A053067 -id:A053067 - cp's OEIS frontend

A077146 a(n) = floor((concatenation of next (n+1) numbers) / (concatenation of next n numbers)).

23, 19, 173, 14093, 145, 137, 131, 127, 124, 121, 119, 117, 116453592, 11416794012, 1141, 1132, 1124, 1116, 1110, 1104, 1099, 1094, 1090, 1086, 1083, 1079, 1076, 1073, 1071, 1068, 1066, 1064, 1062, 1060, 1058, 1057, 1055, 1053, 1052, 1051, 1049, 1048, 1047

Offset: 1

Amarnath Murthy, Oct 30 2002

The numerator numbers to be concatenated are taken from 2,3,4,... and the denominator numbers from 1,2,3,... - Lars Blomberg, Mar 18 2012

			a(1) = floor(23/1) = 23.
a(2) = floor(456/23) = 19.
a(3) = floor(78910/456) = 173.

Michael S. Branicky, Table of n, a(n) for n = 1..1413

Cf. A053067.

def b(n): return int("".join(map(str, range((n-1)*n//2+1, n*(n+1)//2+1))))
def a(n): return b(n+1)//b(n)
print([a(n) for n in range(1, 44)]) # Michael S. Branicky, Jul 31 2022

a(n) = floor(A053067(n+1)/A053067(n)). - Michel Marcus, Jul 31 2022

a(7)-a(36) from Lars Blomberg, Mar 18 2012

a(37) and beyond from Michael S. Branicky, Jul 31 2022

A095222 (Concatenation of T(n)+1..T(n+1)) mod (concatenation of T(n-1)+1..T(n)), where T(k) is the k-th triangular number, A000217(k).

Original entry on oeis.org

0, 19, 22, 52785, 459136846, 77032955851, 17865523926168, 1689623507805373, 118763391490664275, 34129170492806856511, 3616977859402101826424, 614427099275584124069065, 34445463838393113230619233, 2480185781714663029796564541499860

Offset: 1

Amarnath Murthy, Jun 10 2004

			a(3) = 78910 mod 456 = 22.
From _Jon E. Schoenfield_, Feb 23 2021: (Start)
.
      concatenation of  concatenation of
   n   T(n-1)+1..T(n)    T(n)+1..T(n+1)    a(n)
  --  ----------------  ----------------  -----
   1                 1                23      0
   2                23               456     19
   3               456             78910     22
   4             78910        1112131415  52785
(End)

Harvey P. Dale, Table of n, a(n) for n = 1..200

Cf. A000217, A053067.

ccat[a_,b_]:=FromDigits[Flatten[IntegerDigits/@Range[a,b]]]; modc[n_]: = Module[{w=(n(n-1))/2+1,x=(n(n+1))/2,y,z},y=x+1;z=2x-w+2; Mod[ccat[y,z], ccat[w,x]]]; Array[modc,20] (* Harvey P. Dale, Nov 27 2012 *)

a(n) = A053067(n+1) mod A053067(n). - Hagen von Eitzen, Jun 17 2009

More terms from Hagen von Eitzen, Jun 17 2009

a(14) from Harvey P. Dale, Nov 27 2012

A133026 Concatenation of next n Fermat numbers A000215.

Original entry on oeis.org

3, 517, 257655374294967297

Offset: 1

Omar E. Pol, Nov 09 2007

a(4) has 292 digits and is too large to include.

Amiram Eldar, Table of n, a(n) for n = 1..4

Cf. A000215, A053067, A133013.

a[n_] := FromDigits[Flatten[IntegerDigits[Table[2^(2^(k - 1)) + 1, {k, n*(n - 1)/2 + 1, n*(n + 1)/2}]]]]; Array[a, 3] (* Amiram Eldar, Jul 09 2025 *)

A133069 Successive digits of twin primes A001097.

