A053067 -id:A053067 - cp's OEIS frontend

A095219 First differences of A053067.

22, 433, 78454, 1112052505, 160606060606, 22070707070707, 2908080808080808, 370909090909090909, 46101010101010101010, 5611111111111111111111, 671212121212121212121212, 79131313131313131313131313, 9293949517171716261615233314205014, 106107108099816161515133215015015015015015015

Offset: 2

Amarnath Murthy, Jun 10 2004

			a(2) = 456 - 23 = 433.

A000124 := proc(n) n*(n+1)/2+1 ; end proc:
catL := proc(L) local a; a := op(1,L) ; for i from 2 to nops(L) do a := cat2(a,op(i,L)) ; end do; a; end proc:
A053067 := proc(n) local l ; l := A000124(n-1) ; catL([seq(k,k=l..l+n-1)]) ; end proc:
A095219 := proc(n) A053067(n+1)-A053067(n) ; end proc:
seq(A095219(n),n=1..15) ; # R. J. Mathar, Oct 14 2010

Definition clarified by R. J. Mathar, Oct 14 2010

A133013 Concatenation of next n primes.

Original entry on oeis.org

2, 35, 71113, 17192329, 3137414347, 535961677173, 79838997101103107, 109113127131137139149151, 157163167173179181191193197, 199211223227229233239241251257, 263269271277281283293307311313317

Offset: 1

Omar E. Pol, Oct 19 2007

C. K. Caldwell, The Prime Pages.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A053067, A053068. Prime numbers: A000040.

With[{nn=15},FromDigits[Flatten[IntegerDigits/@#]]&/@TakeList[Prime[ Range[ (nn(nn+1))/2]],Range[nn]]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Apr 26 2018 *)

A035333 Concatenation of two or more consecutive positive integers.

Original entry on oeis.org

12, 23, 34, 45, 56, 67, 78, 89, 123, 234, 345, 456, 567, 678, 789, 910, 1011, 1112, 1213, 1234, 1314, 1415, 1516, 1617, 1718, 1819, 1920, 2021, 2122, 2223, 2324, 2345, 2425, 2526, 2627, 2728, 2829, 2930, 3031, 3132, 3233, 3334, 3435, 3456, 3536, 3637, 3738

Offset: 1

Erich Friedman

Paul Tek, Table of n, a(n) for n = 1..10000

For concatenations of exactly k consecutive integers see A000027 (k=1), A127421 (k=2), A001703 (k=3), A279204 (k=4).
See also A007908 for concatenation of 1 through n.
For primes see A052087.
All of A007908, A052087, A053067, A279610 are subsequences.

import heapq
from itertools import islice
def agen():
    c = 12
    h = [(c, 1, 2)]
    nextcount = 3
    while True:
        (v, s, l) = heapq.heappop(h)
        yield v
        if v >= c:
            c = int(str(c) + str(nextcount))
            heapq.heappush(h, (c, 1, nextcount))
            nextcount += 1
        l += 1; v = int(str(v)[len(str(s)):] + str(l)); s += 1
        heapq.heappush(h, (v, s, l))
print(list(islice(agen(), 47))) # Michael S. Branicky, Dec 23 2021

Edited by Charles R Greathouse IV, Apr 28 2010

Corrected by Paul Tek, Jun 08 2013

A080480 Largest number formed by using all the digits (with multiplicity) of next n numbers.

Original entry on oeis.org

1, 32, 654, 98710, 5432111111, 987622111110, 87654322222222, 9654333333332210, 987544444443333210, 98765555555444443210, 9876666666665555543210, 988777777777776666543210, 99998888888888877654321100, 9999999998765544332211111110000000

Offset: 1

Amarnath Murthy, Mar 11 2003

			a(4) = 98710 formed by using digits of 7,8,9 and 10.

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

Cf. A053067 (next n concatenated), A076068 (smallest without zeros), A080479 (smallest).

FromDigits[Sort[Flatten[IntegerDigits/@#],Greater]]&/@Table[ Reverse[ Range[ (n(n+1))/2+1,((n+1)(n+2))/2]],{n,0,15}] (* Harvey P. Dale, May 13 2017 *)

def a(n):
  s = "".join(sorted("".join(map(str, range((n-1)*n//2+1, n*(n+1)//2+1)))))
  return int("".join(sorted(s, reverse=True)))
print([a(n) for n in range(1, 15)]) # Michael S. Branicky, Jan 23 2021

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003

More terms from Harvey P. Dale, May 13 2017

A080479 Smallest number formed by using all the digits (with multiplicity) of next n numbers.

Original entry on oeis.org

1, 23, 456, 10789, 1111112345, 101111226789, 22222222345678, 1022333333334569, 102333344444445789, 10234444455555556789, 1023455555666666666789, 102345666677777777777889, 10012345677888888888889999, 1000000011111122334455678999999999

Offset: 1

Amarnath Murthy, Mar 11 2003

Cf. A053067 (next n concatenated), A080480 (largest).

def a(n):
  s = "".join(sorted("".join(map(str, range((n-1)*n//2+1, n*(n+1)//2+1)))))
  if '0' not in s: return int(s)
  rz = s.rfind('0')
  return int(s[rz+1] + s[:rz+1] + s[rz+2:])
print([a(n) for n in range(1, 15)]) # Michael S. Branicky, Jan 23 2021

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003

A076068 Smallest number that can be formed by using the nonzero digits of the numbers 1+n(n-1)/2 through n(n+1)/2.

