A001026 -id:A001026 - cp's OEIS frontend

A224384 a(n) = 1 + 17^n.

2, 18, 290, 4914, 83522, 1419858, 24137570, 410338674, 6975757442, 118587876498, 2015993900450, 34271896307634, 582622237229762, 9904578032905938, 168377826559400930, 2862423051509815794, 48661191875666868482, 827240261886336764178, 14063084452067724991010

Offset: 0

Philippe Deléham, Apr 05 2013

Sum of n-th powers of divisors of 17.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (18,-17).

Cf. A001026.

17^Range[0,20]+1 (* Harvey P. Dale, Jul 19 2019 *)

a(n) = A001026(n) + 1.
G.f.: 1/(1-x) + 1/(1-17*x).
E.g.f.: exp(x) + exp(17*x).
a(n) = 18*a(n-1) - 17*a(n-2) with a(0) = 2, a(1) = 18.

A264871 Array read by antidiagonals: T(n,m) = (1+2^n)^m; n,m>=0.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 8, 9, 5, 1, 16, 27, 25, 9, 1, 32, 81, 125, 81, 17, 1, 64, 243, 625, 729, 289, 33, 1, 128, 729, 3125, 6561, 4913, 1089, 65, 1, 256, 2187, 15625, 59049, 83521, 35937, 4225, 129, 1, 512, 6561, 78125, 531441, 1419857, 1185921, 274625, 16641, 257

Offset: 0

R. J. Mathar, Nov 27 2015

			       1,       2,       4,       8,      16,      32,
       1,       3,       9,      27,      81,     243,
       1,       5,      25,     125,     625,    3125,
       1,       9,      81,     729,    6561,   59049,
       1,      17,     289,    4913,   83521, 1419857,
       1,      33,    1089,   35937, 1185921,39135393,

Cf. A000079 (row 0), A000244 (row 1), A000351 (row 2), A001019 (row 3), A001026 (row 4), A009977 (row 5), A000051 (column 1), A028400 (column 2), A136516 (main diagonal), A165327 (upper subdiagonal).

Reverse /@ Table[(1 + 2^(n - m))^m, {n, 0, 9}, {m, 0, n}] // Flatten (* Michael De Vlieger, Nov 27 2015 *)

G.f. for row n: 1/(1-(1+2^n)*x). - R. J. Mathar, Dec 15 2015

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).

A333385 a(n) = 3^n + 2 * 17^n for n >= 0.

Original entry on oeis.org

3, 37, 587, 9853, 167123, 2839957, 48275867, 820679533, 13951521443, 237175772677, 4031987859947, 68543792792413, 1165244474990963, 19809156067406197, 336755653123584827, 5724846103033980493, 97322383751376783683, 1654480523772802668517

Offset: 0

Bernard Schott, Mar 18 2020

This sequence was the subject of the 1st problem of the 27th British Mathematical Olympiad in 1991 (see the link BMO).
Proposition: a(n) is never a perfect square.
Proof (by induction): the unit digits of a(n) follow the pattern 3773, 3773, ...
Generalization: Steve Dinh proves that for nonnegative integers k, m, u and v, the numbers (10^k*u + 3)^n + 2*(10^m*v + 7)^n are never a perfect square for n >= 0 (see reference). - Bernard Schott, Dec 27 2021

			a(4) = 3^4 + 2 * 17^4 = 167123 = 7 * 19 * 1031 is not a perfect square.

S. Dinh, The Hard Mathematical Olympiad Problems And Their Solutions, AuthorHouse, 2011, Problem 1 of British Mathematical Olympiad 1991, page 186.
A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Problem 1 pp. 57 and 115 (1991).

Colin Barker, Table of n, a(n) for n = 0..800
British Mathematical Olympiad, Problem 1, 1991.
Index to sequences related to Olympiads.
Index entries for linear recurrences with constant coefficients, signature (20,-51).

Cf. A000244 (3^n), A001026 (17^n), A330770.

Subsequence of A000037.

