A002064 -id:A002064 - cp's OEIS frontend

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A050919 a(n) = n*7^n + 1.

1, 8, 99, 1030, 9605, 84036, 705895, 5764802, 46118409, 363182464, 2824752491, 21750594174, 166095446413, 1259557135292, 9495123019887, 71213422649146, 531726889113617, 3954718737782520, 29311444762388083, 216579008522089718, 1595845325952240021, 11729463145748964148

Offset: 0

N. J. A. Sloane, Dec 30 1999

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Leyland, Factors of Cullen and Woodall numbers.
Paul Leyland, Generalized Cullen and Woodall numbers.
Index entries for linear recurrences with constant coefficients, signature (15,-63,49).

Cf. A002064, A036293.

[ n*7^n+1: n in [0..20]]; // Vincenzo Librandi, Sep 16 2011

Table[n*7^n+1,{n,0,20}] (* or *) LinearRecurrence[{15,-63,49},{1,8,99},20] (* Harvey P. Dale, Nov 23 2013 *)

G.f.: -(42*x^2 - 7*x + 1)/((x-1)*(7*x-1)^2). - Colin Barker, Oct 14 2012
From Elmo R. Oliveira, May 03 2025: (Start)
E.g.f.: exp(x)*(1 + 7*x*exp(6*x)).
a(n) = 15*a(n-1) - 63*a(n-2) + 49*a(n-3).
a(n) = A036293(n) + 1. (End)

A064746 a(n) = n*8^n + 1.

Original entry on oeis.org

1, 9, 129, 1537, 16385, 163841, 1572865, 14680065, 134217729, 1207959553, 10737418241, 94489280513, 824633720833, 7146825580545, 61572651155457, 527765581332481, 4503599627370497, 38280596832649217, 324259173170675713, 2738188573441261569, 23058430092136939521

Offset: 0

N. J. A. Sloane, Oct 19 2001

Harry J. Smith, Table of n, a(n) for n = 0..150
Paul Leyland, Factors of Cullen and Woodall numbers.
Paul Leyland, Generalized Cullen and Woodall numbers.
Index entries for linear recurrences with constant coefficients, signature (17,-80,64).

Cf. A002064, A036294.

Table[n*8^n+1,{n,0,20}] (* or *) LinearRecurrence[{17,-80,64},{1,9,129},20] (* Harvey P. Dale, Jul 24 2012 *)

a(n) = { n*8^n + 1 } \\ Harry J. Smith, Sep 24 2009

a(n) = 17*a(n-1) - 80*a(n-2) + 64*a(n-3), a(0)=1, a(1)=9, a(2)=129. - Harvey P. Dale, Jul 24 2012
G.f.: -(56*x^2-8*x+1)/((x-1)*(8*x-1)^2). - Colin Barker, Oct 15 2012
From Elmo R. Oliveira, May 04 2025: (Start)
E.g.f.: exp(x)*(1 + 8*x*exp(7*x)).
a(n) = A036294(n) + 1. (End)

A064750 a(n) = n*12^n + 1.

Original entry on oeis.org

1, 13, 289, 5185, 82945, 1244161, 17915905, 250822657, 3439853569, 46438023169, 619173642241, 8173092077569, 106993205379073, 1390911669927937, 17974858503684097, 231105323618795521, 2958148142320582657, 37716388814587428865, 479219999055934390273, 6070119988041835610113

Offset: 0

N. J. A. Sloane, Oct 19 2001

Harry J. Smith, Table of n, a(n) for n = 0..150
Paul Leyland, Factors of Cullen and Woodall numbers.
Paul Leyland, Generalized Cullen and Woodall numbers.
Index entries for linear recurrences with constant coefficients, signature (25,-168,144).

Cf. A002064, A064758.

