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 1-4 of 4 results.

A244033 Nonnegative integers that do not appear in A243357.

Original entry on oeis.org

0, 10, 14, 18, 20, 30, 34, 38, 40, 50, 54, 58, 60, 70, 74, 78, 80, 90, 94, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 114, 118, 120, 130, 134, 138, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 154, 158, 160, 170, 174, 178, 180, 181, 182, 183, 184, 185, 186, 187, 188
Offset: 1

Views

Author

N. J. A. Sloane, Jun 26 2014

Keywords

References

  • Giovanni Resta and Hans Havermann, Postings to the Sequence Fans Mailing List, circa Jun 26 2014

Links

Crossrefs

Cf. A243357, A244034.

Programs

A244034 Numbers not containing the digit 0 that do not appear in A243357.

Original entry on oeis.org

14, 18, 34, 38, 54, 58, 74, 78, 94, 98, 114, 118, 134, 138, 141, 142, 143, 144, 145, 146, 147, 148, 149, 154, 158, 174, 178, 181, 182, 183, 184, 185, 186, 187, 188, 189, 194, 198, 214, 218, 228, 234, 238, 254, 258, 268, 274, 278, 294, 298, 313, 314, 316, 318, 319, 323, 326, 329
Offset: 1

Views

Author

N. J. A. Sloane, Jun 26 2014

Keywords

References

  • Giovanni Resta and Hans Havermann, Postings to the Sequence Fans Mailing List, circa Jun 26 2014

Links

Crossrefs

Cf. A243357, A244033.

Programs

  • Mathematica
    Select[Complement[Range[0,26547],Take[A243357,11021]],MemberQ[IntegerDigits[#],0]==False&] (* Hans Havermann, Jun 26 2014 *)

A244471 Lexicographically earliest sequence of integers with property that if a vertical line is drawn between any pair of adjacent digits, the number Z formed by the digits to the left of the line is divisible by the digit to the right of the line.

Original entry on oeis.org

1, 11, 3, 7, 71, 31, 111, 113, 33, 117, 77, 13, 37, 711, 1111, 19, 9, 91, 1117, 73, 311, 131, 1131, 1133, 93, 331, 11111, 39, 99, 97, 119, 333, 911, 133, 931, 1139, 771, 337, 713, 339, 933, 391, 1137, 773, 1113, 991, 11171, 3111, 777, 3311, 79, 17, 191, 171, 11311, 137, 719, 993
Offset: 1

Views

Author

N. J. A. Sloane, Jul 02 2014

Keywords

Comments

"Lexicographically earliest" means in the sense of a sequence of integers, not digits.
No digit can be even or five. - Hans Havermann, Jul 02 2014 [Proof: if not, let d be the first digit in the sequence that is even or 5, and let Z be the concatenation of all earlier digits. But then Z is odd and does not end in 5, so is not divisible by d. Contradiction. - N. J. A. Sloane, Jul 03 2014] So any term must have only the odd digits {1, 3, 7, 9} (see A136333). - Robert G. Wilson v, Jul 02 2014
We choose the next term, a(n), to be the minimal number not already in the sequence such that the property "if a vertical line is drawn between any pair of adjacent digits, the number Z formed by the digits to the left of the line is divisible by the first digit following Z" holds.
So even if Z is prime, the next term can start with a 1.
So if Z is divisible by any d in {2,3,...,9} the next term can start with 1 or d, otherwise it must start with 1.
This sequence is missing A136333 terms 313, 319, 373, 379, 717, 737, 797, 913, 919, 939, 973, 979, 1313, ... The earliest occurrences of n-digit numbers are the repunits at indices 1, 2, 7, 15, 27, 97, 372, 939, 2164, 4781, 10851, 22779, 47056, ... The latest n-digit numbers and their indices are: (9,17), (17,52), (397,290), (1917,867), (19317,2003), (199117,7241), (1999117,17953), (19999997,44173), ... - Hans Havermann, Jul 04 2014, Jul 07 2014, Jul 15 2014

Examples

			After 1,11,3,7, let a(5) = x be the next term. Now 11137 = 7*37*43, so x must begin with 1 or 7. The candidates for x are therefore 12,13,...,19,71,72,,...,79,111,...
If x=12, we would get 1 11 3 7 12 ... but Z = 11371 is prime and is not divisible by 2, ..., 9. So x is not 12, ...,19. The next candidate is x=71, and this works. So a(5)=71.

References

  • Eric Angelini, Posting to Sequence Fans Mailing List, Jun 26 2014

Links

Crossrefs

A sister sequence to A243357 and A244496. A subsequence of A136333.

