A319075 -id:A319075 - cp's OEIS frontend

A062457 a(n) = prime(n)^n.

2, 9, 125, 2401, 161051, 4826809, 410338673, 16983563041, 1801152661463, 420707233300201, 25408476896404831, 6582952005840035281, 925103102315013629321, 73885357344138503765449, 12063348350820368238715343, 3876269050118516845397872321

Offset: 1

Labos Elemer, Jul 09 2001

Heinz numbers of square integer partitions, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Apr 14 2018
Main diagonal of A182944. - Omar E. Pol, Sep 12 2018
Second diagonal of A319075. - Omar E. Pol, Sep 13 2018

T. D. Noe, Table of n, a(n) for n = 1..100

Cf. A000040, A000312, A000720, A000961, A001222, A051674, A062006, A076146, A093358, A201614, A275024, A302242, A302493, A302498, A319075.

[NthPrime(n)^n : n in [1..20]]; // Wesley Ivan Hurt, Jan 18 2016

A062457:=n->ithprime(n)^n: seq(A062457(n), n=1..20); # Wesley Ivan Hurt, Jan 18 2016

Table[Prime[n]^n, {n,20}] (* Harvey P. Dale, Jun 22 2011 *)

a(n) = prime(n)^n; \\ Harry J. Smith, Aug 07 2009

a(n) = A062006(n) - 1. - Wesley Ivan Hurt, Jan 18 2016
From Amiram Eldar, Nov 16 2020: (Start)
Sum_{n>=1} 1/a(n) = A093358.
Sum_{n>=1} (-1)^(n+1)/a(n) = A201614. (End)

A329332 Table of powers of squarefree numbers, powers of A019565(n) in increasing order in row n. Square array A(n,k) n >= 0, k >= 0 read by descending antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 9, 6, 1, 1, 16, 27, 36, 5, 1, 1, 32, 81, 216, 25, 10, 1, 1, 64, 243, 1296, 125, 100, 15, 1, 1, 128, 729, 7776, 625, 1000, 225, 30, 1, 1, 256, 2187, 46656, 3125, 10000, 3375, 900, 7, 1, 1, 512, 6561, 279936, 15625, 100000, 50625, 27000, 49, 14

Offset: 0

Peter Munn, Nov 10 2019

The A019565 row order gives the table neat relationships with A003961, A003987, A059897, A225546, A319075 and A329050. See the formula section.

Transposition of this table, that is reflection about its main diagonal, has subtle symmetries. For example, consider the unique factorization of a number into powers of distinct primes. This can be restated as factorization into numbers from rows 2^n (n >= 0) with no more than one from each row. Reflecting about the main diagonal, this factorization becomes factorization (of a related number) into numbers from columns 2^k (k >= 0) with no more than one from each column. This is also unique and is factorization into powers of squarefree numbers with distinct exponents that are powers of two. See the example section.

			Square array A(n,k) begins:
n\k |  0   1     2      3        4          5           6             7
----+------------------------------------------------------------------
   0|  1   1     1      1        1          1           1             1
   1|  1   2     4      8       16         32          64           128
   2|  1   3     9     27       81        243         729          2187
   3|  1   6    36    216     1296       7776       46656        279936
   4|  1   5    25    125      625       3125       15625         78125
   5|  1  10   100   1000    10000     100000     1000000      10000000
   6|  1  15   225   3375    50625     759375    11390625     170859375
   7|  1  30   900  27000   810000   24300000   729000000   21870000000
   8|  1   7    49    343     2401      16807      117649        823543
   9|  1  14   196   2744    38416     537824     7529536     105413504
  10|  1  21   441   9261   194481    4084101    85766121    1801088541
  11|  1  42  1764  74088  3111696  130691232  5489031744  230539333248
  12|  1  35  1225  42875  1500625   52521875  1838265625   64339296875
Reflection of factorization about the main diagonal: (Start)
The canonical (prime power) factorization of 864 is 2^5 * 3^3 = 32 * 27. Reflecting the factors about the main diagonal of the table gives us 10 * 36 = 10^1 * 6^2 = 360. This is the unique factorization of 360 into powers of squarefree numbers with distinct exponents that are powers of two.
Reflection about the main diagonal is given by the self-inverse function A225546(.). Clearly, all positive integers are in the domain of A225546, whether or not they appear in the table. It is valid to start from 360, observe that A225546(360) = 864, then use 864 to derive 360's factorization into appropriate powers of squarefree numbers as above.
(End)

