A001026 -id:A001026 - cp's OEIS frontend

A319075 Square array T(n,k) read by antidiagonal upwards in which row n lists the n-th powers of primes, hence column k lists the powers of the k-th prime, n >= 0, k >= 1.

1, 2, 1, 4, 3, 1, 8, 9, 5, 1, 16, 27, 25, 7, 1, 32, 81, 125, 49, 11, 1, 64, 243, 625, 343, 121, 13, 1, 128, 729, 3125, 2401, 1331, 169, 17, 1, 256, 2187, 15625, 16807, 14641, 2197, 289, 19, 1, 512, 6561, 78125, 117649, 161051, 28561, 4913, 361, 23, 1, 1024, 19683, 390625, 823543, 1771561, 371293

Offset: 0

Omar E. Pol, Sep 09 2018

If n = p - 1 where p is prime, then row n lists the numbers with p divisors.

The partial sums of column k give the column k of A319076.

			The corner of the square array is as follows:
         A000079 A000244 A000351  A000420    A001020    A001022     A001026
A000012        1,      1,      1,       1,         1,         1,          1, ...
A000040        2,      3,      5,       7,        11,        13,         17, ...
A001248        4,      9,     25,      49,       121,       169,        289, ...
A030078        8,     27,    125,     343,      1331,      2197,       4913, ...
A030514       16,     81,    625,    2401,     14641,     28561,      83521, ...
A050997       32,    243,   3125,   16807,    161051,    371293,    1419857, ...
A030516       64,    729,  15625,  117649,   1771561,   4826809,   24137569, ...
A092759      128,   2187,  78125,  823543,  19487171,  62748517,  410338673, ...
A179645      256,   6561, 390625, 5764801, 214358881, 815730721, 6975757441, ...
...

Rows 0-13: A000012, A000040, A001248, A030078, A030514, A050997, A030516, A092759, A179645, A179665, A030629, A079395, A030631, A138031.
Other rows n: A030635 (n=16), A030637 (n=18), A137486 (n=22), A137492 (n=28), A139571 (n=30), A139572 (n=36), A139573 (n=40), A139574 (n=42), A139575 (n=46), A173533 (n=52), A183062 (n=58), A183085 (n=60), A261700 (n=100).
Columns 1-15: A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.
Main diagonal gives A093360.
Second diagonal gives A062457.
Third diagonal gives A197987.
Removing the 1's we have A182944/ A182945.
Cf. A006093, A319074, A319076.

PARI
```
T(n, k) = prime(k)^n;
```

T(n,k) = A000040(k)^n, n >= 0, k >= 1.

A133356 a(n) = 2a(n-1) + 16a(n-2) for n>1, a(0)=1, a(1)=1.

Original entry on oeis.org

1, 1, 18, 52, 392, 1616, 9504, 44864, 241792, 1201408, 6271488, 31765504, 163874816, 835997696, 4293992448, 21963948032, 112631775232, 576686718976, 2955481841664, 15137951186944, 77563611840512, 397334442672128

Offset: 0

Philippe Deléham, Dec 21 2007

Binomial transform of A001026 (powers of 17), with interpolated zeros .

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,16).

First differences of A161007.

Cf. A001026, A098158, A161007.

[n le 2 select 1 else 2*(Self(n-1) +8*Self(n-2)): n in [1..41]]; // G. C. Greubel, Oct 15 2022

LinearRecurrence[{2,16},{1,1},30] (* Harvey P. Dale, Dec 12 2012 *)

Vec((1-x)/(1-2*x-16*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 12 2012

A133356=BinaryRecurrenceSequence(2,16,1,1)
[A133356(n) for n in range(41)] # G. C. Greubel, Oct 15 2022

G.f.: (1-x)/(1-2*x-16*x^2).
a(n) = Sum_{k=0..n} A098158(n,k)*17^(n-k). - Philippe Deléham, Dec 26 2007
If p[1]=1, and p[i]=17, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j] = p[j-i+1], (i<=j), A[i,j] = -1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. - Milan Janjic, Apr 29 2010
a(n) = (4*i)^(n-1)*(4*i*ChebyshevU(n, -i/4) - ChebyshevU(n-1, -i/4)) = A161007(n) - A161007(n-1). - G. C. Greubel, Oct 15 2022

A159530 Numerator of Hermite(n, 2/17).

Original entry on oeis.org

1, 4, -562, -6872, 947020, 19676144, -2658183224, -78869600288, 10439530923152, 406451155424320, -52680635240539424, -2560010219314727296, 324703437982090748608, 19055044633095311519488, -2363601454465048638962560, -163647826988867455371547136

Offset: 0

N. J. A. Sloane, Nov 12 2009

G. C. Greubel, Table of n, a(n) for n = 0..404

The denominators are the powers of 17, A001026.

Cf. A159521, A159529.

