A019522 -id:A019522 - cp's OEIS frontend

A067103 a(n) = floor(X/Y), where X = concatenation of cubes and Y = concatenation of natural numbers.

1, 1, 14, 148, 14804, 1480398, 148039049, 14803895356, 1480389427723, 148038942652481, 14803894265116205, 1480389426511476635, 148038942651147507639, 14803894265114750596056, 1480389426511475059425814, 148038942651147505942389607, 14803894265114750594238756940

Offset: 1

Robert G. Wilson v, Jan 09 2002

a(n) -> 148038942651147505942387547594667814093751032610233441970375...

			a(6) = floor(182764125216/123456) = floor(1480398.888802...) = 1480398.

Cf. A000578, A007908, A019522.
Cf. A067091, A067092, A067093, A067094, A067095, A067096, A067097, A067098, A067099, A067100, A067101, A067102.
See also A066700.

a:= n-> floor(parse(cat(i^3$i=1..n))/parse(cat($1..n))):
seq(a(n), n=1..17);  # Alois P. Heinz, May 25 2022

f[n_] := (k = 1; x = y = "0"; While[k < n + 1, x = StringJoin[x, ToString[k^3]]; y = StringJoin[y, ToString[k]]; k++ ]; Return[ Floor[ ToExpression[x] / ToExpression[y]]] ); Table[ f[n], {n, 1, 20} ]
nn=20;With[{c=Table[IntegerDigits[n^3],{n,nn}],s=Table[IntegerDigits[n],{n,nn}]}, Table[Floor[FromDigits[Flatten[Take[c,i]]]/FromDigits[Flatten[Take[s,i]]]],{i,nn}]] (* Harvey P. Dale, Feb 10 2013 *)

c1(n) = my(s=""); for(k=1, n, s=Str(s, k)); eval(s); \\ A007908
c3(n) = my(s=""); for(k=1, n, s=Str(s, k^3)); eval(s); \\ A019522
a(n) = c3(n)\c1(n); \\ Michel Marcus, May 25 2022

A133820 Triangle whose rows are sequences of increasing cubes: 1; 1,8; 1,8,27; ... .

Original entry on oeis.org

1, 1, 8, 1, 8, 27, 1, 8, 27, 64, 1, 8, 27, 64, 125, 1, 8, 27, 64, 125, 216, 1, 8, 27, 64, 125, 216, 343, 1, 8, 27, 64, 125, 216, 343, 512, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

Offset: 1

Peter Bala, Sep 25 2007

Reading the triangle by rows produces the sequence 1,1,8,1,8,27,1,8,27,64,..., analogous to A002260.

			Triangle starts
1;
1, 8;
1, 8, 27;
1, 8, 27, 64;
1, 8, 27, 64, 125;

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Cf. A000537 (row sums), A002260, A019522, A133819, A133821, A133823.

a133820 n k = a133820_tabl !! (n-1) !! (k-1)
a133820_row n = a133820_tabl !! (n-1)
a133820_tabl = map (`take` (tail a000578_list)) [1..]
-- Reinhard Zumkeller, Nov 11 2012

Module[{nn=10,c},c=Range[nn]^3;Flatten[Table[Take[c,n],{n,10}]]] (* Harvey P. Dale, Mar 05 2014 *)

O.g.f.: (1+4qx+q^2x^2)/((1-x)(1-qx)^4) = 1 + x(1 + 8q) + x^2(1 + 8q + 27q^2) + ... .

Offset changed by Reinhard Zumkeller, Nov 11 2012

A066700 The leading digits in the terms in A067103 converge; dividing by a suitable power of 10 they converge to the number shown below; sequence gives continued fraction for this number.

Original entry on oeis.org

1, 2, 12, 4, 34, 1, 22, 1, 4, 1, 1, 2, 1, 1, 17, 16, 3, 1, 1, 2, 1, 1, 1, 5, 1, 1, 3, 3, 14, 2, 107, 1, 1, 8, 5, 4, 7, 1, 4, 1, 6, 3, 19, 3, 1, 1, 2, 3, 5, 76, 1, 1, 2, 1, 1, 90, 2, 2, 48717, 1, 1, 1, 3, 1, 1, 2, 4, 1, 1, 1, 14, 2, 1, 1, 2, 4, 28, 2, 3, 46, 1, 1, 3, 1, 1, 1, 2, 1, 5, 12, 1, 1, 3, 3, 1, 2, 3, 1, 78, 1, 1, 1, 3, 2, 4, 1, 6, 1, 1, 1048, 1, 3, 1, 1, 2, 3, 4, 1, 2, 4, 3, 8, 1, 12, 5, 1, 1, 7, 1, 11, 11, 1, 118, 6, 1, 2, 1, 5, 3, 1, 1, 1, 2, 3, 1, 2, 1, 1, 2, 2, 3, 5, 4, 1, 12, 147838832589501802758390, 1, 10, 1, 1, 1, 2, 4, 6, 10, 2, 8, 1, 2, 1, 1, 7, 1, 1, 1, 3, 9, 1, 1, 1, 55, 1, 1, 1, 1, 2