The range of values is A072774.
Rows (abbreviated list): A000079(1), A000244(2), A000400(3), A000351(4), A011557(5), A001024(6), A009974(7), A000420(8), A001023(9), A009965(10), A001020(16), A001022(32), A001026(64).
A019565 is column 1, A334110 is column 2, and columns that are sorted in increasing order (some without the 1) are: A005117(1), A062503(2), A062838(3), A113849(4), A113850(5), A113851(6), A113852(7).
Other subtables: A182944, A319075, A329050.
Re-ordered subtable of A297845, A306697, A329329.
A000290, A003961, A003987, A059897 and A225546 are used to express relationships between terms of this sequence.
Cf. A285322.

A(n,k) = A019565(n)^k.
A(k,n) = A225546(A(n,k)).
A(n,2k) = A000290(A(n,k)) = A(n,k)^2.
A(2n,k) = A003961(A(n,k)).
A(n,2k+1) = A(n,2k) * A(n,1).
A(2n+1,k) = A(2n,k) * A(1,k).
A(A003987(n,m), k) = A059897(A(n,k), A(m,k)).
A(n, A003987(m,k)) = A059897(A(n,m), A(n,k)).
A(2^n,k) = A319075(k,n+1).
A(2^n, 2^k) = A329050(n,k).
A(n,k) = A297845(A(n,1), A(1,k)) = A306697(A(n,1), A(1,k)), = A329329(A(n,1), A(1,k)).
Sum_{n>=0} 1/A(n,k) = zeta(k)/zeta(2*k), for k >= 2. - Amiram Eldar, Dec 03 2022

A182944 Square array A(i,j), i >= 1, j >= 1, of prime powers prime(i)^j, by descending antidiagonals.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 14 2010

We alternatively refer to this sequence as a triangle T(.,.), with T(n,k) = A(k,n-k+1) = prime(k)^(n-k+1).
The monotonic ordering of this sequence, prefixed by 1, is A000961.
The joint-rank array of this sequence is A182869.
Main diagonal gives A062457. - Omar E. Pol, Sep 11 2018

			Square array A(i,j) begins:
  i \ j: 1      2      3      4      5  ...
  ---\-------------------------------------
  1:     2,     4,     8,    16,    32, ...
  2:     3,     9,    27,    81,   243, ...
  3:     5,    25,   125,   625,  3125, ...
  4:     7,    49,   343,  2401, 16807, ...
  ...
The triangle T(n,k) begins:
  n\k:  1     2     3     4     5     6  ...
  1:    2
  2:    4     3
  3:    8     9     5
  4:   16    27    25     7
  5:   32    81   125    49    11
  6:   64   243   625   343   121    13
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1..150, flattened)
Michael De Vlieger, Diagram showing row n of triangle in a semicircle as noted, with a color function associated with the magnitude of T(n,k) compared to 2^n in light blue, where prime(n) is the smallest and the prime power indicated in red the largest in the row.

Cf. A000961, A006939 (row products of triangle), A062457, A182945, A332979 (row maxima of triangle).
Columns: A000040 (1), A001248 (2), A030078 (3), A030514 (4), A050997 (5), A030516 (6), A092759 (7), A179645 (8), A179665 (9), A030629 (10).
A319075 extends the array with 0th powers.
Subtable of A242378, A284457, A329332.

TableForm[Table[Prime[n]^j,{n,1,14},{j,1,8}]]

From Peter Munn, Dec 29 2019: (Start)
A(i,j) = A182945(j,i) = A319075(j,i).
A(i,j) = A242378(i-1,2^j) = A329332(2^(i-1),j).
A(i,i) = A062457(i).
(End)

Clarified in respect of alternate reading as a triangle by Peter Munn, Aug 28 2022

A093360 a(n) = prime(n)^(n-1).