[n*12^n + 1: n in [0..30]]; // Vincenzo Librandi, Jun 21 2018

Table[n*12^n+1,{n,0,20}] (* or *) LinearRecurrence[{25,-168,144},{1,13,289},20] (* Harvey P. Dale, Apr 30 2015 *)
CoefficientList[Series[(1 - 12 x + 142 x^2 - 250 x^3 + 1680 x^4 - 1440 x^5) / ((1 - 12 x)^2 (1 - x)), {x, 0, 33}], x] (* Vincenzo Librandi, Jun 21 2018 *)

a(n) = { n*12^n + 1 } \\ Harry J. Smith, Sep 24 2009

a(n) = 25*a(n-1) - 168*a(n-2) + 144*a(n-3), a(0)=1, a(1)=13, a(2)=289. - Harvey P. Dale, Apr 30 2015
G.f.: (1 - 12*x + 142*x^2 - 250*x^3 + 1680*x^4 - 1440*x^5)/((1 - 12*x)^2*(1 - x)). - Vincenzo Librandi, Jun 21 2018
From Elmo R. Oliveira, May 03 2025: (Start)
E.g.f.: exp(x)*(1 + 12*x*exp(11*x)).
a(n) = A064758(n) + 2 for n >= 1. (End)

A242175 Numbers k such that k*2^k + 1 is a semiprime.

Original entry on oeis.org

2, 3, 4, 5, 8, 9, 11, 16, 21, 33, 35, 101, 105, 131, 158, 165, 191, 234, 251, 435, 453, 459, 561, 579, 604, 671, 744, 753, 933, 963, 1041, 1146, 1168, 1254, 1794

Offset: 1

Vincenzo Librandi, May 07 2014

The semiprimes of this form are 9, 25, 65, 161, 2049, 4609, 22529, ... (A242116).
a(35) >= 1528. Below 2000, 1794 and 1961 are in the sequence. Unknown factorization for 1528, 1576, 1908. - Hugo Pfoertner, Jul 29 2019
The k*2^k + 1 corresponding to 1528 and 1576 each have at least three prime factors. - Tyler Busby, Mar 16 2025

FactorDB, Status of 1908*2^1908+1.

Cf. A001358, A002064, A005849, A242116, A242273.

IsSemiprime:=func; [n: n in [2..230] | IsSemiprime(s) where s is n*2^n+1]; // Bruno Berselli, May 08 2014

Mathematica

Select[Range[165], Plus@@Last/@FactorInteger[# * 2^# + 1]==2&]

Formula

A002064(a(n)) = A242116(n). - Amiram Eldar, Nov 27 2019

Extensions

a(17) from Bruno Berselli, May 08 2014

a(18)-a(30) from Luke March, Aug 13 2015

a(31)-a(34) from Hugo Pfoertner, Jul 29 2019

Wrong term 941 removed by Amiram Eldar, Nov 27 2019

a(35) from Tyler Busby, Mar 16 2025

A248917 a(n) = 2^n * n^2 + 1.

Original entry on oeis.org

1, 3, 17, 73, 257, 801, 2305, 6273, 16385, 41473, 102401, 247809, 589825, 1384449, 3211265, 7372801, 16777217, 37879809, 84934657, 189267969, 419430401, 924844033, 2030043137, 4437573633, 9663676417, 20971520001, 45365592065, 97844723713, 210453397505, 451508436993

Offset: 0

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Author

Paul Curtz, Oct 22 2014

Keywords

nonn
easy

Comments

Binomial transform of A118239 (Engel expansion of cosh(1)).

Table of successive differences of a(n):

1, 3, 17, 73, 257, 801, 2305,...

2, 14, 56, 184, 544, 1504,...

12, 42, 128, 360, 960,...

30, 86, 232, 600,...

56, 146, 368,...

90, 222,...

132,...

etc.

Via b(n) = 0, 0, 0 followed by A055580(n), i.e., 0, 0, 0, 1, 7, 31, 111, ... (the main sequence for the recurrence), a link can be found between a(n) and A002064(n): c(n) = b(n+1) - 2*b(n) = 0, 0, 1, 5, 17, 49, 129, ... (the main sequence for the signature (5, -8, 4)).