Maple
```
S:=seq(3^n+2*17^n, n=0..40);
```

a[n_] := 3^n + 2 * 17^n ; Array[a, 18, 0] (* Amiram Eldar, Mar 18 2020 *)

Vec((3 - 23*x) / ((1 - 3*x)*(1 - 17*x)) + O(x^20)) \\ Colin Barker, Mar 18 2020

a(n) = A000244(n) + 2 * A001026(n).
From Colin Barker, Mar 18 2020: (Start)
G.f.: (3 - 23*x) / ((1 - 3*x)*(1 - 17*x)).
a(n) = 20*a(n-1) - 51*a(n-2) for n>1.
(End)

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

A356242 a(n) is the number of Fermat numbers dividing n, counted with multiplicity.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 0, 2, 0, 1, 2, 0, 1, 1, 0, 0, 1, 2, 0, 3, 0, 0, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 0, 1, 0, 0, 3, 0, 0, 1, 0, 2, 2, 0, 0, 3, 1, 0, 1, 0, 0, 2, 0, 0, 2, 0, 1, 1, 0, 1, 1, 1, 0, 2, 0, 0, 3, 0, 0, 1, 0, 1, 4, 0, 0, 1, 2, 0, 1

Offset: 1

Amiram Eldar, Jul 30 2022

nonn

The multiplicity of a divisor d (not necessarily a prime) of n is defined in A169594 (see also the first formula).
A000244(n) is the least number k such that a(k) = n.
The asymptotic density of occurrences of 0 is 1/2.
The asymptotic density of occurrences of 1 is (1/2) * Sum_{k>=0} 1/(2^(2^k)+1) = (1/2) * A051158 = 0.2980315860... .

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fermat Number.
Wikipedia, Fermat number.

Cf. A000244, A000215, A007404, A051158, A051179, A169594, A356241.

Cf. A080307 (positions of nonzeros), A080308 (positions of 0's).

f = Table[(2^(2^n) + 1), {n, 0, 5}]; a[n_] := Total[IntegerExponent[n, f]]; Array[a, 100]

a(n) = Sum_{k>=1} v(A000215(k), n), where v(m, n) is the exponent of the largest power of m that divides n.
a(A000215(n)) = 1.
a(A000244(n)) = a(A000351(n)) = a(A001026(n)) = n.
a(A003593(n)) = A112754(n).
a(n) >= A356241(n).
a(A051179(n)) = n.
a(A080307(n)) > 0 and a(A080308(n)) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} 1/(2^(2^k)) = 0.8164215090... (A007404).

A013885 a(n) = 17^(5*n + 4).

Original entry on oeis.org

83521, 118587876497, 168377826559400929, 239072435685151324847153, 339448671314611904643504117121, 481968572106750915091411825223071697, 684326450885775034048946719925754910487329, 971645701575323882519635342913622589939807491953

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 (1419857).

Cf. A001026.

[17^(5*n+4): n in [0..10]]; // Vincenzo Librandi, May 27 2011

17^(5*Range[0,20]+4) (* or *) NestList[1419857#&,83521,20] (* Harvey P. Dale, May 26 2011 *)

a(n) = 1419857*a(n-1). - Harvey P. Dale, May 26 2011

A017955 Powers of sqrt(17) rounded down.

Original entry on oeis.org

1, 4, 17, 70, 289, 1191, 4913, 20256, 83521, 344365, 1419857, 5854220, 24137569, 99521746, 410338673, 1691869691, 6975757441, 28761784747, 118587876497, 488950340714, 2015993900449, 8312155792152, 34271896307633, 141306648466586, 582622237229761, 2402213023931974

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A000196, A001026 (even bisection), A010473 (sqrt(17)).

[Floor(Sqrt(17^n)): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

a(n)=sqrtint(17^n) \\ Charles R Greathouse IV, Nov 18 2011

a(n) = floor(sqrt(17^n)) = A000196(A001026(n)). - Vincenzo Librandi, Jun 24 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

A224384 a(n) = 1 + 17^n.

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A264871 Array read by antidiagonals: T(n,m) = (1+2^n)^m; n,m>=0.

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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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A333385 a(n) = 3^n + 2 * 17^n for n >= 0.

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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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A356242 a(n) is the number of Fermat numbers dividing n, counted with multiplicity.

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A013885 a(n) = 17^(5*n + 4).

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