Offset: 1

Randall L Rathbun, Jan 12 2002

			1.480389426511475059423875475946678140937510326102334419703757169...

Cf. A007908, A019522, A067103.

For a more dramatic continued fraction see A030167.

{A067103(n)= c=0; d=0; for(i=1,n, c=c*10^(1+floor(3*log(i)/log(10)))+i^3; d=d*10^(1+floor(log(i)/log(10)))+i; ); floor(c/d) }

c1(n) = my(s=""); for(k=1, n, s=Str(s, k)); eval(s); \\ A007908
c3(n) = my(s=""); for(k=1, n, s=Str(s, k^3)); eval(s); \\ A019522
lista() = my(nn=1000); default(realprecision, 1000); my(x=c3(nn)\c1(nn)); x = x/10.^(#Str(x)-1); contfrac(x); \\ Michel Marcus, May 25 2022

A284377 Concatenation of the first n positive 4th powers.

Original entry on oeis.org

1, 116, 11681, 11681256, 11681256625, 116812566251296, 1168125662512962401, 11681256625129624014096, 116812566251296240140966561, 11681256625129624014096656110000, 1168125662512962401409665611000014641, 116812566251296240140966561100001464120736

Offset: 1

XU Pingya, Mar 25 2017

Eric Weisstein's World of Mathematics, Consecutive Number Sequences

Cf. A007908, A019521, A019522, A359403.

a(n) = my(s=""); for (k=1, n, s = concat(s, Str(k^4))); eval(s); \\ Michel Marcus, Dec 30 2022

a(8) corrected by Georg Fischer, Dec 30 2022

A359403 Primes that are the concatenation of the first m consecutive k-th powers.

Original entry on oeis.org

149, 11681, 164729, 1102459049, 1262144387420489, 1472236648286964521369622528399544939174411840147874772641, 1755578637259143234191361824800363140073127359051977856583921

Offset: 1

Michel Marcus, Dec 30 2022

The following comments were moved from A284377 when this sequence was created.
From XU Pingya, Mar 25 2017: (Start)
There are 2 primes in the first 1000 terms of A284377, a(3) = 11681 and a(637) = 11681256...164648481361 (which has 6368 digits).
Let cp(k, n) denote the concatenation of the first n k-th powers. Note that there are no primes in cp(1, n) = A007908(n) up to n = 64000 or cp(3, n) = A019522(n) up to n = 31152. But there is a prime in cp(2, n) = A019521(n): cp(2, 3) = A019521(3) = 149. Conjecture: if k is odd, no "smaller primes" (< 10000 digits, or non-gigantic primes) exist in cp(k, n); if k is even, then such primes may occur. So far, the expanded search supports this conjecture. For k from 5 to 500, when k is an odd number, we have found no "smaller primes" in cp(k, n); for even k, the following primes have been found:
k = 6, cp(6, 3) = 164729.
k = 10, cp(10, 3) = 1102459049.
k = 18, cp(18, 3) = 1262144387420489.
k = 72, cp(72, 3) = 1472236648286964521369622528399544939174411840147874772641.
k = 76, cp(76, 3) = 1755578637259143234191361824800363140073127359051977856583921.
k = 108, cp(108, 3) = 13245185536...198527451848561 (a 86-digit prime).
k = 124, cp(124, 7) = 12126764793...315965558474401 (a 463-digit prime).
k = 432, cp(432, 7) = 11109067877...198797825539201 (a 1605-digit prime).
However, the next search contradicted the above conjecture. For k from 501 to 1100, we get two counterexamples:
k = 543, cp(543,13) = 12879304828...134856329859397 (a 5325-digit prime).
k = 815, cp(815,7) = 12184974969...385767566149943 (a 3021-digit prime).
And finally, by testing all cp(k, n) less than 10^10000 (using Mathematica's "PrimeQ[*]" function), it is confirmed that there are 15 "smaller primes" in all concatenations of k-th powers (k >= 1), of which 13 are listed above, and the other two are
k = 2140, cp(2140, 3) = 11600260630...045844060490801 (a 1608-digit prime) and
k = 2176, cp(2176, 3) = 11099690731...621473725585921 (a 1696-digit prime). (End)

Cf. A007908, A019521, A019522, A284377.