Examples

a(3) = 9 * 8 + 1 = 73. a(4) = 16 * 16 + 1 = 257. a(5) = 25 * 32 + 1 = 801.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8).

Crossrefs

Cf. A000225, A002064 (Cullen numbers), A006784, A007758, A055580, A118239, A168298.

Programs

Magma

[2^n*n^2+1: n in [0..30]]; // Vincenzo Librandi, Oct 29 2016

Mathematica

Table[n^2 * 2^n + 1, {n, 0, 31}] (* Alonso del Arte, Oct 22 2014 *) LinearRecurrence[{7,-18,20,-8}, {1,3,17,73}, 25] (* G. C. Greubel, Oct 28 2016 *)

PARI

Vec(-(12*x^3-14*x^2+4*x-1)/((x-1)*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Oct 22 2014

PARI

a(n)=n^2<Charles R Greathouse IV, Oct 22 2014

Formula

a(n) = 4*a(n-1) - 4*a(n-2) + 2^(n+1) + 1.

a(n) = A007758(n) + 1.

a(n) = 7*a(n-1) - 18*a(n-2) + 20*a(n-3) - 8*a(n-4). - Jean-François Alcover, Oct 22 2014

G.f.: -(12*x^3-14*x^2+4*x-1) / ((x-1)*(2*x-1)^3). - Colin Barker, Oct 22 2014

E.g.f.: exp(x) + 2*x*(1 + 2*x)*exp(2*x). - G. C. Greubel, Oct 28 2016

A367005 a(n) is the largest prime factor of n*2^n+1 for n>0, and a(0)=1.

Original entry on oeis.org

1, 3, 3, 5, 13, 23, 11, 23, 683, 419, 19, 1733, 199, 11833, 487, 997, 61681, 4691, 211, 5279, 7541, 1914791, 7177, 607, 5233, 6689, 2373919, 336823, 8937209, 6051013, 409, 11681, 25781083, 6031230671, 18803, 32502455213, 934861, 339016085231, 55586743

Offset: 0

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Author

Sean A. Irvine, Oct 31 2023

Keywords

nonn

Links

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

Crossrefs

Cf. A002064, A006530, A005420, A367003, A367004.

Programs

Mathematica

A367005[n_]:=FactorInteger[n*2^n+1][[-1,1]];Array[A367005,50,0] (* Paolo Xausa, Nov 09 2023 *)

Formula

a(n) = A006530(A002064(n)).

Extensions

Name edited by Michel Marcus, Nov 10 2023

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

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Author

N. J. A. Sloane

Keywords

nonn
tabl
easy

Examples

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)

References

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

Links

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

Crossrefs

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.

Programs

Mathematica

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 *)

PARI

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

Formula

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

Extensions

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

A049069 Array T read by antidiagonals: T(k,n) = k*n*2^(n-1) + 1, n >= 0, k >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 5, 1, 4, 9, 13, 1, 5, 13, 25, 33, 1, 6, 17, 37, 65, 81, 1, 7, 21, 49, 97, 161, 193, 1, 8, 25, 61, 129, 241, 385, 449, 1, 9, 29, 73, 161, 321, 577, 897, 1025, 1, 10, 33, 85, 193, 401, 769, 1345, 2049, 2305, 1, 11, 37, 97, 225, 481, 961, 1793, 3073, 4609, 5121

Offset: 0

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Author

N. J. A. Sloane, Clark Kimberling, Michael Somos

Keywords

nonn
tabl
easy

Examples

Antidiagonals: {1}; {1,2}; {1,3,5}; ...

Crossrefs

Transpose of the array in A048472.

Row 1 = (1, 2, 5, 13, 33, ...) = A005183.

Row 2 = (1, 3, 9, 25, 65, ...) = A002064.

Cf. A049513.

Essentially the same as A049513.

