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.

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A378308 Array read by antidiagonals: For k >= 3, row k-2 consists of the numbers with no digits 0 in bases 3 to k.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 5, 5, 2, 1, 7, 7, 7, 2, 1, 8, 13, 13, 7, 2, 1, 13, 14, 14, 13, 13, 2, 1, 14, 22, 22, 14, 22, 13, 2, 1, 16, 23, 23, 22, 23, 22, 13, 2, 1, 17, 25, 41, 23, 43, 23, 22, 13, 2, 1, 22, 26, 43, 43, 121, 43, 23, 22, 13, 2, 1, 23, 41, 121, 121, 122, 121, 43, 23, 23, 13, 2, 1, 25, 43, 122
Offset: 1

Views

Author

Robert Israel, Nov 22 2024

Keywords

Comments

Row k-2 consists of 1, 2 and numbers x with A270027(x) > k.

Examples

			Array starts
  1,   2,   4,   5,    7,    8,   13,   14,   16,   17,   22,   23, ...
  1,   2,   5,   7,   13,   14,   22,   23,   25,   26,   41,   43, ...
  1,   2,   7,  13,   14,   22,   23,   41,   43,  121,  122,  157, ...
  1,   2,   7,  13,   14,   22,   23,   43,  121,  122,  157,  158, ...
  1,   2,  13,  22,   23,   43,  121,  122,  157,  158,  214,  607, ...
  1,   2,  13,  22,   23,   43,  121,  122,  157,  158,  214,  607, ...
  1,   2,  13,  22,   23,   43,  121,  122,  157,  158,  214,  607, ...
  1,   2,  13,  22,   23,   43,  121,  122,  157,  158,  214,  619, ...
  1,   2,  13,  23,   43,  157,  158,  214,  619, 5471, 5557, 5561, ...
  1,   2,  13,  23,   43,  157,  158,  214,  619, 5471, 5557, 5561, ...
  1,   2,  23,  43,  157,  158,  214,  619, 5471, 5557, 5561, 5791, ...
  1,   2,  23,  43,  157,  158,  214,  619, 5471, 5791, 5821, 5822, ...
  1,   2,  23,  43,  157,  158,  214,  619, 5471, 5791, 5821, 5822, ...
		

Crossrefs

Cf. A270027.

Programs

  • Maple
    g:= proc(x,d) local L,i;
      L:= convert(2^d+x,base,2);
      add((L[i]+1)*3^(i-1),i=1..d)
    end proc:
    T3:= NULL:
    for d from 1 to 14 do T3:= T3, seq(g(x,d),x=0..2^d-1); od:
    T[3]:= [T3]:
    for b from 4 to 14 do T[b]:= remove(t -> member(0, convert(t,base,b)),T[b-1]); if nops(T[b]) = 0 then printf("b=%d\n",b); break fi; od:
    for b from 3 to 14 do T[b][1..14] od;

A319033 a(n) is the (conjectured) largest number k that is zeroless in every base b such that n <= b < k.

Original entry on oeis.org

7, 619, 26237, 698531, 3979433, 3979433, 29643151199, 29643151199, 29643151199, 29643151199, 260621258159, 260621258263, 260621258263, 296126238241, 296126238241, 296126238241, 296126238241, 556715917481, 971156053631, 971156053631, 971156053631, 971156053631
Offset: 2

Views

Author

Jon E. Schoenfield, Oct 08 2018

Keywords

Comments

All terms are necessarily prime.
It seems nearly certain that there is no k > 7 that is zeroless in every base from 2 through k-1; if such a k exists, it exceeds 2^(10^9).
Up to 10^5000 (see A069575), no number k > 619 is zeroless in every base from 3 through k-1.
a(4) = 26237 or > 10^1000; a(5) = 698531 or > 10^1000; a(6) = a(7) = 3979433 unless a(7) > 10^1000; a(8) = a(9) = a(10) = a(11) = 29643151199 unless a(11) > 10^1000; it seems extremely unlikely that any of these terms could actually exceed 10^1000.

Examples

			a(2) = 7 because k = 7 = 111_2 = 21_3 = 13_4 = 12_5 = 11_6, with no zero digits in any base from 2 through k-1, and this is almost certainly (see Comments) the largest such number having this property.
a(3) = 619 because k = 619 = 211221_3 = 21223_4 = 4434_5 = 2511_6 = 1543_7 = 1153_8 = 757_9 = 619_10 = 513_11 = 437_12 = 388_13 = 323_14 = 2B4_15 = ... = 11_(k-1), and this is almost certainly (see Comments) the largest number having this property.
		

Crossrefs

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