A152607 -id:A152607

A152610

Primes appearing in A152607.

+20
1

13, 37, 79, 97, 71, 17, 73, 37, 79, 97, 71, 11, 17, 71, 13

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

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..15.

KEYWORD

nonn,base,more

AUTHOR

N. J. A. Sloane, Sep 23 2009

STATUS

approved

A152136

a(0) = 0; thereafter a(n) is always the smallest integer > a(n-1) not leading to a contradiction, such that the concatenation of any two consecutive digits in the sequence is a prime.

+10
10

0, 2, 3, 7, 9, 71, 73, 79, 711, 713, 717, 971, 973, 1111, 1113, 1117, 1119, 7111, 7113, 7117, 9711, 9713, 11111, 11113, 11117, 11119, 71111, 71113, 71117, 97111, 97113, 111111, 111113, 111117, 111119, 711111, 711113, 711117, 971111

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

OFFSET

0,2

COMMENTS

A variant of A152607 suggested by Zak Seidov, Sep 24 2009.

Computed by Jean-Marc Falcoz.

Comment from Jean-Marc Falcoz: (Start)

The sequence is infinite since it has the following structure:

9713, 11111, 11113, 11117, 11119, 71111, 71113, 71117, 97111,

97113, 111111, 111113, 111117, 111119, 711111, 711113, 711117, 971111,

971113, 1111111, 1111113, 1111117, 1111119, 7111111, 7111113, 7111117, 9711111,

9711113, 11111111, 11111113, 11111117, 11111119, 71111111, 71111113, 71111117, 97111111,

97111113, 111111111, 111111113, 111111117, 111111119, 711111111, 711111113, 711111117, 971111111,

971111113, 1111111111, 1111111113, 1111111117, 1111111119, 7111111111, 7111111113, 7111111117, 9711111111,

9711111113, ... (End)

LINKS

Table of n, a(n) for n=0..38.

Eric Angelini, Chiffres consecutifs dans quelques suites

E. Angelini, Chiffres consecutifs dans quelques suites [Cached copy, with permission]

CROSSREFS

Cf. A152607, A158652, A152604-A152609.

KEYWORD

nonn,base

AUTHOR

N. J. A. Sloane, Sep 24 2009

EXTENSIONS

Offset changed by N. J. A. Sloane, Jun 16 2021

STATUS

approved

A152603

a(1) = 1; thereafter, a(n) is always the smallest integer > a(n-1) not leading to a contradiction, such that any three consecutive digits in the sequence sum up to a prime.

+10
5

1, 2, 4, 5, 8, 41, 60, 70, 410, 412, 416, 418, 452, 454, 458, 470, 472, 476, 478, 812, 814, 818, 830, 832, 836, 838, 872, 874, 878, 2101, 2210, 2300, 2302, 3002, 3003, 4011, 5110, 6101, 6410, 6500, 7002, 9020, 9200, 20020, 30020, 30021, 40110

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

OFFSET

1,2

COMMENTS

Computed by Jean-Marc Falcoz.

From a(34)=3002 on, there starts a pattern [ 3(002){n}, ..., 2(002){n+1} ] of length 52 which then repeats forever. This allows us to write an explicit formula for any term a(n) of the sequence. - M. F. Hasler, Oct 16 2009

LINKS

Table of n, a(n) for n=1..47.

Eric Angelini, Chiffres consecutifs dans quelques suites

E. Angelini, Chiffres consecutifs dans quelques suites [Cached copy, with permission]

FORMULA

a(n) = b(n)*10^[3n/52] = c(n)*10^(3n/52) with (except for smaller initial terms) 20 < b(n) < 611 and c(52k+23) = 9.89... < c(n) < c(52k) = 91.1... for all integers k > 0. - M. F. Hasler, Oct 16 2009

PROG

(PARI) A152603(n, show_all=0)={ my(a); for(i=1, n, if(i<4, a=2^i/2, my( l2d=a%100+if(i<7, 10*[1, 2, 4, 5][i-2])); while(a++, my(t=a+l2d*10^#Str(a)); forstep(d=#Str(a)-1, 0, -1, isprime(z=t\10^d%10+t\10^(d+1)%10+t\10^(d+2)%10) & next; a+=10^d-a%10^d-1; next(2)); break)); show_all&print1(a", ")); a} \\ M. F. Hasler, Oct 16 2009

CROSSREFS

Cf. A158652, A152604, A152605, A152606, A152607, A152608, A152609.

KEYWORD

nonn,base

AUTHOR

N. J. A. Sloane, Sep 23 2009

STATUS

approved

A152605

a(1) = 1; thereafter a(n) is always the smallest integer > a(n-1) not leading to a contradiction, such that any five consecutive digits in the sequence sum up to a prime.

+10
3

1, 2, 3, 4, 7, 12, 30, 51, 83, 231, 232, 312, 323, 327, 413, 414, 530, 541, 701, 811, 812, 1101, 2110, 3011, 6301, 7030, 7103, 8110, 9011, 21011, 21013, 21017, 21019, 21053, 21055, 21059, 21071, 21073, 21077, 21079, 21413, 21415, 21419

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

OFFSET

1,2

COMMENTS

Computed by Jean-Marc Falcoz.