Programs

PARI

T(k,n)=k*n*2^(n-1)+1

A049513 Array T by antidiagonals: T(k,n) = k*n*2^(n-1) + 1, n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 9, 13, 1, 1, 5, 13, 25, 33, 1, 1, 6, 17, 37, 65, 81, 1, 1, 7, 21, 49, 97, 161, 193, 1, 1, 8, 25, 61, 129, 241, 385, 449, 1, 1, 9, 29, 73, 161, 321, 577, 897, 1025, 1, 1, 10, 33, 85, 193, 401, 769, 1345, 2049, 2305, 1, 1, 11, 37, 97, 225, 481

Offset: 0

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Author

Michael Somos, Sep 25 1999

Keywords

nonn
tabl
easy

Examples

Antidiagonals: 1; 1,1; 1,2,1; 1,3,5,1; 1,4,9,13,1; ...

Crossrefs

Cf. A000337, A002064, A005183, A016813, A017533, A048474, A049069, A048472.

Essentially the same as A049069.

Programs

PARI

{T(k, n) = k * n * 2^(n-1) + 1}

Formula

A005183(n) = T(1, n), A002064(n) = T(2, n), A048474(n) = T(3, n), A000337(n) = T(4, n), A016813(n) = T(n, 2), A017533(n) = T(n, 3).

A064749 a(n) = n*11^n + 1.

Original entry on oeis.org

1, 12, 243, 3994, 58565, 805256, 10629367, 136410198, 1714871049, 21221529220, 259374246011, 3138428376722, 37661140520653, 448795257871104, 5316497670165375, 62658722541234766, 735195677817154577, 8592599484487994108, 100078511642860166659, 1162022718519876379530

Offset: 0

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Author

N. J. A. Sloane, Oct 19 2001

Keywords

nonn
easy

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Paul Leyland, Factors of Cullen and Woodall numbers.

Paul Leyland, Generalized Cullen and Woodall numbers.

Index entries for linear recurrences with constant coefficients, signature (23,-143,121).

Crossrefs

For a(n)=n*k^n+1: A000012 (k=0), A000027(n+1) (k=1), A002064 (k=2), A050914 (k=3), A050915 (k=4), A050916 (k=5), A050917 (k=6), A050919 (k=7), A064746 (k=8), A064747 (k=9), A064748 (k=10), this sequence (k=11), A064750 (k=12).

Cf. A064757.

Programs

Magma

[n*11^n+1: n in [0..20]]; // Vincenzo Librandi, Sep 16 2011

Maple

k:= 11; f:= gfun:-rectoproc({-1 - (k-1)*n + k*n*a(n-1) - (n-1)*a(n) = 0, a(0) = 1, a(1) = k+1}, a(n), remember): map(f, [$0..20]); # Georg Fischer, Feb 19 2021

Formula

a(n) = A064757(n) + 2 for n>=1. - Georg Fischer, Feb 19 2021

G.f.: -(110*x^2-11*x+1)/((x-1)*(11*x-1)^2). - Alois P. Heinz, Feb 19 2021

From Elmo R. Oliveira, May 03 2025: (Start)

E.g.f.: exp(x)*(1 + 11*x*exp(10*x)).

a(n) = 23*a(n-1) - 143*a(n-2) + 121*a(n-3). (End)

Previous Showing 31-40 of 74 results. Next

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

A050919 a(n) = n*7^n + 1.

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Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A064746 a(n) = n*8^n + 1.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A064750 a(n) = n*12^n + 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A242175 Numbers k such that k*2^k + 1 is a semiprime.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A248917 a(n) = 2^n * n^2 + 1.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A367005 a(n) is the largest prime factor of n*2^n+1 for n>0, and a(0)=1.

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Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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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Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A049069 Array T read by antidiagonals: T(k,n) = kn2^(n-1) + 1, n >= 0, k >= 1.

A049513 Array T by antidiagonals: T(k,n) = kn2^(n-1) + 1, n >= 0, k >= 0.