[Numerator((&+[(-1)^k*Factorial(n)*(4/17)^(n-2*k)/( Factorial(k) *Factorial(n-2*k)): k in [0..Floor(n/2)]])): n in [0..30]]; // G. C. Greubel, Jun 09 2018

Numerator[Table[HermiteH[n,2/17],{n,0,50}]] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2011 *)

/* needs version >= 2.4 */
A159530(n)=numerator(polhermite(n,2/17)); /* Joerg Arndt, Apr 30 2011 */

From G. C. Greubel, Jun 02 2018: (Start)
a(n) = 17^n * Hermite(n,2/17).
E.g.f.: exp(4*x-289*x^2).
a(n) = numerator(Sum_{k=0..floor(n/2)} (-1)^k*n!*(4/17)^(n-2*k)/(k!*(n-2*k)!)). (End)

A073213 Sum of two powers of 17.

Original entry on oeis.org

2, 18, 34, 290, 306, 578, 4914, 4930, 5202, 9826, 83522, 83538, 83810, 88434, 167042, 1419858, 1419874, 1420146, 1424770, 1503378, 2839714, 24137570, 24137586, 24137858, 24142482, 24221090, 25557426, 48275138, 410338674, 410338690, 410338962, 410343586, 410422194, 411758530, 434476242, 820677346

Offset: 0

Jeremy Gardiner, Jul 20 2002

			T(2,0) = 17^2 + 17^0 = 290.
Table T(n,m) begins:
      2;
     18,    34;
    290,   306,   578;
   4914,  4930,  5202,  9826;
  83522, 83538, 83810, 88434, 167042;
  ...

T. D. Noe, Rows n = 0..100 of triangle, flattened

Cf. A001026 (powers of 17).
Equals twice A073221.
Sums of two powers of n: A073423 (0), A007395 (1), A173786 (2), A055235 (3), A055236 (4), A055237 (5), A055257 (6), A055258 (7), A055259 (8), A055260 (9), A052216 (10), A073211 (11), A194887 (12), A072390 (13), A055261 (16), A073214 (19), A073215 (23).

Flatten[Table[Table[17^n + 17^m, {m, 0, n}], {n, 0, 7}]] (* T. D. Noe, Jun 18 2013 *)
Union[Total/@Tuples[17^Range[0,10],2]] (* Harvey P. Dale, Apr 09 2015 *)

from math import isqrt
def A073213(n): return 17**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+17**(n-1-(a*(a+1)>>1)) # Chai Wah Wu, Apr 09 2025

T(n,m) = 17^n + 17^m, n = 0, 1, 2, 3, ..., m = 0, 1, 2, 3, ... n.

Bivariate g.f.: (2 - 18*x)/((1 - x)*(1 - 17*x)*(1 - 17*x*y)). - J. Douglas Morrison, Jul 26 2021

A383809 Consecutive states of a linear congruential pseudo-random number generator for Lisp 1985 when started at 1.

Original entry on oeis.org

1, 17, 38, 144, 189, 201, 154, 108, 79, 88, 241, 81, 122, 66, 118, 249, 217, 175, 214, 124, 100, 194, 35, 93, 75, 20, 89, 7, 119, 15, 4, 68, 152, 74, 3, 51, 114, 181, 65, 101, 211, 73, 237, 13, 221, 243, 115, 198, 103, 245, 149, 23, 140, 121, 49, 80, 105, 28

Offset: 1

Sean A. Irvine, May 17 2025

An example of a terrible random number generator.

Periodic with period 125 (well below the modulus of 251).

Sean A. Irvine, Table of n, a(n) for n = 1..125
Richard P. Gabriel, Performance and Evaluation of Lisp Systems, MIT, 1985 (see p. 140).
Stephen K. Park and Keith W. Miller, Random number generators: good ones are hard to find, Communications of the ACM, Vol 31, 10 (1988), 192-201.
Index entries for sequences related to pseudo-random numbers.
Index entries for linear recurrences with constant coefficients, order 125.

Cf. A001026.

Cf. A096550-A096561 other pseudo-random number generators.

a:= proc(n) option remember; `if`(n<2, n,
      irem(17*a(n-1), 251))
    end:
seq(a(n), n=1..58);  # Alois P. Heinz, May 21 2025

NestList[Mod[17*#, 251] &, 1, 100] (* Paolo Xausa, May 21 2025 *)

a(n) = 17 * a(n-1) mod 251.

A013722 a(n) = 17^(2*n + 1).

Original entry on oeis.org

17, 4913, 1419857, 410338673, 118587876497, 34271896307633, 9904578032905937, 2862423051509815793, 827240261886336764177, 239072435685151324847153, 69091933913008732880827217

Offset: 0

N. J. A. Sloane

Sum_{n>=0} 1/a(n) = 17/288. - Jaume Oliver Lafont, Feb 04 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (289).

Bisection of A001026 (17^n).

Cf. A013708-A013721, A013723-A013729.

[17^(2*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 26 2011

seq(17^(2*n+1),n=0..10); # Nathaniel Johnston, Jun 25 2011

17^(2*Range[0,10]+1) (* or *) NestList[289#&,17,10] (* Harvey P. Dale, Aug 15 2012 *)

a(n)=17^(2*n+1) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 28 2008: (Start)
a(n) = 289*a(n-1), a(0)=17.
G.f.: 17/(1-289*x). (End)

A009978 Powers of 34.

