A158652 -id:A158652 - cp's OEIS frontend

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.

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

A152604 a(1) = 1; thereafter a(n) is always the smallest integer > a(n-1) not leading to a contradiction, such that any four consecutive digits in the sequence sum up to a prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 51, 83, 110, 111, 211, 301, 310, 311, 631, 703, 710, 911, 2111, 2113, 2117, 2119, 2153, 2155, 2159, 2171, 2173, 2177, 2179, 2513, 2515, 2519, 2531, 2533, 2537, 2539, 2573, 2575, 2579, 8513, 8515, 8519, 8573, 8579, 8591

Offset: 1

N. J. A. Sloane, Sep 23 2009

Computed by Jean-Marc Falcoz.

From a(69)=1100110 onward starts a repeating pattern of length 54. - 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..A152609.

A152604(n,show_all=0)={my(a);for(i=1,n,if( i<8,a=i+(i>3)+(i>4), my(l3d=if(a>99,a%1000,[789,951,183][i-7])); while( a++, my(t=a+l3d*10^#Str(a));forstep(d=#Str(a)-1,0,-1, isprime(sum(j=d,d+3,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

A152609 a(1) = 1; thereafter a(n) is always the smallest integer > a(n-1) not leading to a contradiction, such that the concatenation of any four consecutive digits in the sequence is a prime.

Original entry on oeis.org

1, 2, 3, 7, 19, 31, 91, 373, 931, 1931, 1933, 7193, 11931, 19311, 93119, 311931, 913733, 1373313, 7331373, 9311931, 19311931, 19311933, 71931193, 119311931, 193119311, 931193119, 3119311931, 9137331373, 9311931193

Offset: 1

N. J. A. Sloane, Sep 23 2009

Computed by Jean-Marc Falcoz.

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

Cf. A158652, A152604-A152609.

A152607 a(1) = 1; 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

1, 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: 1

N. J. A. Sloane, Sep 23 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
Eric Angelini, Chiffres consecutifs dans quelques suites [Cached copy, with permission]

Cf. A158652, A152604-A152609. See A152136 for another version.

from itertools import count, islice
def cgen(seed, digits, geq="0"): # numbers satisfying the condition
    allowed = {"1": "1379", "3": "17", "7": "139", "9": "7"}
    if digits == 1:
        yield from (c for c in allowed[seed] if c >= geq); return
    for f in (c for c in allowed[seed] if c >= geq):
        yield from (f + r for r in cgen(f, digits-1))
def nextc(k): # next element of cgen greater than k
    s = str(k)
    for d in count(len(s)):
        geq = s[0] if d == len(s) else "0"
        for c in map(int, cgen(s[-1], d, geq=geq)):
            if c > k: return c
def agen():
    an = 1
    for n in count(1): yield an; an = nextc(an)
print(list(islice(agen(), 40))) # Michael S. Branicky, Jul 12 2022

# alternate using pattern from comments
from itertools import count, islice
def agen():
    yield from [1, 3, 7, 9, 71, 73, 79, 711, 713, 717, 971]
    for i in count(0):
        i1 = "1"*i
        yield from map(int, ("97"+i1+"3", i1+"1111", i1+"1113", i1+"1117", i1+"1119", "7111"+i1, "711"+i1+"3", "711"+i1+"7", "9711"+i1))
print(list(islice(agen(), 40))) # Michael S. Branicky, Jul 12 2022

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

A152608 a(1) = 1; thereafter a(n) is always the smallest integer > a(n-1) not leading to a contradiction, such that the concatenation of any three consecutive digits in the sequence is a prime.

Original entry on oeis.org

1, 2, 7, 19, 37, 39, 71, 91, 93, 733, 739, 773, 971, 977, 3311, 3733, 7331, 7337, 9719, 9773, 31131, 31137, 33113, 73311, 311311, 311313, 733113, 733131, 733137, 971911, 3113113, 7331131, 7331137, 9719113, 9719191, 9719193

Offset: 1

N. J. A. Sloane, Sep 23 2009

Computed by Jean-Marc Falcoz.

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

Cf. A158652, A152604-A152609.

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

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

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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A152604 a(1) = 1; thereafter a(n) is always the smallest integer > a(n-1) not leading to a contradiction, such that any four consecutive digits in the sequence sum up to a prime.

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A152609 a(1) = 1; thereafter a(n) is always the smallest integer > a(n-1) not leading to a contradiction, such that the concatenation of any four consecutive digits in the sequence is a prime.

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A152607 a(1) = 1; 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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A152608 a(1) = 1; thereafter a(n) is always the smallest integer > a(n-1) not leading to a contradiction, such that the concatenation of any three consecutive digits in the sequence is 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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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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