A019521 -id:A019521 - cp's OEIS frontend

A038397 Concatenate first n squares in reverse order.

1, 41, 941, 16941, 2516941, 362516941, 49362516941, 6449362516941, 816449362516941, 100816449362516941, 121100816449362516941, 144121100816449362516941, 169144121100816449362516941, 196169144121100816449362516941, 225196169144121100816449362516941

Offset: 1

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

Mihaly Bencze [Beneze], L. Tutescu, Some Notions and Questions in Number Theory, Sequence 5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..225

Cf. A000290, A019521.

a038397 n = a038397_list !! (n-1)
a038397_list = f "" $ tail a000290_list where
   f xs (q:qs) = (read ys :: Integer) : f ys qs
     where ys = show q ++ xs
-- Reinhard Zumkeller, Mar 01 2014

a:= proc(n) a(n):= `if`(n=1, 1, parse(cat(n^2, a(n-1)))) end:
seq(a(n), n=1..20);  # Alois P. Heinz, Jan 13 2021

Join[{s = 1}, Table[s = FromDigits[Flatten[IntegerDigits[{n^2, s}]]], {n, 2, 12}]] (* Jayanta Basu, Jul 14 2013 *)

A067095 a(n) = floor(X/Y) where X is the concatenation in increasing order of the first n even numbers and Y is that of the first n odd numbers.

Original entry on oeis.org

2, 1, 1, 1, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 181, 181, 181, 181, 181, 181, 181, 181, 181, 181, 181, 181, 181, 181, 181, 181

Offset: 1

Amarnath Murthy, Jan 07 2002

For n > 1, the sequence is increasing and tends to infinity. Proof: for k>=1, when the last concatenated integer at the numerator A019520(n) has k digits, then a(n) > 10^(k-1) (see Krusemeyer reference). - Bernard Schott, Dec 06 2021

Values taken by this function are in A349960. - Bernard Schott, Dec 18 2021

			a(4) = floor(2468/1357) = floor(1.81871775976418570375829034635225) = 1.
a(20000) = 18175.

Mark I. Krusemeyer, George T. Gilbert, and Loren C. Larson, A Mathematical Orchard, Problems and Solutions, MAA, 2012, Problem 87, pp. 159-161.

Cf. A067088, A067089, A067090, A067091, A067092, A067093, A067094.

Cf. A019519, A019520, A349960.

f[n_] := (k = 1; x = y = "0"; While[k < n + 1, x = StringJoin[x, ToString[2k]]; y = StringJoin[y, ToString[2k - 1]]; k++ ]; Return[ Floor[ ToExpression[x] / ToExpression[y]]] ); Table[ f[n], {n, 1, 75} ]
With[{ev=Range[2,140,2],od=Range[1,139,2]},Table[Floor[FromDigits[ Flatten[ IntegerDigits/@ Take[ev,n]]]/FromDigits[Flatten[ IntegerDigits/@ Take[od,n]]]],{n,70}]] (* Harvey P. Dale, Aug 19 2011 *)

ae(n)=my(s=""); for(k=1, n, s=Str(s, 2*k)); eval(s); \\ A019520
ao(n)=my(s=""); for(k=1, n, s=Str(s, 2*k-1)); eval(s); \\ A019521
a(n) = ae(n)\ao(n); \\ Michel Marcus, Dec 07 2021

a(n) = floor(A019520(n)/A019519(n)).

More terms from Robert G. Wilson v, Jan 09 2002

A019522 Concatenate cubes.

Original entry on oeis.org

1, 18, 1827, 182764, 182764125, 182764125216, 182764125216343, 182764125216343512, 182764125216343512729, 1827641252163435127291000, 18276412521634351272910001331, 182764125216343512729100013311728

Offset: 1

R. Muller

S. Smarandoiu, Convergence of Smarandache continued fractions, Abstract 96T-11-195, Abstracts Amer. Math. Soc., 17 (No. 4, 1996), 680.

Y. Guo and M. Le, Smarandache concatenated power decimals and their irrationality, Smarandache Notions Journal, Vol. 9, No. 1-2. 1998, 100-102.
Eric Weisstein's World of Mathematics, Consecutive Number Sequences

Cf. A007908, A000578, A019521.

A067102 Floor[ X/Y] where X = concatenation of the squares and Y = concatenation of natural numbers.

