A001029 -id:A001029 - 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.

A073214 Sum of two powers of 19.

Original entry on oeis.org

2, 20, 38, 362, 380, 722, 6860, 6878, 7220, 13718, 130322, 130340, 130682, 137180, 260642, 2476100, 2476118, 2476460, 2482958, 2606420, 4952198, 47045882, 47045900, 47046242, 47052740, 47176202, 49521980, 94091762, 893871740, 893871758, 893872100, 893878598, 894002060, 896347838, 940917620, 1787743478

Offset: 0

Jeremy Gardiner, Jul 20 2002

			T(2,0) = 19^2 + 19^0 = 362.
Table begins:
       2;
      20,     38;
     362,    380,    722;
    6860,   6878,   7220,  13718;
  130322, 130340, 130682, 137180, 260642;
  ...

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

Cf. A001029.
Equals twice A073222.
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), A073213 (17), A073215 (23).

Flatten[Table[Table[19^n + 19^m, {m, 0, n}], {n, 0, 7}]] (* T. D. Noe, Jun 18 2013 *)
Total/@Tuples[19^Range[0,10],2]//Union (* Harvey P. Dale, Jan 04 2019 *)

from math import isqrt
def A073214(n): return 19**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+19**(n-1-(a*(a+1)>>1)) # Chai Wah Wu, Apr 09 2025

T(n,m) = 19^n + 19^m for n >= 0 and m in [0..n].

Bivariate g.f.: (2 - 20*x) / ((1 - x) * (1 - 19*x) * (1 - 19*x*y)). - J. Douglas Morrison, Jul 28 2021

A013724 a(n) = 19^(2*n + 1).

Original entry on oeis.org

19, 6859, 2476099, 893871739, 322687697779, 116490258898219, 42052983462257059, 15181127029874798299, 5480386857784802185939, 1978419655660313589123979, 714209495693373205673756419

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (361).

Bisection of A001029.

Cf. A013708-A013723, A013725-A013729.

[19^(2*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 26 2011

seq(19^(2*n+1),n=0..10); # Nathaniel Johnston, Jun 25 2011

NestList[361#&,19,20] (* Harvey P. Dale, Jun 12 2017 *)

a(n)=19^(2*n+1) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 28 2008: (Start)
a(n) = 361*a(n-1); a(0)=19.
G.f.: 19/(1-361*x). (End)

A009982 Powers of 38.

Original entry on oeis.org

1, 38, 1444, 54872, 2085136, 79235168, 3010936384, 114415582592, 4347792138496, 165216101262848, 6278211847988224, 238572050223552512, 9065737908494995456, 344498040522809827328, 13090925539866773438464, 497455170514937390661632, 18903296479567620845142016

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 38), L(1, 38), P(1, 38), T(1, 38). Essentially same as Pisot sequences E(38, 1444), L(38, 1444), P(38, 1444), T(38, 1444). 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 38-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011. [See A000244 for a proof.]

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 (38).

Cf. A000079, A000244, A001029, A008776.

[38^n: n in [0..20]]; // Vincenzo Librandi, Nov 21 2010

38^Range[0, 19] (* Alonso del Arte, Feb 18 2017 *)

PARI
```
a(n)=38^n \\ M. F. Hasler, Feb 21 2017
```

G.f.: 1/(1 - 38*x). - Philippe Deléham, Nov 24 2008
a(n) = 38^n; a(n) = 38*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(38*x).
a(n) = A000079(n)*A001029(n). (End)

A180705 Smallest power of 19 that begins with n.

Original entry on oeis.org

1, 2476099, 361, 47045881, 5480386857784802185939, 6859, 799006685782884121, 893871739, 93076495688256089536609610280499, 104127350297911241532841, 116490258898219, 12129821994589221844500501021364910179, 130321

Offset: 1

Daniel Mondot, Sep 18 2010

Cf. A001029, A180703, A180706, A180725.

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).

A339794 a(n) is the least integer k satisfying rad(k)^2 < sigma(k) and whose prime factors set is the same as the prime factors set of A005117(n+1).