Original entry on oeis.org

3, 5, 7, 1, 1, 1, 3, 1, 7, 1, 9, 2, 9, 3, 1, 4, 1, 4, 3, 5, 9, 6, 1, 7, 1, 7, 3, 1, 0, 1, 1, 0, 3, 1, 0, 7, 1, 0, 9, 1, 3, 7, 1, 3, 9, 1, 4, 9, 1, 5, 1, 1, 7, 9, 1, 8, 1, 1, 9, 1, 1, 9, 3, 1, 9, 7, 1, 9, 9, 2, 2, 7, 2, 2, 9, 2, 3, 9, 2, 4, 1, 2, 6, 9, 2, 7, 1, 2, 8, 1, 2, 8, 3, 3, 1, 1, 3, 1, 3

Offset: 1

Omar E. Pol, Nov 10 2007

Also, with offset 0, decimal expansion of the constant by concatenate twin primes. Example: 0.35711131719293141435961... Hence, subsequence of A033308. - Omar E. Pol, Oct 21 2017

G. C. Greubel, Table of n, a(n) for n = 1..5000
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A001097, A033308, A053067, A133013, A133014.

Flatten[IntegerDigits/@Union[Flatten[Select[Partition[Prime[ Range[ 100]],2,1], Last[#]- First[#] == 2&]]]] (* Harvey P. Dale, May 29 2014 *)

A133074 Concatenation of next n isolated primes.

Original entry on oeis.org

2, 2337, 475367, 79838997, 113127131157163, 167173211223233251, 257263277293307317331, 337353359367373379383389, 397401409439443449457467479, 487491499503509541547557563577

Offset: 1

Omar E. Pol, Nov 10 2007

Harvey P. Dale, Table of n, a(n) for n = 1..166
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A007510, A053067, A133013, A133014.

Module[{ip=Select[Prime[Range[200]],NoneTrue[#+{2,-2},PrimeQ]&],c},c = Floor[ (Sqrt[8 Length[ip]+1]-1)/2];FromDigits[Flatten[IntegerDigits/@#]]&/@TakeList[ ip,Range[c]]] (* Harvey P. Dale, Jun 09 2022 *)

A134714 Concatenation of next n Catalan numbers.

Original entry on oeis.org

1, 12, 51442, 13242914304862, 16796587862080127429002674440, 96948453535767012964479047763870017672631906564120420, 2446626702091482563640343059613650128990414732448619464014521836735307215269533550916004

Offset: 1

Omar E. Pol, Nov 12 2007

Harvey P. Dale, Table of n, a(n) for n = 1..15

Cf. A053067, A053068, A133013. Catalan numbers: A000108.

Module[{nn=50,cn},cn=CatalanNumber[Range[0,nn]];Table[ FromDigits[ Flatten[ IntegerDigits/@ Take[cn,{(n(n+1))/2+1,((n+1)(n+2))/2}]]],{n,0,Floor[ (Sqrt[ 8*nn+1]-1)/2]-1}]] (* Harvey P. Dale, May 15 2014 *)
Module[{nn=8},FromDigits[Flatten[IntegerDigits/@#]]&/@Rest[TakeList[CatalanNumber[Range[0,(nn(nn+1))/2-1]],Range[0,nn-1]]]] (* Harvey P. Dale, Oct 06 2024 *)

Edited by Charles R Greathouse IV, Apr 25 2010

A134733 Concatenation of next n imperfect numbers.

Original entry on oeis.org

1, 23, 457, 891011, 1213141516, 171819202122, 23242526272930, 3132333435363738, 394041424344454647, 48495051525354555657, 5859606162636465666768, 697071727374757677787980

Offset: 1

Omar E. Pol, Nov 12 2007

Harvey P. Dale, Table of n, a(n) for n = 1..200

Cf. A053067, A132943, A133013. Imperfect numbers: A132999.

Module[{nn=120,in,len},in=Select[Range[nn],DivisorSigma[1,#]!=2#&];len= Floor[x/.Last[Solve[(x(x+1))/2==Length[in]]]];FromDigits[Flatten[ IntegerDigits/@ #]]&/@TakeList[in,Range[len]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 01 2020 *)

A134794 Concatenation of next n composite numbers.

