cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A076072 Largest multiple of n that can be formed by using the digits of the next n numbers, or 0 if no such number exists.

Original entry on oeis.org

1, 32, 654, 97180, 4321111115, 987622111110, 87654322222222, 9654333333332120, 987544444443333210, 98765555555444443210, 9876666666665555524130, 988777777777776666543120, 99998888888888877654321100
Offset: 1

Views

Author

Amarnath Murthy, Oct 05 2002

Keywords

Comments

Conjecture: No entry is zero. (At least one multiple of n can be formed by using the digits of the next n numbers.)
"Next n numbers" means {1+n(n-1)/2,...,n(n+1)/2}. - M. F. Hasler, Jul 22 2011

Crossrefs

Programs

  • PARI
    precperm(p)={ my(t); forstep(i=#p-1,1,-1, p[i]>p[i+1] && for( j=1, #t=vecsort(vecextract(p,2^#p-2^(i-1)),,4), t[j]A076072(n)={ my(d=vecsort(Vec(concat(vector(n,i,Str(i+n*(n-1)/2)))),,4)); for( i=1, (#d)!, eval(concat(d))%n || break; d=precperm(d)); eval(concat(d))}  \\ M. F. Hasler, Jul 24 2011

Extensions

More terms from David Wasserman, Mar 23 2005

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

Views

Author

Amarnath Murthy, Mar 11 2003

Keywords

Crossrefs

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

Programs

  • Python
    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

Extensions

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

A192392 Largest multiple of n which can be formed as concatenation of the next n numbers, {1+n(n-1)/2,...,n(n+1)/2} (written in decimal), or 0 if no such number exists.

Original entry on oeis.org

1, 32, 654, 97108, 1413121115, 212019181716, 28272625222324, 3635343331293032, 454443424140393837, 55545352514948474650, 6665646362616059585756, 787776757473727170696768, 91908988878685848382817980, 9998979695949392105104103101100102
Offset: 1

Views

Author

M. F. Hasler, Jun 29 2011

Keywords

Comments

Equal to A076069(n) whenever A076069(n) is a multiple of n. Less than or equal to A076072(n), where permutations of individual digits are allowed.

Crossrefs

Programs

  • PARI
    precperm(p)={ my(t); forstep( i=#p-1,1,-1, p[i]>p[i+1] && for( j=1,#t=vecsort( vecextract( p, 2^#p-2^(i-1) ),,4), t[j]A192392(n)={ my( d=vecsort( vector( n,i,Str( i+n*(n-1)/2 )),,4 ), t );
     for( i=1,n!, eval(concat(d))%n || break; d=precperm(d)); eval(concat(d))}

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

Views

Author

Amarnath Murthy, Oct 05 2002

Keywords

Comments

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

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • Python
    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

Extensions

More terms from David Wasserman, Mar 19 2005

A076069 Largest number formed as concatenation of a permutation of the n consecutive numbers 1+n(n-1)/2, ..., n(n+1)/2 (written in decimal).

Original entry on oeis.org

1, 32, 654, 98710, 1514131211, 212019181716, 28272625242322, 3635343332313029, 454443424140393837, 55545352515049484746, 6665646362616059585756, 787776757473727170696867, 91908988878685848382818079, 9998979695949392105104103102101100
Offset: 1

Views

Author

M. F. Hasler, Jun 29 2011

Keywords

Comments

Nontrivial (i.e., different from concatenation in reverse order) whenever a number of the form 10^k is in the considered range of n numbers; then a(n) is given as concatenation (10^k-1)...(1+n(n-1)/2)(n(n+1)/2)...(10^k).

Examples

			1, then a concatenation of a permutation of 2 and 3, then a concatenation of a permutation of 4, 5 and 6.
		

Crossrefs

Extensions

Definition modified to obtain a new sequence different from (earlier duplicate) A080480. M. F. Hasler, Jun 29 2011.

A193380 A192392(n)/n = largest m such that m*n is a concatenation of the numbers n(n-1)/2+1,...,n(n+1)/2.

Original entry on oeis.org

1, 16, 218, 24277, 282624223, 35336530286, 4038946460332, 454417916411629, 50493713793377093, 5554535251494847465, 605967851146914507796, 65648063122810597558064, 7069922221437372952524460, 714212835424956578936007364364293
Offset: 1

Views

Author

M. F. Hasler, Jul 24 2011

Keywords

Crossrefs

Programs

  • PARI
    a(n) = A192392(n)/n  /* see there */

Formula

a(n) = A192392(n)/n.
Showing 1-6 of 6 results.