Original entry on oeis.org

1, 3, 25, 343, 14641, 371293, 24137569, 893871739, 78310985281, 14507145975869, 819628286980801, 177917621779460413, 22563490300366186081, 1718264124282290785243, 256666986187667409334369

Offset: 1

Jorge Coveiro, Apr 28 2004

nonn

Main diagonal of A319075. - Omar E. Pol, Sep 10 2018

Robert Israel, Table of n, a(n) for n = 1..303

Cf. A062457.

Maple
```
seq(ithprime(x)^(x-1),x=1..20);
```

Table[Prime[n]^(n-1),{n,20}] (* Harvey P. Dale, Jun 11 2014 *)

a(n) = prime(n)^(n-1);
vector(20, n, a(n)) \\ Michel Marcus, Sep 13 2018

A182945 Array of prime powers p^j, as transpose of A182944.

Original entry on oeis.org

2, 3, 4, 5, 9, 8, 7, 25, 27, 16, 11, 49, 125, 81, 32, 13, 121, 343, 625, 243, 64, 17, 169, 1331, 2401, 3125, 729, 128, 19, 289, 2197, 14641, 16807, 15625, 2187, 256, 23, 361, 4913, 28561, 161051, 117649, 78125, 6561, 512, 29, 529, 6859, 83521, 371293, 1771561, 823543, 390625, 19683, 1024

Offset: 1

Clark Kimberling, Dec 14 2010

The monotonic ordering of this sequence, with 1 prefixed, is A000961.

The joint-rank array of this sequence is A182869.

			Northwest corner:
   2    3     5     7
   4    9    25    49
   8   27   125   343
  16   81   625  2401

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000961, A182944, A000040 (row 1), A001248 (row 2), A030078 (row 3).
Antidiagonal products give A006939.
Cf. A319075 (extends the array with 0th powers).

[NthPrime(n-i)^i: i in [1..n-1], n in [2..15]]; // Vincenzo Librandi, Jul 28 2015

seq(seq(ithprime(n-i)^i,i=1..n-1),n=2..20); # Robert Israel, Jul 27 2015

width=9;Table[Table[Prime[n]^j,{n,1,width},{j,1,width}]]; Flatten[Table[Table[%[[z-k+1]][[k]],{k,1,z}],{z,1,width}]]

A197987 a(n) = prime(n)^(n+1).

Original entry on oeis.org

4, 27, 625, 16807, 1771561, 62748517, 6975757441, 322687697779, 41426511213649, 12200509765705829, 787662783788549761, 243569224216081305397, 37929227194915558802161, 3177070365797955661914307, 566977372488557307219621121, 205442259656281392806087233013

Offset: 1

Bruno Berselli, Oct 20 2011

Subsequence of A000961, A120458.
First five elements are also consecutive members of A133018. - Omar E. Pol, Oct 20 2011
Third diagonal of A319075. - Omar E. Pol, Sep 13 2018

			The fourth prime number is 7, so a(4) = 7^(4+1) = 7^5 = 16807. - _Omar E. Pol_, Oct 20 2011

Bruno Berselli, Table of n, a(n) for n = 1..50

Cf. A000961, A062457, A093360, A120458, A133018, A319075.

Magma
```
[NthPrime(n)^(n+1): n in [1..16]];
```

Table[Prime[n]^(n+1),{n,20}] (* Harvey P. Dale, Dec 16 2012 *)

for(n=1, 16, print1(prime(n)^(n+1)", "));

a(n) = A000040(n)^(n+1). - Omar E. Pol, Oct 20 2011

A319074 a(n) is the sum of the first n nonnegative powers of the n-th prime.

Original entry on oeis.org

1, 4, 31, 400, 16105, 402234, 25646167, 943531280, 81870575521, 15025258332150, 846949229880161, 182859777940000980, 23127577557875340733, 1759175174860440565844, 262246703278703657363377, 74543635579202247026882160, 21930887362370823132822661921, 2279217547342466764922495586798

Offset: 1

Omar E. Pol, Sep 11 2018

nonn

			For n = 4 the 4th prime is 7 and the sum of the first four nonnegative powers of 7 is 7^0 + 7^1 + 7^2 + 7^3 = 1 + 7 + 49 + 343 = 400, so a(4) = 400.

