A152607 -id:A152607 - cp's OEIS frontend

A152610 Primes appearing in A152607.

13, 37, 79, 97, 71, 17, 73, 37, 79, 97, 71, 11, 17, 71, 13, 37, 71, 17, 79, 97, 71, 19, 97, 73, 31, 11, 11, 11, 11, 11, 11, 13, 31, 11, 11, 17, 71, 11, 11, 19, 97, 71, 11, 11, 17, 71, 11, 13, 37, 71, 11, 17, 79, 97, 71, 11, 19, 97, 71, 13, 31, 11, 11, 11, 11, 11

Offset: 1

N. J. A. Sloane, Sep 23 2009

Amiram Eldar, Table of n, a(n) for n = 1..153

Cf. A152607.

More terms from Amiram Eldar, Apr 11 2025

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.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Sep 24 2009

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)

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

Cf. A152607, A158652, A152604-A152609.

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

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.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Sep 23 2009

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

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

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

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

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

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.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Sep 23 2009

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

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

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

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

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.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Sep 23 2009

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

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

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

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

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.

Original entry on oeis.org

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

Offset: 1

Eric Angelini, Aug 17 2023

			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.

Tyler Busby, Table of n, a(n) for n = 1..10000
Eric Angelini, Prime welds, Personal blog.

Cf. A152607.

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.

Original entry on oeis.org

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

Offset: 0

Carole Dubois, Jun 08 2022

			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.

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

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

cp's OEIS Frontend

A152610 Primes appearing in A152607.

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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.

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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.

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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.

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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.

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PARI

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.

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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.

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Python