From a(116)=6100011 on, there starts a pattern of 75 terms which then repeats indefinitely (with duplication of a substring of 5 digits in the middle of each term). - M. F. Hasler, Oct 16 2009

LINKS

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

Eric Angelini, Chiffres consecutifs dans quelques suites

E. Angelini, Chiffres consecutifs dans quelques suites [Cached copy, with permission]

PROG

(PARI) A152605(n, show_all=0, s=[1, 2, 3, 4, 7, 12, 30, 51, 83, 231, 232, 312, 323, 327, 413, 414, 530, 541, 701, 811, 812, 1101])={ my(a); for(i=1, n, if(i<=#s, a=s[i], my(ld=a%10^4); while(a++, my(t=a+ld*10^#Str(a)); forstep(d=#Str(a)-1, 0, -1, isprime(sum(j=d, d+4, t\10^j%10))&next; a+=10^d-a%10^d-1; next(2)); break)); show_all&print1(a", ")); a } \\ M. F. Hasler, Oct 16 2009

CROSSREFS

Cf. A158652, A152603, A152604, A152606, A152607, A152608, A152609.

KEYWORD

nonn,base

AUTHOR

N. J. A. Sloane, Sep 23 2009

STATUS

approved

A152606

a(1) = 1; thereafter a(n) is always the smallest integer > a(n-1) not leading to a contradiction, such that any six consecutive digits in the sequence sum up to a prime.

+10
2

1, 2, 3, 4, 5, 8, 9, 21, 45, 83, 89, 450, 503, 630, 701, 810, 901, 2101, 2103, 4121, 6301, 6303, 6503, 6901, 43030, 70103, 81010, 90101, 210101, 210103, 210107, 210109, 210143, 210145, 210149, 210161, 210163, 210167, 210169, 210503

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

OFFSET

1,2

COMMENTS

Computed by Jean-Marc Falcoz.

From a(269) = 1010001010 on, there starts a pattern of 104 terms, which then repeats indefinitely (with 6 digits in the middle of each term duplicated). - M. F. Hasler, Oct 16 2009

LINKS

Table of n, a(n) for n=1..40.

Eric Angelini, Chiffres consecutifs dans quelques suites

E. Angelini, Chiffres consecutifs dans quelques suites [Cached copy, with permission]

PROG

(PARI) a(n, show_all=0, s=[1, 2, 3, 4, 5, 8, 9, 21, 45, 83, 89, 450, 503, 630, 701, 810, 901, 2101, 2103, 4121, 6301, 6303, 6503, 6901, 43030])={ my(a, nd=#Str(s[ #s])); for(i=1, n, if( i<=#s, a=s[i], my(ld=a%10^nd); while(a++, my(t=a+ld*10^#Str(a)); forstep(d=#Str(a)-1, 0, -1, isprime(sum(j=d, d+nd, t\10^j%10))&next; a+=10^d-a%10^d-1; next(2)); break)); show_all & print1(a", ")); a} \\ M. F. Hasler, Oct 16 2009

CROSSREFS

Cf. A158652, A152604, A152605, A152607, A152608, A152609.

KEYWORD

nonn,base

AUTHOR

N. J. A. Sloane, Sep 23 2009

STATUS

approved

A363572

Lexicographically earliest sequence of distinct terms > 0 such that the concatenation of the rightmost digit of a(n) and the leftmost digit of a(n+1) forms a prime number. The rightmost digit of a(n) cannot be 0.

+10
1

1, 3, 7, 9, 71, 11, 12, 31, 13, 14, 15, 32, 33, 16, 17, 18, 34, 19, 72, 35, 36, 73, 74, 37, 38, 39, 75, 91, 76, 77, 92, 93, 78, 94, 79, 701, 95, 96, 101, 97, 98, 99, 702, 301, 102, 302, 303, 103, 104, 105, 304, 106, 107, 108, 305, 306, 109, 703, 111, 112, 307, 113, 114, 115

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

OFFSET

1,2

LINKS

Tyler Busby, Table of n, a(n) for n = 1..10000

Eric Angelini, Prime welds, Personal blog.

EXAMPLE

a(1) = 1 and a(2) = 3 form 13, a prime number;

a(2) = 3 and a(3) = 7 form 37, a prime number;

a(3) = 7 and a(4) = 9 form 79, a prime number;

a(4) = 9 and the leftmost digit of a(5) = 71 form 97, a prime number;

a(5) = 71 and its rightmost digit, concatenated to the leftmost digit of a(6) = 11, form 11, a prime number; etc.

CROSSREFS

Cf. A152607.

KEYWORD

base,nonn

AUTHOR

Eric Angelini, Aug 17 2023

STATUS

approved

A354839

Beginning with 0, smallest positive integer not yet in the sequence such that the concatenation of two digits of the sequence separated by a comma is prime.

+10
0

0, 2, 3, 1, 7, 9, 70, 5, 30, 20, 21, 10, 22, 31, 11, 12, 32, 33, 13, 14, 15, 34, 16, 17, 18, 35, 36, 19, 71, 37, 38, 39, 72, 90, 23, 73, 74, 75, 91, 76, 77, 92, 93, 78, 94, 79, 700, 24, 100, 25, 95, 96, 101, 97, 98, 99, 701, 102, 300, 26, 103, 104, 105, 301

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

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..63.

EXAMPLE

a(4)=1 because this is the first number not in the sequence whose first digit is 3 (last digit of a(3)), concatenated with its first digit 1, is prime: 31.

a(14)=31 because this is the first number not in the sequence whose first digit is 2 (last digit of a(13)), concatenated with its first digit 3, is prime: 23.

PROG

(Python)

from sympy import isprime

from itertools import count, islice

def agen(): # generator of terms

aset, k, mink = {0}, 0, 1; yield 0

for n in count(2):

k, prevdig = mink, str(k%10)

while k in aset or not isprime(int(prevdig+str(k)[0])): k += 1

aset.add(k); yield k

while mink in aset: mink += 1

print(list(islice(agen(), 64))) # Michael S. Branicky, Jun 09 2022

CROSSREFS

Cf. A074721, A158652, A152604-A152609, A152136, A152607.

KEYWORD

nonn,easy,base

AUTHOR

Carole Dubois, Jun 08 2022

STATUS

approved