Main diagonal of A319076.
Cf. A000040, A000203, A006093, A062457, A069459, A093360, A319075.
Cf. A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.
Cf. A126646, A003462, A003463, A023000, A016123, A091030, A091045, A218722, A218726, A218732, A218734, A218740, A218744, A218746, A218750.

a(n) = sum(k=0, n-1, prime(n)^k); \\ Michel Marcus, Sep 13 2018

a(n) = Sum_{k=0..n-1} A000040(n)^k.
a(n) = Sum_{k=0..n-1} A319075(k,n).
a(n) = (A000040(n)^n - 1)/(A000040(n) - 1).
a(n) = (A062457(n) - 1)/A006093(n).
a(n) = A069459(n)/A006093(n).
a(n) = A000203(A000040(n)^(n-1)).
a(n) = A000203(A093360(n)).

A319076 Square array T(n,k) read by antidiagonal upwards in which column k lists the partial sums of the powers of the k-th prime, n >= 0, k >= 1.

Original entry on oeis.org

1, 3, 1, 7, 4, 1, 15, 13, 6, 1, 31, 40, 31, 8, 1, 63, 121, 156, 57, 12, 1, 127, 364, 781, 400, 133, 14, 1, 255, 1093, 3906, 2801, 1464, 183, 18, 1, 511, 3280, 19531, 19608, 16105, 2380, 307, 20, 1, 1023, 9841, 97656, 137257, 177156, 30941, 5220, 381, 24, 1, 2047, 29524, 488281, 960800, 1948717

Offset: 0

Omar E. Pol, Sep 09 2018

T(n,k) is also the sum of the divisors of the n-th nonnegative power of the k-th prime, n >= 0, k >= 1.

			The corner of the square array is as follows:
         A126646 A003462 A003463  A023000    A016123    A091030     A091045
A000012        1,      1,      1,       1,         1,         1,          1, ...
A008864        3,      4,      6,       8,        12,        14,         18, ...
A060800        7,     13,     31,      57,       133,       183,        307, ...
A131991       15,     40,    156,     400,      1464,      2380,       5220, ...
A131992       31,    121,    781,    2801,     16105,     30941,      88741, ...
A131993       63,    364,   3906,   19608,    177156,    402234,    1508598, ...
.......      127,   1093,  19531,  137257,   1948717,   5229043,   25646167, ...
.......      255,   3280,  97656,  960800,  21435888,  67977560,  435984840, ...
.......      511,   9841, 488281, 6725601, 235794769, 883708281, 7411742281, ...
...

Rows 0-5: A000012, A008864, A060800, A131991, A131992, A131993.
Columns 1-15: A126646, A003462, A003463, A023000, A016123, A091030, A091045, A218722, A218726, A218732, A218734, A218740, A218744, A218746, A218750.
Main diagonal gives A319074.
Cf. A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.
Cf. A000040, A000203, A006093, A319075.

T(n, k) = sigma(prime(k)^n); \\ Michel Marcus, Sep 13 2018

T(n,k) = A000203(A000040(k)^n).
T(n,k) = Sum_{j=0..n} A000040(k)^j.
T(n,k) = Sum_{j=0..n} A319075(j,k).
T(n,k) = (A000040(k)^(n+1) - 1)/(A000040(k) - 1).
T(n,k) = (A000040(k)^(n+1) - 1)/A006093(k).

cp's OEIS Frontend

A062457 a(n) = prime(n)^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A329332 Table of powers of squarefree numbers, powers of A019565(n) in increasing order in row n. Square array A(n,k) n >= 0, k >= 0 read by descending antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A182944 Square array A(i,j), i >= 1, j >= 1, of prime powers prime(i)^j, by descending antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A093360 a(n) = prime(n)^(n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A182945 Array of prime powers p^j, as transpose of A182944.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

A197987 a(n) = prime(n)^(n+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A319074 a(n) is the sum of the first n nonnegative powers of the n-th prime.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A319076 Square array T(n,k) read by antidiagonal upwards in which column k lists the partial sums of the powers of the k-th prime, n >= 0, k >= 1.