A038398 Concatenate first n cubes in reverse order.

Original entry on oeis.org

1, 81, 2781, 642781, 125642781, 216125642781, 343216125642781, 512343216125642781, 729512343216125642781, 1000729512343216125642781, 13311000729512343216125642781, 172813311000729512343216125642781

Offset: 1

M. I. Petrescu (mipetrescu(AT)yahoo.com)

Mihaly Bencze [Beneze], L. Tutescu, Some Notions and Questions in Number Theory, Sequence 6.

Cf. A000578, A019522, A038397.

nn=20;With[{c=Reverse[Range[nn]^3]},Table[FromDigits[Flatten[ IntegerDigits/@ Take[ c,-n]]],{n,nn}]] (* Harvey P. Dale, Sep 28 2013 *)

More terms from Andrew Gacek (andrew(AT)dgi.net), Feb 21 2000

A266368 The smallest prime formed by the concatenation of consecutive n-th powers beginning with "1," plus a trailing "1.".

Original entry on oeis.org

11, 1231, 149161, 181, 1168125662512961, 1321, 1647294096156251, 11282187163847812527993682354320971524782969100000001948717135831808627485171, 125665616553639062516796161, 15121

Offset: 0

Thomas S. Pedigo, Dec 28 2015

a(10) has 437 and a(11) has 1810 decimal digits respectively. - Michael De Vlieger, Jan 05 2016

a(13) is a bit more manageable, with 65 decimal digits - 18192159432367108864122070312513060694016968890104075497558138881. a(14) and a(15) are even shorter, with 23 and 15 decimal digits, respectively - 11638447829692684354561 and 132768143489071. - Thomas S. Pedigo, Jan 06 2016

			a(6)=1647294096156251; 1=1^6; 64=2^6; 729=3^6; 4096=4^6; 15625=5^6; 1647294096156251 is prime.

Cf. A019521, A019522.

f[k_, n_] := FromDigits@ Flatten@ Map[IntegerDigits, Append[Range[k + 1]^n, 1], 1]; Table[If[n == 0, k = 0, k = 1]; While[! PrimeQ@ f[k, n], k++]; f[k, n], {n, 0, 9}] (* Michael De Vlieger, Jan 05 2016 *)

A326120 a(n) is the concatenation of n^1, n^2, ..., n^n.

Original entry on oeis.org

1, 24, 3927, 41664256, 5251256253125, 6362161296777646656, 749343240116807117649823543, 864512409632768262144209715216777216, 981729656159049531441478296943046721387420489, 10100100010000100000100000010000000100000000100000000010000000000

Offset: 1

Trevor Mulindi, Sep 10 2019

a(100) has 10200 digits.

a(n) is the concatenation of the n terms of the n-th row of A075363 triangle. - Michel Marcus, Sep 15 2019

Cf. A007908, A019521, A019522, A053062, A075363.

a:= n-> parse(cat(n^j$j=1..n)):
seq(a(n), n=1..14);  # Alois P. Heinz, Jan 24 2021

a[n_] := FromDigits@ Flatten@ IntegerDigits[n^Range[n]]; Array[a, 10] (* Giovanni Resta, Sep 12 2019 *)

def a(n): return int("".join(str(n**i) for i in range(1, n+1)))
print([a(n) for n in range(1, 11)]) # Michael S. Branicky, Jan 24 2021

More terms from Giovanni Resta, Sep 12 2019

cp's OEIS Frontend

A067103 a(n) = floor(X/Y), where X = concatenation of cubes and Y = concatenation of natural numbers.

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A133820 Triangle whose rows are sequences of increasing cubes: 1; 1,8; 1,8,27; ... .

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A066700 The leading digits in the terms in A067103 converge; dividing by a suitable power of 10 they converge to the number shown below; sequence gives continued fraction for this number.

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A284377 Concatenation of the first n positive 4th powers.

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A359403 Primes that are the concatenation of the first m consecutive k-th powers.

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A038398 Concatenate first n cubes in reverse order.

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A266368 The smallest prime formed by the concatenation of consecutive n-th powers beginning with "1," plus a trailing "1.".

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Mathematica

A326120 a(n) is the concatenation of n^1, n^2, ..., n^n.

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