Original entry on oeis.org

1, 23, 456, 1789, 1111112345, 11111226789, 22222222345678, 122333333334569, 12333344444445789, 1234444455555556789, 123455555666666666789, 12345666677777777777889, 112345677888888888889999, 111111122334455678999999999, 111111111111111111111111112234566778899

Offset: 1

Amarnath Murthy, Oct 05 2002

Is there any r and s such that a(r) = a(s)? Probably not.

			a(4) = 1789 (=01789) formed by using digits of 7,8,9 and 10.

Cf. A053067 (next n concatenated), A080479 (smallest with zeros), A080480 (largest with zeros).

sncbf[n_]:=Sort[Flatten[IntegerDigits/@Range[(n(n-1))/2+1,(n(n+1))/2]]/.(0->Nothing)]//FromDigits; Array[sncbf,15] (* Harvey P. Dale, Nov 26 2019 *)

def a(n):
  s = "".join(sorted("".join(map(str, range((n-1)*n//2+1, n*(n+1)//2+1)))))
  if '0' not in s: return int(s)
  return int(s[s.rfind('0')+1:])
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Jan 23 2021

More terms from David Wasserman, Mar 19 2005

A133014 Concatenation of next n Mersenne primes.

Original entry on oeis.org

3, 731, 1278191131071, 52428721474836472305843009213693951618970019642690137449562111

Offset: 1

Omar E. Pol, Oct 19 2007

Cf. A053067, A053068. Mersenne primes: A000668.

A279610 a(n) = concatenate n consecutive integers, starting with the last number of the previous batch.

Original entry on oeis.org

1, 12, 234, 4567, 7891011, 111213141516, 16171819202122, 2223242526272829, 293031323334353637, 37383940414243444546, 4647484950515253545556, 565758596061626364656667, 67686970717273747576777879, 7980818283848586878889909192

Offset: 1

José de Jesús Camacho Medina, Dec 09 2016, and Paolo Iachia, Dec 15 2016

A variant of A053067. The first number of the concatenation a(n) is A152947(n) = (n-2)*(n-1)/2+1 and the last is (n-1)*n/2+1.
The fourth term, 4567, is a prime. When is the next prime, if there is another? - N. J. A. Sloane, Dec 16 2016
a(n) is the concatenation of the terms of the n-th row of A122797 when seen as a triangle. - Michel Marcus, Dec 17 2016

			a(4) is the concatenation of 4 numbers beginning with the last number (4) that was used to build a(3), so a(4) = 4 5 6 7 = 4567. Then a(5) is the concatenation of 5 numbers beginning with the last number of a(4), which is 7, so a(5) = 7 8 9 10 11 = 7891011. And so on.
For n = 3, n^2/2 - n/2 + 1 = 4; a(3) = 4 + 3*10^1 + 2*10^(1+1) = 234.

Chai Wah Wu, Table of n, a(n) for n = 1..200

Cf. A053067, A122797, A152947.

A subsequence of A035333. For primes in latter, see A052087.

Table[FromDigits[Flatten[IntegerDigits /@ Range[(n(n - 1))/2 + 1, (n(n + 1))/2 + 1 ]]], {n, 0, 20}]

from _future_ import division
def A279610(n):
    return int(''.join(str(d) for d in range((n-1)*(n-2)//2+1,n*(n-1)//2+2))) # Chai Wah Wu, Dec 17 2016

a(n) = n^2/2 - n/2 + 1 + Sum{k=1..n-1} ((n^2/2 - n/2 + 1 - k)*10^Sum{j=0..k-1} (floor(1+log_10(n^2/2 - n/2 + 1 - j)))).

A134800 Concatenation of next n partition numbers A000041.

Original entry on oeis.org

1, 12, 357, 11152230, 425677101135, 176231297385490627, 792100212551575195824363010, 37184565560468428349101431231014883, 179772163726015311853733844583531746326175175

Offset: 1

Omar E. Pol, Nov 12 2007

Harvey P. Dale, Table of n, a(n) for n = 1..37

Cf. A000041, A053067, A132926, A133013. See A134801 for another version.

FromDigits[Flatten[IntegerDigits/@#]]&/@With[{nn=10},TakeList[ PartitionsP[ Range[0,(nn(nn+1))/2]],Range[nn]]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Jul 25 2019 *)

Edited by Charles R Greathouse IV, Apr 28 2010

A134801 Concatenation of next n partition numbers of positive integers.

Original entry on oeis.org

1, 23, 5711, 15223042, 5677101135176, 231297385490627792, 1002125515751958243630103718, 456556046842834910143123101488317977, 216372601531185373384458353174632617517589134

Offset: 1

Omar E. Pol, Nov 12 2007

Harvey P. Dale, Table of n, a(n) for n = 1..37

Cf. A053067, A132926, A133013. Partition numbers: A000041. See A134800 for another version of the concatenation of next n partition numbers.

Module[{nn=10},FromDigits[Flatten[IntegerDigits/@#]]&/@TakeList[ PartitionsP[ Range[ (nn(nn+1))/2]],Range[nn]]] (* Harvey P. Dale, Apr 18 2022 *)

Edited by Charles R Greathouse IV, Apr 28 2010

cp's OEIS Frontend

A095219 First differences of A053067.

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A133013 Concatenation of next n primes.

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A035333 Concatenation of two or more consecutive positive integers.

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A080480 Largest number formed by using all the digits (with multiplicity) of next n numbers.

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A080479 Smallest number formed by using all the digits (with multiplicity) of next n numbers.

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A076068 Smallest number that can be formed by using the nonzero digits of the numbers 1+n(n-1)/2 through n(n+1)/2.

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A133014 Concatenation of next n Mersenne primes.

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A279610 a(n) = concatenate n consecutive integers, starting with the last number of the previous batch.

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A134800 Concatenation of next n partition numbers A000041.

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