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 1-7 of 7 results.

A245193 Smallest prime having in decimal representation A136333(n) as suffix.

Original entry on oeis.org

11, 3, 7, 19, 11, 13, 17, 19, 31, 233, 37, 139, 71, 73, 277, 79, 191, 193, 97, 199, 2111, 113, 1117, 3119, 131, 4133, 137, 139, 1171, 173, 4177, 179, 191, 193, 197, 199, 311, 313, 317, 1319, 331, 2333, 337, 2339, 2371, 373, 2377, 379, 3391, 2393, 397, 1399
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 18 2014

Keywords

Comments

a(n) = A136333(n) iff A136333(n) itself is a prime number, cf. A091633.

Examples

			.    n  |   a(n)  | A136333(n)
. ------+---------+-----------
.   10  |    233  |      33
.   11  |     37  |      37
.   12  |    139  |      39
.   13  |     71  |      71
.   14  |     73  |      73
.   15  |    277  |      77
.   16  |     79  |      79
.   17  |    191  |      91
.   18  |    193  |      93
.   19  |     97  |      97
.   20  |    199  |      99
.   21  |   2111  |     111
.   22  |    113  |     113
.   23  |   1117  |     117
.   24  |   3119  |     119
.   25  |    131  |     131
.   26  |   4133  |     133
.   27  |    137  |     137
.   28  |    139  |     139
.   29  |   1171  |     171
.   30  |    173  |     173 .

Links

Crossrefs

Cf. A000040, A091633, A136333.

Programs

  • Haskell
    import Data.List (isSuffixOf); import Data.Function (on)
a245193 n = head [p | p <- a000040_list,
                      (isSuffixOf `on` show) (a136333 n) p]
  • PARI
    isok(m) = my(d=digits(m)); apply(x->gcd(x, 10), d) == vector(#d, k, 1); \\ A136333
f(m) = my(p=nextprime(m), s=10^#Str(m)); while ((p-m) % s, p = nextprime(p+1)); p;
lista(nn) = apply(x->f(x), select(isok, [1..nn]));
lista(1000) \\ Michel Marcus, Feb 25 2022

A163401 Numbers m such that all 10*m+d_1, 100*m+d_2 and 1000*m+d_3 are composite, where d_i are any i-digit numbers of A136333.

Original entry on oeis.org

487856, 694103, 771084, 836254, 1051886, 1119347, 1122734, 1157014, 1181077, 1591742, 1638820, 1646819, 1820743, 1921148, 1945355, 2001782, 2026571
Offset: 1

Views

Author

Vladislav-Stepan Malakhovsky and Juri-Stepan Gerasimov, Jul 26 2009

Keywords

Comments

A subsequence A163398 satisfying an additional requirement on the 1000*m+d_3 compositions.

Examples

			m=487856 is in the sequence because 4878561, 4878563, 4878567, 4878569 from attachment of
d_1 = 1 to 9, and 48785611, 48785613, .., 48785697, 48785699 from attachment of
d_2 = 11 to 99, and 487856111, 487856113,..., 487856997, 487856999 from
attachment of d_3=111 to 999 are all composite.

Crossrefs

Cf. A002808.

Extensions

13 more terms from R. J. Mathar, Aug 07 2009

A091633 Primes having only {1, 3, 7, 9} as digits.

Original entry on oeis.org

3, 7, 11, 13, 17, 19, 31, 37, 71, 73, 79, 97, 113, 131, 137, 139, 173, 179, 191, 193, 197, 199, 311, 313, 317, 331, 337, 373, 379, 397, 719, 733, 739, 773, 797, 911, 919, 937, 971, 977, 991, 997, 1117, 1171, 1193, 1319, 1373, 1399, 1733, 1777, 1913, 1931, 1933
Offset: 1

Views

Author

Enoch Haga, Jan 26 2004

Keywords

Comments

Some primes of sufficient length might be termed DNA primes if the sequence of digits 1,3,7,9 in any order happens to be an appropriate analog of the DNA bases A, G, C, T. It would be interesting to know if it is possible for any DNA sequence to match a DNA prime.

