A136333 -id:A136333

Search: a136333 -id:a136333

Displaying 1-7 of 7 results found.

page 1

A245193

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

+20
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

EXAMPLE

. 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 .

PROG

(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

CROSSREFS

Cf. A000040, A091633, A136333.

KEYWORD

nonn,base

AUTHOR

Reinhard Zumkeller, Jul 18 2014

STATUS

approved

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.

+20
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..17.

EXAMPLE

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.

KEYWORD

nonn,base

AUTHOR

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

EXTENSIONS

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

STATUS

approved

A091633

Primes whose digits are restricted to 1,3,7,9 (same as terminal digits of primes).

+10
13

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

Pierre Cami and Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 5058 terms from Pierre Cami)

FORMULA

Select primes having digits 1, 3, 7, 9 only.

a(n) = A000040(A091871(n)). - R. J. Mathar, Aug 29 2018

MATHEMATICA

Select[Flatten[Table[FromDigits/@Tuples[{1, 3, 7, 9}, n], {n, 4}]], PrimeQ] (* Harvey P. Dale, Jun 26 2015 *)

PROG

(Haskell)

a091633 n = a091633_list !! (n-1)

a091633_list = filter ((== 1) . a010051') a136333_list

-- Reinhard Zumkeller, Jul 17 2014

CROSSREFS

Subsequence of A136333, A245193, and A030096.

A091871 gives prime index.

Cf. A010051.

KEYWORD

easy,nonn,base

AUTHOR

Enoch Haga, Jan 26 2004

STATUS

approved

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.

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

REFERENCES

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

LINKS

Robert G. Wilson v and Hans Havermann (Robert G. Wilson v to 1000), Table of n, a(n) for n = 1..10850

EXAMPLE

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.

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 *)

CROSSREFS

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

KEYWORD

nonn,base

AUTHOR

N. J. A. Sloane, Jul 02 2014

EXTENSIONS

Corrected and extended by Hans Havermann, Jul 02 2014

STATUS

approved

A091871

A091633 indexed by A000040.

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

FORMULA

a(n)=k such that A000040(k) = A091633(n).

a(n) = A049084(A091633(n)). - Reinhard Zumkeller, Jul 18 2014

PROG

(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

CROSSREFS

Cf. A049084, A091633, A136333.

KEYWORD

easy,nonn,base

AUTHOR

Ray Chandler, Feb 07 2004

STATUS

approved

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.

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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.

LINKS

Table of n, a(n) for n=1..43.

EXAMPLE

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.

KEYWORD

nonn,base,less

AUTHOR

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

EXTENSIONS

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

STATUS

approved

A287534

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

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..42.

MATHEMATICA

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

CROSSREFS

Cf. A091633, A136333.

KEYWORD

nonn,base

AUTHOR

Luke Zieroth, May 26 2017

STATUS

approved

page 1

Search completed in 0.005 seconds