481898, a(481888)=481900,..., a(1354815)=1354827, a(1354816)=1354829,..., a(6328534)=6328547, a(6328535)=6328549,...
Otherwise said, this includes all nonnegative integers except for the "prime Cullen numbers" (more precisely, indices of primes in A002064): 1, 140, 4713, 5795, ... listed in A005849. - M. F. Hasler, Jan 18 2015

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A005849, A045344, A050920, A087156, A141468.

nnnpQ[n_]:=Module[{c=n 2^n+1},!PrimeQ[c]&&c>=0]; Select[Range[0,100], nnnpQ] (* Harvey P. Dale, Aug 23 2011 *)

Corrected and edited by M. F. Hasler, Jan 18 2015

Name edited by Michel Marcus, Nov 02 2017

A181527 Binomial transform of A113127; (1, 1, 3, 7, 15, 31, ...) convolved with (1, 3, 7, 15, 31, 63, ...).

Original entry on oeis.org

1, 4, 13, 38, 103, 264, 649, 1546, 3595, 8204, 18445, 40974, 90127, 196624, 426001, 917522, 1966099, 4194324, 8912917, 18874390, 39845911, 83886104, 176160793, 369098778, 771751963, 1610612764, 3355443229, 6979321886, 14495514655, 30064771104, 62277025825

Offset: 0

Gary W. Adamson, Oct 26 2010

A181527 = Partial sums of (A002064 Cullen numbers: n*2^n+1). - Vladimir Joseph Stephan Orlovsky, Jul 09 2011

Form a triangle with T(1,1) = n, T(2,1) = T(2,2) = n-1, T(3,1) = T(3,3) = n-2, ..., T(n,1) = T(n,n) = 1. The interior members are T(i,j) = T(i-1,j-1) + T(i-1,j). The sum of all members for a triangle of size n is a(n-1). Example for n = 5: row(1) = 5; row(2) = 4, 4; row(3) = 3, 8, 3; row(4) = 2, 11, 11, 2; row(5) = 1, 13, 22, 13, 1. The sum of all members is 103 = a(4). - J. M. Bergot, Oct 16 2012

			a(4) = 103 = (1, 1, 3, 7, 15) dot (31, 15, 7, 3, 1) = (31 + 15 + 21, + 21 + 15)
a(3) = 38 = (1, 3, 3, 1) dot (1, 3, 6, 10) = (1 + 9 + 18 + 10).

Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Cf. A002064, A113127.

Accumulate[Table[n*2^n + 1, {n, 0, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jul 09 2011 *)
LinearRecurrence[{6,-13,12,-4},{1,4,13,38},40] (* Harvey P. Dale, Apr 14 2016 *)

Binomial transform of A113127; (1, 3, 7, 15, 31, ...) convolved with (1, 1, 3, 7, 15, 31, ...).
From R. J. Mathar, Oct 30 2010: (Start)
a(n) = 3+  n + 2^(n+1)*(n-1) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4).
G.f.: ( 1-2*x+2*x^2 ) / ( (2*x-1)^2*(x-1)^2 ). (End)

A242116 Cullen semiprimes: Semiprimes of the form k*2^k + 1.

Original entry on oeis.org

9, 25, 65, 161, 2049, 4609, 22529, 1048577, 44040193, 283467841537, 1202590842881, 256065421246102339102334047485953, 4259306016766850789028922770063361, 356615920533143509709616588588493085605889, 57729314674570665269045550892293179276409335447553

Offset: 1

K. D. Bajpai, May 04 2014

nonn

The k-th Cullen number Cullen(k) = k*2^k + 1.
If Cullen(k) is semiprime, it is in the sequence.
The next term, a(16), has 52 digits.

			a(4) = 161 = (5*2^5+1) is 5th Cullen number and 161 = 7 * 23 is semiprime.
a(5) = 2049 = (8*2^8+1) is 8th Cullen number and 2049 = 3 * 683 is semiprime.

Tyler Busby, Table of n, a(n) for n = 1..35 (terms 1..16 from K. D. Bajpai, terms 17..34 from Amiram Eldar)

Cf. A001358, A002064, A003261, A005849, A242175.