Programs

  • Mathematica
    r=f=e={1,3,7,9};Do[e=10*e;f=Flatten[Table[e[[i]]+f,{i,4}]];r=Join[r,f],{9}];r=Select[r,Intersection[Partition[IntegerDigits[#],3,1],IntegerDigits[{313,319,373,379,717,737,797,913,919,939,973,979}]]=={}&];t=0;Do[c=1;While[d=IntegerDigits[r[[c]]];Union[Table[IntegerQ[(10^i*t+FromDigits[Take[d,i]])/d[[i+1]]],{i,0,Length[d]-1}]]!={True},c++];Print[r[[c]]];t=10^Length[d]*t+r[[c]];r=Delete[r,c],{10850}] (* Hans Havermann, Jul 04 2014 *)

Extensions

Corrected and extended by Hans Havermann, Jul 02 2014

A244496 Lexicographically earliest sequence S of integers with property that if a vertical line is drawn between any pair of adjacent digits p and q, say, the number Z formed by the p digits to the left of the line is divisible by p.

Original entry on oeis.org

1, 2, 3, 11, 5, 6, 4, 8, 12, 13, 15, 21, 22, 24, 17, 16, 25, 19, 7, 23, 27, 9, 28, 41, 51, 31, 26, 42, 32, 43, 52, 44, 45, 35, 55, 59, 111, 53, 29, 56, 48, 46, 112, 57, 36, 33, 115, 71, 61, 121, 116, 81, 122, 123, 124, 39, 125, 91, 62, 119, 117, 126, 128, 82, 64, 47, 151, 37, 129, 152, 84, 83, 153
Offset: 1

Views

Author

N. J. A. Sloane, Jul 06 2014

Keywords

Comments

"Lexicographically earliest" means in the sense of a sequence of integers, not digits.
S is infinite, of course, as it can always be extended with an integer (not yet present) containing only 1's.
Apart from numbers containing the digit zero, the first numbers that cannot appear as terms are 14, 18, 34, 38, 54, 58, 74, 78, 94, 98, 113, 114, 118, 133, 134, 138, 141, 142, 143, 144, 145, 146, 147, 148, 149, 154, 158, 163, 173, 174, 178, 181, 182, 183, 184, 185, 186, 187, 188, 189, 193, 194, 198, 214, 218, 223, 228, 233, 234, 238, 253, 254, 258, 263, 268, 274, 278, 283, 293, 294, 298, 313, 314, 318, 323, 334, ... - Hans Havermann, Jul 14 2014

Examples

			Example:a) draw a line between 6 and 4, for instance -- thus p = 6:
   S = 1,2,3,11,5,6|,4,
b) concatenate the last 6 digits before the line (to get Z):
   Z = 231156
c) Z/p is an integer (indeed, Z/6 = 38526)
Here are notes on the initial terms:
         Z / p = integer   (Z ends in p and has digit-length p)
         1 / 1 = 1
        12 / 2 = 6
       123 / 3 = 41
         1 / 1 = 1
         1 / 1 = 1
     23115 / 5 = 4623
    231156 / 6 = 38526
      1564 / 4 = 391
  23115648 / 8 = 2889456
         1 / 1 = 1
        12 / 2 = 6
         1 / 1 = 1
       213 / 3 = 71
         1 / 1 = 1
     21315 / 5 = 4263
        52 / 2 = 26
         1 / 1 = 1
        12 / 2 = 6
        22 / 2 = 11
        22 / 2 = 11
      2224 / 4 = 556
         1 / 1 = 1
   1222417 / 7 = 174631
         1 / 1 = 1
    241716 / 6 = 40286
        62 / 2 = 31
     71625 / 5 = 14325
         1 / 1 = 1
417162519 / 9 = 46351391
   1625197 / 7 = 232171
        72 / 2 = 36
       723 / 3 = 241
        32 / 2 = 16
   1972327 / 7 = 281761
...

References

  • Eric Angelini, Posting to Sequence Fans Mailing List, Jun 26 2014

Links

Crossrefs

Cf. A243357, A244471.

Programs

  • Mathematica
    s={1,2,3,11,5,6,4};t=Flatten[IntegerDigits[s]];r=Select[Complement[Select[Range[60000],MemberQ[IntegerDigits[#],0]==False&],s],Intersection[Partition[IntegerDigits[#],2,1],IntegerDigits[{14,18,34,38,54,58,74,78,94,98}]]=={}&];Do[c=1;While[d=IntegerDigits[r[[c]]];Union[Table[IntegerQ[FromDigits[Take[Join[t,Take[d,i]],-d[[i]]]]/d[[i]]],{i,Length[d]}]]!={True},c++];AppendTo[s,r[[c]]];r=Delete[r,c];t=Take[Join[t,d],-9],{10002}];s
(* Hans Havermann, Jul 12 2014 *)

Extensions

More terms from Jean-Marc Falcoz, Jul 05 2014
Showing 1-4 of 4 results.

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