Links

Crossrefs

Subsequence of A136333, A245193, and A030096.
A091871 gives prime index.
Cf. A010051.

Programs

  • Haskell
    a091633 n = a091633_list !! (n-1)
a091633_list = filter ((== 1) . a010051') a136333_list
-- Reinhard Zumkeller, Jul 17 2014
  • Mathematica
    Select[Flatten[Table[FromDigits/@Tuples[{1,3,7,9},n],{n,4}]],PrimeQ] (* Harvey P. Dale, Jun 26 2015 *)

Formula

Select primes having digits 1, 3, 7, 9 only.
a(n) = A000040(A091871(n)). - R. J. Mathar, Aug 29 2018

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

A091871 A091633 indexed by A000040.

Original entry on oeis.org

2, 4, 5, 6, 7, 8, 11, 12, 20, 21, 22, 25, 30, 32, 33, 34, 40, 41, 43, 44, 45, 46, 64, 65, 66, 67, 68, 74, 75, 78, 128, 130, 131, 137, 139, 156, 157, 159, 164, 165, 167, 168, 187, 193, 196, 215, 220, 222, 270, 275, 293, 294, 295, 298, 299, 301, 302, 303, 444, 446
Offset: 1

Views

Author

Ray Chandler, Feb 07 2004

Keywords

Links

Crossrefs

Cf. A049084, A091633, A136333.

Programs

  • Haskell
    a091871 n = a091871_list !! (n-1)
a091871_list = f [1..] a000040_list where
   f (i:is) (p:ps) = if (null $ show p `intersect` "024568")
                        then i : f is ps else f is ps
-- Reinhard Zumkeller, Jul 18 2014

Formula

a(n)=k such that A000040(k) = A091633(n).
a(n) = A049084(A091633(n)). - Reinhard Zumkeller, Jul 18 2014

A163398 Numbers m such that all numbers 10*m+(odd single-digit number) and 100*m+(any 2-digit, digits coprime to 10) are composite.

Original entry on oeis.org

167, 176, 403, 513, 761, 935, 1037, 1218, 1307, 1559, 1865, 1932, 1995, 2057, 2123, 2255, 2288, 2340, 2414, 2852, 3152, 3483, 3581, 3734, 3914, 4136, 4169, 4226, 4238, 4265, 4373, 4390, 4433, 4436, 4443, 4460, 4466, 4482, 4631, 4706, 4806, 4842, 4850
Offset: 1

Views

Author

Vladislav-Stepan Malakhovsky and Juri-Stepan Gerasimov, Jul 26 2009

Keywords

Comments

The first requirement is that 10m+1, 10m+3, 10m+5, 10m+7 and 10m+9 are all composite; for 10*m+5 with the divisor 5 this is redundant. The second requirement is that 100*m plus any 2-digit number of A136333 is also composite.

Examples

			m=167 is in the sequence because 1671, 1673, 1677, 1679, 16711, 16713, 16717, 16719, 16731,
16733, 16737, 16739, 16771, 16773, 16777, 16779, 16791, 16793, 16797, 16799 are composites.

Crossrefs

Cf. A002808.

Extensions

Rephrased in terms of A136333 and extended by R. J. Mathar, Aug 02 2009

A287534 Composite numbers whose digits are restricted to 1, 3, 7, and 9.

Original entry on oeis.org

9, 33, 39, 77, 91, 93, 99, 111, 113, 117, 119, 133, 171, 177, 319, 333, 339, 371, 377, 391, 393, 399, 711, 713, 717, 731, 737, 771, 777, 779, 791, 793, 799, 913, 917, 931, 933, 939, 973, 979, 993, 999
Offset: 1

Views

Author

Luke Zieroth, May 26 2017

Keywords

Crossrefs

Cf. A091633, A136333.

Programs

  • Mathematica
    Select[Flatten@ Array[FromDigits /@ Tuples[{1, 3, 7, 9}, #] &, 3], CompositeQ] (* Michael De Vlieger, Jun 01 2017 *)
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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