IsSemiprime:=func; [s: n in [1..200] | IsSemiprime(s) where s is n*2^n+1]; // // Vincenzo Librandi, May 07 2014

Maple

with(numtheory): A242116:= proc(); if bigomega(x*2^x+1) = 2 then RETURN (x*2^x+1); fi; end: seq(A242116 (), x=1..200);

Mathematica

cullen[n_] := n * 2^n + 1; Select[cullen[Range[35]], PrimeOmega[#] == 2 &] (* Amiram Eldar, Nov 27 2019 *)

PARI

select(n->bigomega(n)==2, vector(90,n,n<Charles R Greathouse IV, May 06 2014

Formula

a(n) = A002064(A242175(n)). - Amiram Eldar, Nov 27 2019

A278910 Triangle of order m: C(n,k) = k*(n-k+1)^(k+m)+n-k, 0 <= k <= n, m = 0, read by rows.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 5, 9, 3, 4, 7, 20, 25, 4, 5, 9, 35, 83, 65, 5, 6, 11, 54, 195, 326, 161, 6, 7, 13, 77, 379, 1027, 1217, 385, 7, 8, 15, 104, 653, 2504, 5123, 4376, 897, 8, 9, 17, 135, 1035, 5189, 15629, 24579, 15311, 2049, 9, 10, 19, 170, 1543, 9610, 38885, 93754, 114691, 52490, 4609, 10

Offset: 0

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Author

Juri-Stepan Gerasimov, Nov 30 2016

Keywords

nonn
tabl
easy

Examples

As an infinite triangular array: 0 1 1 2 3 2 3 5 9 3 4 7 20 25 4 5 9 35 83 65 5 As an infinite square array (matrix): 0 1 2 3 4 5 1 3 9 25 65 161 2 5 20 83 326 1217 3 7 35 195 1027 5123 4 9 54 379 2504 15629 5 11 77 653 5189 38885

Crossrefs

Cf. A002064.

Cf. Triangles of order m: A003056 (m = -k), A059036 (m = 1-k).

Programs

Magma

/* As triangle */ [[k*(n-k+1)^k+n-k: k in [0..n]]: n in [0..10]];

Maple

A278910 := proc(n,k) k*(n-k+1)^k+n-k ; end proc: seq(seq(A278910(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Dec 02 2016

A321123 a(n) = 2^n + 2*n^2 + 2*n + 1.

Original entry on oeis.org

2, 7, 17, 33, 57, 93, 149, 241, 401, 693, 1245, 2313, 4409, 8557, 16805, 33249, 66081, 131685, 262829, 525049, 1049417, 2098077, 4195317, 8389713, 16778417, 33555733, 67110269, 134219241, 268437081, 536872653, 1073743685, 2147485633, 4294969409, 8589936837

Offset: 0

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Author

Franck Maminirina Ramaharo, Dec 17 2018

Keywords

nonn
easy

Comments

For n >= 2, a(n) is the number of evaluation points on the n-dimensional cube in Genz and Malik's degree 7 cubature rule.

Links

Marius A. Burtea, Table of n, a(n) for n = 0..201

Ronald Cools, Encyclopaedia of Cubature Formulas

Ronald Cools and Philip Rabinowitz, Monomial cubature rules since "Stroud": a compilation, Journal of Computational and Applied Mathematics Vol. 48 (1993), 309-326.

Alan C. Genz and Awais A. Malik, Remarks on algorithm 006: An adaptive algorithm for numerical integration over an N-dimensional rectangular region, Journal of Computational and Applied Mathematics Vol. 6 (1980), 295-302.

Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Crossrefs

Cf. A000051, A000079, A001787, A001844, A002064, A005126, A058331, A100314, A131830, A100314, A132750, A176691, A322593.

Programs

Magma

[2^n + 2*n^2 + 2*n + 1: n in [0..33]]; // Marius A. Burtea, Dec 28 2018

Mathematica

Table[2^n + 2*n^2 + 2*n + 1, {n, 0, 50}]

Maxima

makelist(2^n + 2*n^2 + 2*n + 1, n, 0, 50);

Formula

a(n) = A000079(n) + A001844(n).

a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4), n >= 4.

G.f.: (2 - 3*x - 3*x^3)/((1 - 2*x)*(1 - x)^3).

E.g.f.: exp(2*x) + (1 + 4*x + 2*x^2)*exp(x).

A322593 a(n) = 2^n + 2*n^2 + 1.