Original entry on oeis.org

1, 1, 1, 12, 120, 1208, 12082, 120821, 1208216, 12082165, 120821655, 1208216555, 12082165556, 120821655562, 1208216555626, 12082165556267, 120821655562672, 1208216555626728, 12082165556267282, 120821655562672822

Offset: 1

Amarnath Murthy, Jan 07 2002

			a(5) = floor [1491625/12345]=floor[] = floor[120.828270554880518428513568246254]=120.

Cf. A067091, A067092, A067093, A067094, A067095, A067096, A067097, A067098, A067099, A067100, A067101.

floor[A019521(n)/A007908(n)]

f[n_] := (k = 1; x = y = "0"; While[k < n + 1, x = StringJoin[x, ToString[k^2]]; y = StringJoin[y, ToString[k]]; k++ ]; Return[ Floor[ ToExpression[x] / ToExpression[y]]] ); Table[ f[n], {n, 1, 20} ]

More terms from Robert G. Wilson v, Jan 09 2002

A283560 Concatenation of the first n nonsquares (A000037).

Original entry on oeis.org

2, 23, 235, 2356, 23567, 235678, 23567810, 2356781011, 235678101112, 23567810111213, 2356781011121314, 235678101112131415, 23567810111213141517, 2356781011121314151718, 235678101112131415171819, 23567810111213141517181920

Offset: 1

XU Pingya, Mar 10 2017

There are 5 primes in the first 6100 terms of this sequence, see A283561.

Robert Israel, Table of n, a(n) for n = 1..365
Eric Weisstein's World of Mathematics, Consecutive Number Sequences

Cf. A000037, A007908, A019521, A283561.

A[1]:= 2:
x:= 2:
for n from 2 to 20 do
  if issqr(x+1) then x:= x+2 else x:= x+1 fi;
  A[n]:= A[n-1]*10^(1+ilog10(x))+x;
od:
seq(A[n],n=1..20); # Robert Israel, Mar 13 2017

cns[n_]:=FromDigits[Flatten[IntegerDigits[Table[k+Floor[1/2+Sqrt[k]],{k,1,n}]]]]
Module[{nn=20,sqf},sqf=Select[Range[2,nn],!IntegerQ[Sqrt[#]]&];Table[FromDigits[Flatten[ IntegerDigits/@ Take[sqf,n]]],{n,Length[sqf]}]] (* Harvey P. Dale, Sep 07 2024 *)

a(n)=my(s=""); for(k=1,n, s=Str(s, (sqrtint(4*k)+1)\2 + k)); eval(s) \\ Charles R Greathouse IV, Mar 10 2017

A032343 a(n) = 10*a(n-1)+n^2, a(0)=0.

Original entry on oeis.org

0, 1, 14, 149, 1506, 15085, 150886, 1508909, 15089154, 150891621, 1508916310, 15089163221, 150891632354, 1508916323709, 15089163237286, 150891632373085, 1508916323731106, 15089163237311349, 150891632373113814

Offset: 0

Henry Bottomley, Jul 04 2000

a(n)/10^n converges to 110/729=0.15089163237311...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-33,31,-10).

Cf. A014824, A019521.

[0] cat [n le 1 select n else 10*Self(n-1)+n^2: n in [1..20]]; // Vincenzo Librandi, Mar 10 2013

CoefficientList[Series[(x^2+x)/((x-1)^3 (10x-1)),{x,0,30}],x]  (* Harvey P. Dale, Mar 20 2011 *)
RecurrenceTable[{a[0] == 0, a[n] == 10*a[n-1] + n^2}, a, {n, 100}] (* Vincenzo Librandi, Mar 10 2013 *)
LinearRecurrence[{13,-33,31,-10},{0,1,14,149},20] (* Harvey P. Dale, May 25 2024 *)

a(n) =(10^n-1)*(110/729)-n^2/9-n*(20/81).

G.f.: (x^2+x)/((x-1)^3*(10*x-1)). - Harvey P. Dale, Mar 20 2011

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.

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

A038397 Concatenate first n squares in reverse order.

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Maple

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A067095 a(n) = floor(X/Y) where X is the concatenation in increasing order of the first n even numbers and Y is that of the first n odd numbers.

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A019522 Concatenate cubes.

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A067102 Floor[ X/Y] where X = concatenation of the squares and Y = concatenation of natural numbers.

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Mathematica

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A283560 Concatenation of the first n nonsquares (A000037).

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Maple

Mathematica

PARI

A032343 a(n) = 10*a(n-1)+n^2, a(0)=0.

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Magma

Mathematica

Formula

A284377 Concatenation of the first n positive 4th powers.

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Programs

PARI

Extensions

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

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Comments

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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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