Original entry on oeis.org

1, 34, 1156, 39304, 1336336, 45435424, 1544804416, 52523350144, 1785793904896, 60716992766464, 2064377754059776, 70188843638032384, 2386420683693101056, 81138303245565435904, 2758702310349224820736, 93795878551873643905024, 3189059870763703892770816

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 34), L(1, 34), P(1, 34), T(1, 34). Essentially same as Pisot sequences E(34, 1156), L(34, 1156), P(34, 1156), T(34, 1156). See A008776 for definitions of Pisot sequences.

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 34-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

T. D. Noe, Table of n, a(n) for n=0..100
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (34).

Cf. A000079, A001026, A008776.

[34^n: n in [0..100]]; // Vincenzo Librandi, Nov 21 2010

34^Range[0,20] (* or *) NestList[34#&,1,20] (* Harvey P. Dale, May 03 2025 *)

G.f.: 1/(1-34*x). - Philippe Deléham, Nov 24 2008
a(n) = 34^n; a(n) = 34*a(n-1), n > 0; a(0)=1. - Vincenzo Librandi, Nov 21 2010
From Elmo R. Oliveira, Jul 10 2025: (Start)
E.g.f.: exp(34*x).
a(n) = A000079(n)*A001026(n). (End)

A180701 Smallest power of 17 that begins with n.

Original entry on oeis.org

1, 289, 34271896307633, 4913, 582622237229761, 6975757441, 7961145753492658188015880378976844387030440651052782229932477774154576998240582422097, 83521, 9904578032905937

Offset: 1

Daniel Mondot, Sep 18 2010

Cf. A001026, A180699, A180702, A180703.

With[{s=17^Range[0,80]},Table[First[Select[s,First[IntegerDigits[#]]==n&]], {n,9}]] (* Harvey P. Dale, Mar 24 2011 *)

A013806 a(n) = 17^(4*n+1).

Original entry on oeis.org

17, 1419857, 118587876497, 9904578032905937, 827240261886336764177, 69091933913008732880827217, 5770627412348402378939569991057, 481968572106750915091411825223071697, 40254497110927943179349807054456171205137

Offset: 0

N. J. A. Sloane

As phi(a(n)) = (2*17^n)^4 is a perfect biquadrate (where phi is the Euler totient A000010), this is a subsequence of A078164 and A307690. - Bernard Schott, Mar 29 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (83521).

Cf. A000010, A013776, A307690.

Intersection of A001026 and A078164.

[17^(4*n+1): n in [0..15]]; // Vincenzo Librandi, Jul 06 2011

17^(4Range[0,10]+1) (* or *) NestList[83521#&,17,20] (* Harvey P. Dale, May 21 2013 *)

a(0)=17, a(n)=83521*a(n-1). - Harvey P. Dale, May 21 2013
Sum_{n>=0} 1/a(n) = 4913/83520. - Bernard Schott, Mar 29 2022
Sum_{n>=0} (-1)^n/a(n) = 4913/83522. - Bernard Schott, Apr 08 2022

A197351 a(0)=0, a(1)=1, a(2n)=17*a(n), a(2n+1)=a(2n)+1.

Original entry on oeis.org

0, 1, 17, 18, 289, 290, 306, 307, 4913, 4914, 4930, 4931, 5202, 5203, 5219, 5220, 83521, 83522, 83538, 83539, 83810, 83811, 83827, 83828, 88434, 88435, 88451, 88452, 88723, 88724, 88740, 88741, 1419857, 1419858, 1419874, 1419875

Offset: 0

Philippe Deléham, Oct 14 2011

Numbers whose set of base 17 digits is {0,1}.
Sums of distinct powers of 17.
a(n) modulo 2 is the Prouhet-Thue-Morse sequence A010060. - Philippe Deléham, Oct 17 2011.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 45.

Cf. A001026, A104257.

[n: n in [0..1500000] | Set(IntegerToSequence(n, 17)) subset {0, 1}]; // Vincenzo Librandi, Jun 05 2012

Take[Union[Total/@Subsets[17^Range[0,20],5]],40] (* Harvey P. Dale, Dec 17 2011 *)
FromDigits[#,17]&/@Tuples[{0,1},5] (* Vincenzo Librandi, Jun 05 2012 *)

a(n)=Sum_k>=0 {A030308(n,k)*17^k}.

G.f.: (1/(1 - x))*Sum_{k>=0} 17^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

cp's OEIS Frontend

A319075 Square array T(n,k) read by antidiagonal upwards in which row n lists the n-th powers of primes, hence column k lists the powers of the k-th prime, n >= 0, k >= 1.

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A133356 a(n) = 2*a(n-1) + 16*a(n-2) for n>1, a(0)=1, a(1)=1.

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A159530 Numerator of Hermite(n, 2/17).

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A073213 Sum of two powers of 17.

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A383809 Consecutive states of a linear congruential pseudo-random number generator for Lisp 1985 when started at 1.

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Maple

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A013722 a(n) = 17^(2*n + 1).

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A009978 Powers of 34.

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A133356 a(n) = 2a(n-1) + 16a(n-2) for n>1, a(0)=1, a(1)=1.