Original entry on oeis.org

2, 5, 13, 27, 49, 83, 137, 227, 385, 675, 1225, 2291, 4385, 8531, 16777, 33219, 66049, 131651, 262793, 525011, 1049377, 2098035, 4195273, 8389667, 16778369, 33555683, 67110217, 134219187, 268437025, 536872595, 1073743625, 2147485571, 4294969345, 8589936771

Offset: 0

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Author

Franck Maminirina Ramaharo, Dec 18 2018

Keywords

nonn
easy

Comments

For n = 3..7, a(n) is the number of evaluating points on the n-dimensional sphere (also n-space with weight function exp(-r^2) or exp(-r)) in a degree 7 cubature rule.

References

Arthur H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall, 1971.

Links

Marius A. Burtea, Table of n, a(n) for n = 0..200

Ronald Cools, Encyclopaedia of Cubature Formulas

Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Crossrefs

Cf. A000051, A000079, A001580, A001787, A002064, A005126, A058331, A100314, A131830, A100314, A132750, A176691, A321123.

Programs

Magma

[2^n + 2*n^2 + 1: n in [0..33]]; // Marius A. Burtea, Dec 28 2018

Mathematica

Table[2^n + 2*n^2 + 1, {n, 0, 50}] LinearRecurrence[{5,-9,7,-2},{2,5,13,27},50] (* Harvey P. Dale, Mar 23 2021 *)

Maxima

makelist(2^n + 2*n^2 + 1, n, 0, 50);

Formula

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n >= 4.

a(n) = a(n-1) + A100315(n-1), n >= 2.

G.f.: (2 - 5*x + 6*x^2 - 7*x^3)/((1 - 2*x)*(1 - x)^3)

E.g.f.: exp(2*x) + (1 + 2*x + 2*x^2)*exp(x).

A367008 Number of prime factors of n*2^n + 1, counted with multiplicity.

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 3, 3, 2, 2, 4, 2, 3, 3, 3, 3, 2, 4, 4, 4, 5, 2, 4, 4, 4, 4, 5, 4, 3, 3, 6, 6, 3, 2, 5, 2, 4, 3, 4, 3, 3, 4, 4, 5, 4, 4, 5, 5, 4, 5, 6, 3, 6, 3, 5, 4, 4, 5, 5, 4, 4, 7, 3, 3, 7, 5, 9, 5, 4, 5, 5, 6, 5, 5, 5, 5, 5, 4, 4, 6, 4, 4, 4, 5, 4, 7, 6

Offset: 0

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Author

Sean A. Irvine, Oct 31 2023

Keywords

nonn

Comments

The numbers n*2^n+1 are called Cullen numbers.

Links

Amiram Eldar, Table of n, a(n) for n = 0..865

Crossrefs

Cf. A002064, A001222, A366899, A367007.

Programs

Mathematica

Table[PrimeOmega[n*2^n + 1], {n, 0, 100}] (* Amiram Eldar, Jan 06 2024 *)

PARI

a(n) = bigomega(n*2^n + 1); \\ Amiram Eldar, Jan 06 2024

Formula

a(n) = bigomega(n*2^n + 1) = A001222(A002064(n)).

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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.
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cp's OEIS Frontend

A128001 Numbers k such that (k-1)*2^k + 1 is prime.

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Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Extensions

A129765 Triangle, (1, 1, 2, 2, 2, ...) in every column.

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Comments

Examples

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Maple

Mathematica

Formula

Extensions

A130197 Binomial transform of A130196.

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Formula

Extensions

A173474 Numbers n such that n*2^n + 1 is not prime.

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Crossrefs

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Mathematica

Extensions

A181527 Binomial transform of A113127; (1, 1, 3, 7, 15, 31, ...) convolved with (1, 3, 7, 15, 31, 63, ...).

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Crossrefs

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Mathematica

Formula

A242116 Cullen semiprimes: Semiprimes of the form k*2^k + 1.

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Magma

Maple

Mathematica

PARI

Formula

A278910 Triangle of order m: C(n,k) = k*(n-k+1)^(k+m)+n-k, 0 <= k <= n, m = 0, read by rows.

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Examples

Crossrefs

Programs

Magma

Maple

A321123 a(n) = 2^n + 2*n^2 + 2*n + 1.

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A321123 a(n) = 2^n + 2n^2 + 2n + 1.