Original entry on oeis.org

4, 9, 25, 18, 49, 80, 121, 169, 112, 135, 289, 361, 441, 352, 529, 416, 841, 360, 961, 891, 1088, 875, 1369, 1216, 1053, 1681, 672, 1849, 1472, 2209, 2601, 2809, 3025, 3249, 1856, 3481, 3721, 1984, 4225, 1584, 4489, 4761, 1960, 5041, 5329, 4736, 5929, 2496, 6241

Offset: 1

Michel Marcus, Dec 17 2020

nonn

Equivalently, subsequence of terms of A339744 excluding terms whose prime factor set has already been encountered.
a(n) = A005117(n + 1)^2 when A005117(n + 1) is prime. Proof: if A005117(n + 1) is a prime p then rad(A005117(n + 1))^2 = rad(p)^2 = p^2 and so integers whose prime factors set is the same as the prime factors set of A005117(n + 1) = p are p^m where m >= 1. p^2 > sigma(p^1) = p + 1 but p^2 < sigma(p^2) = p^2 + p + 1. Q.E.D. - David A. Corneth, Dec 19 2020
From Bernard Schott, Jan 19 2021: (Start)
Indeed, a(n) satisfies the double inequality A005117(n+1) < a(n) <= A005117(n+1)^2.
It is also possible that a(n) = A005117(n+1)^2, even when A005117(n+1) is not prime; the smallest such example is for a(13) = 441 = 21^2 = A005117(14)^2. (End)

			   n  a(n) prime factor set
   1    4  [2]           A000079
   2    9  [3]           A000244
   3   25  [5]           A000351
   4   18  [2, 3]        A033845
   5   49  [7]           A000420
   6   80  [2, 5]        A033846
   7  121  [11]          A001020
   8  169  [13]          A001022
   9  112  [2, 7]        A033847
  10  135  [3, 5]        A033849
  11  289  [17]          A001026
  12  361  [19]          A001029
  13  441  [3, 7]        A033850
  14  352  [2, 11]       A033848
  15  529  [23]          A009967
  16  416  [2, 13]       A288162
  17  841  [29]          A009973
  18  360  [2, 3, 5]     A143207

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Michel Marcus)

Cf. A000203 (sigma), A007947 (rad).
Cf. A005117 (squarefree numbers), A027748, A265668, A339744.
Subsequence: A001248 (squares of primes).

u(n) = {my(fn=factor(n)[,1]); for (k = n, n^2, my(fk = factor(k)); if (fk[,1] == fn, if (factorback(fk[,1])^2 < sigma(fk), return (k));););}
lista(nn) = {for (n=2, nn, if (issquarefree(n), print1(u(n), ", ");););}

a(n) <= A005117(n+1)^2. - David A. Corneth, Dec 19 2020

A013810 a(n) = 19^(4*n + 1).

Original entry on oeis.org

19, 2476099, 322687697779, 42052983462257059, 5480386857784802185939, 714209495693373205673756419, 93076495688256089536609610280499, 12129821994589221844500501021364910179, 1580770532156861979997149793605296459437459

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (130321).

Cf. A001029, A016813.

[19^(4*n+1): n in [0..15]]; // Vincenzo Librandi, Jul 06 2011

A013810:=n->19^(4*n + 1); seq(A013810(n), n=0..15); # Wesley Ivan Hurt, Jan 28 2014

Table[19^(4n + 1), {n, 0, 15}] (* Wesley Ivan Hurt, Jan 28 2014 *)
NestList[130321#&,19,20] (* Harvey P. Dale, Apr 12 2023 *)

a(n) = 19^A016813(n) = A001029(A016813(n)). - Wesley Ivan Hurt, Jan 28 2014

A013890 a(n) = 19^(5*n + 1).

Original entry on oeis.org

19, 47045881, 116490258898219, 288441413567621167681, 714209495693373205673756419, 1768453418076865701195582595329481, 4378865740046709085864680868712732574619, 10842505080063916320800450434338728415281531281, 26847115986241183138017674520015691090350184323352819

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..50
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2476099).

Cf. A001029.

[19^(5*n+1): n in [0..10]]; // Vincenzo Librandi, May 27 2011

NestList[2476099#&,19,20] (* Harvey P. Dale, Dec 23 2016 *)

a(n) = 2476099*a(n-1), a(0)=19. - Vincenzo Librandi, May 27 2011

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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A073214 Sum of two powers of 19.

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Mathematica

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A013724 a(n) = 19^(2*n + 1).

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Magma

Maple

Mathematica

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A009982 Powers of 38.

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Magma

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A180705 Smallest power of 19 that begins with n.

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

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PARI

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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.

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Formula

A339794 a(n) is the least integer k satisfying rad(k)^2 < sigma(k) and whose prime factors set is the same as the prime factors set of A005117(n+1).

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