Original entry on oeis.org

4, 68, 91012, 14151618, 2021222425, 262728303233, 34353638394042, 4445464849505152, 545556575860626364, 65666869707274757677, 7880818284858687889091

Offset: 1

Omar E. Pol, Nov 26 2007

Harvey P. Dale, Table of n, a(n) for n = 1..200

Cf. A053067, A133013. Composite numbers: A002808.

cc[x_]:=FromDigits[Flatten[IntegerDigits/@x]]; Module[{nn= 100,cmps,tr,len},cmps = Select[Range[nn],CompositeQ];len=Length[cmps];tr = Floor[ (Sqrt[1+8 len]-1)/2];cc/@TakeList[cmps,Range[tr]]] (* Harvey P. Dale, Sep 26 2021 *)

A286706 Primes in A077309.

Original entry on oeis.org

23, 4567, 78910111213, 25262728293031323334353637383940414243444546474849

Offset: 1

XU Pingya, May 12 2017

That is, primes formed from concatenation of k numbers starting with k.
Subsequence of A052087.
a(5) must be larger than A077309(2500).
A077309(k) is prime for k = {2, 4, 7, 25, ...}. - Michael De Vlieger, May 13 2017

			The prime 4567 is a term since it is the concatenation of 4 decimal numbers beginning with the number 4. - _Michael De Vlieger_, May 13 2017

Eric Weisstein's World of Mathematics, Consecutive Number Sequences

Cf. A077309, A052087, A035333, A053067.

Select[Table[FromDigits@ Flatten@ Take[#, {n, n + (n - 1)}], {n, Ceiling[Length[#]/2]}], PrimeQ] &@ Array[IntegerDigits, 10^3] (* Michael De Vlieger, May 13 2017 *)

A366585 Write down the positive integers. To obtain the terms of the sequence, concatenate groups of these so the first digit of each term is the number of consecutive numbers that are concatenated.

Original entry on oeis.org

1, 23, 4567, 89101112131415, 16, 17, 18, 19, 2021, 2223, 2425, 2627, 2829, 303132, 333435, 363738, 394041, 42434445, 46474849, 5051525354, 5556575859, 606162636465, 666768697071, 72737475767778, 79808182838485, 8687888990919293, 949596979899100101102, 103, 104, 105, 106, 107

Offset: 1

Tamas Sandor Nagy, Oct 14 2023

Terms beginning with 1 will appear in runs of long linear succession. These are interspersed with clusters of values that tend to increase at other, uneven rates in the sequence.

Cf. A366610, A053067.

lista(nn) = my(list=List(), k=1); while (k < nn, my(s="", n=digits(k)[1]); for (j=1, n, s = concat(s, Str(k+j-1));); listput(list, eval(s)); k += n;); Vec(list); \\ Michel Marcus, Oct 14 2023

from itertools import islice
def A366585_gen(): # generator of terms
    a = 1
    while True:
        c = a+int(str(a)[0])
        yield int(''.join(str(i) for i in range(a,c)))
        a = c
A366585_list = list(islice(A366585_gen(),20)) # Chai Wah Wu, Nov 04 2023

cp's OEIS Frontend

A077146 a(n) = floor((concatenation of next (n+1) numbers) / (concatenation of next n numbers)).

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A095222 (Concatenation of T(n)+1..T(n+1)) mod (concatenation of T(n-1)+1..T(n)), where T(k) is the k-th triangular number, A000217(k).

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A133026 Concatenation of next n Fermat numbers A000215.

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A133069 Successive digits of twin primes A001097.

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A133074 Concatenation of next n isolated primes.

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A134714 Concatenation of next n Catalan numbers.

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A134733 Concatenation of next n imperfect numbers.

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Mathematica

A134794 Concatenation of next n composite numbers.

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Mathematica

A286706 Primes in A077309.

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