A009967 -id:A009967 - cp's OEIS frontend

A180731 Smallest power of 23 that begins with n.

1, 23, 3404825447, 41426511213649, 529, 6436343, 78310985281, 865004941741938633917747707002884268046728983, 952809757913927, 10524515126174167358877236351104092889324551536161

Offset: 1

Daniel Mondot, Sep 18 2010

Cf. A009967, A180729, A180732.

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

A013773 a(n) = 23^(3*n + 2).

Original entry on oeis.org

529, 6436343, 78310985281, 952809757913927, 11592836324538749809, 141050039560662968926103, 1716155831334586342923895201, 20880467999847912034355032910567, 254052654154149545721997685422868689

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 (12167).

Subsequence of A009967.

[23^(3*n+2): n in [0..10]]; // Vincenzo Librandi, Jun 27 2011

23^(3*Range[0,20]+2) (* or *) NestList[12167#&,529,20] (* Harvey P. Dale, Nov 22 2021 *)

A013906 a(n) = 23^(5*n + 1).

Original entry on oeis.org

23, 148035889, 952809757913927, 6132610415680998648961, 39471584120695485887249589623, 254052654154149545721997685422868689, 1635170022196481349560959748587682926364327, 10524515126174167358877236351104092889324551536161, 67739389260745218861137988047774370539553852007909099223

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 (6436343).

Cf. A009967, A016861.

[23^(5*n+1): n in [0..10]]; // Vincenzo Librandi, May 29 2011

A013906:=n->23^(5*n + 1); seq(A013906(n), n=0..10); # Wesley Ivan Hurt, Jan 30 2014

Table[23^(5n + 1), {n, 0, 10}] (* Wesley Ivan Hurt, Jan 30 2014 *)

a(n) = 6436343*a(n-1), a(0)=23. - Vincenzo Librandi, May 29 2011
a(n) = A009967(A016861(n)). - Wesley Ivan Hurt, Jan 30 2014
E.g.f.: 23*exp(6436343*x). - Alejandro J. Becerra Jr., Jun 27 2021

A013907 a(n) = 23^(5*n + 2).

Original entry on oeis.org

529, 3404825447, 21914624432020321, 141050039560662968926103, 907846434775996175406740561329, 5843211045545439551605946764725979847, 37608910510519071039902074217516707306379521, 242063847902005849254176436075394136454464685331703

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 (6436343).

Cf. A009967.

[23^(5*n+2): n in [0..10]]; // Vincenzo Librandi, May 29 2011

23^(5*Range[0,10]+2) (* or *) NestList[6436343#&,529,10] (* Harvey P. Dale, Aug 17 2017 *)

a(n) = 6436343*a(n-1), a(0)=529. - Vincenzo Librandi, May 29 2011

E.g.f.: 529*exp(6436343*x). - Alejandro J. Becerra Jr., Jun 27 2021

A013908 a(n) = 23^(5*n + 3).

Original entry on oeis.org

12167, 78310985281, 504036361936467383, 3244150909895248285300369, 20880467999847912034355032910567, 134393854047545109686936775588697536481, 865004941741938633917747707002884268046728983, 5567468501746134532846058029734065138452687762629169

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 (6436343).

Cf. A009967.

[23^(5*n+3): n in [0..10]]; // Vincenzo Librandi, May 29 2011

a(n) = 6436343*a(n-1), a(0)=12167. - Vincenzo Librandi, May 29 2011

A013909 a(n) = 23^(5*n + 4).

Original entry on oeis.org

279841, 1801152661463, 11592836324538749809, 74615470927590710561908487, 480250763996501976790165756943041, 3091058643093537522799545838540043339063, 19895113660064588580108197261066338165074766609, 128051775540161094255459334683883498184411818540470887

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 (6436343).

Cf. A009967.

[23^(5*n+4): n in [0..10]]; // Vincenzo Librandi, May 29 2011

23^(5*Range[0,10]+4) (* or *) NestList[6436343#&,279841,10] (* Harvey P. Dale, Jan 15 2016 *)

a(n) = 6436343*a(n-1), a(0)=279841. - Vincenzo Librandi, May 29 2011

A086874 Seventh power of odd primes.

Original entry on oeis.org

2187, 78125, 823543, 19487171, 62748517, 410338673, 893871739, 3404825447, 17249876309, 27512614111, 94931877133, 194754273881, 271818611107, 506623120463, 1174711139837, 2488651484819, 3142742836021, 6060711605323

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Sep 16 2003

Cf. A000040, A001248, A030078, A030514, A050997, A030516, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.

Prime[Range[2,20]]^7 (* Harvey P. Dale, Sep 16 2013 *)

a(n) = prime(n+1)^7 \\ Michel Marcus, Jul 17 2013

cp's OEIS Frontend

A180731 Smallest power of 23 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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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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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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PARI

Formula

A013773 a(n) = 23^(3*n + 2).

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Magma

Mathematica

A013906 a(n) = 23^(5*n + 1).

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Magma

Maple

Mathematica

Formula

A013907 a(n) = 23^(5*n + 2).

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Magma

Mathematica

Formula

A013908 a(n) = 23^(5*n + 3).

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Magma

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A013909 a(n) = 23^(5*n + 4).

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