A052089 -id:A052089 - cp's OEIS frontend

A341702 a(n) is the smallest k < n such that the decimal concatenation n||n-1||n-2||...||n-k is prime, or -1 if no such prime exists.

-1, -1, 0, 0, 1, 0, -1, 0, -1, -1, 1, 0, -1, 0, -1, -1, -1, 0, -1, 0, -1, -1, 1, 0, 1, 12, -1, 4, -1, 0, -1, 0, -1, -1, 1, -1, -1, 0, -1, -1, -1, 0, 1, 0, -1, -1, 7, 0, 7, -1, -1, 4, 15, 0, -1, 12, 9, -1, 1, 0, 13, 0, -1, -1, -1, -1, -1, 0, 57, -1, 1, 0, -1, 0

Offset: 0

Chai Wah Wu, Feb 23 2021

A variation of A341716. a(n) = n-1 for n = 82. Are there other n such that a(n) = n-1?

Similar argument as in A341716 shows that if n > 3 and a(n) >= 0, then n-a(n) is odd, a(n) !== 2 (mod 3) and 2n-a(n) !== 0 (mod 3).

			a(10) = 1 since 109 is prime. a(22) = 1 since 2221 is prime.

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A052088, A052089, A054211, A341701, A341715, A341716, A341717.

tcat:= proc(x,y) x*10^(1+ilog10(y))+y end proc:
f:= proc(n) local x,k;
  x:= n;
  for k from 0 to n-1 do
    if isprime(x) then return k fi;
    x:= tcat(x,n-k-1)
  od;
  -1
end proc:
map(f, [$0..100]); # Robert Israel, Mar 02 2022

a(n) = my(k=0, s=Str(n)); while (!isprime(eval(s)), k++; n--; if (k>=n, return(-1)); s = concat(s, Str(n-k))); return(k); \\ Michel Marcus, Mar 02 2022

from sympy import isprime
def A341702(n):
    k, m = n, n-1
    while not isprime(k) and m > 0:
        k = int(str(k)+str(m))
        m -= 1
    return n-m-1 if isprime(k) else -1

a(n) = n-A341701(n).

a(p) = 0 if and only if p is prime.

A084551 Primes which are a concatenation of five consecutive numbers.

Original entry on oeis.org

1516171819, 3940414243, 5758596061, 6566676869, 7778798081, 8384858687, 8990919293, 129130131132133, 153154155156157, 197198199200201, 213214215216217, 239240241242243, 269270271272273, 387388389390391, 399400401402403, 443444445446447, 459460461462463

Offset: 1

Zak Seidov, Jun 27 2003

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A030458, A030471, A052087, A052088, A052089.

Select[Table[FromDigits[Flatten[IntegerDigits[n+Range[-2,2]]]],{n,2,500}],PrimeQ] (* Jayanta Basu, May 24 2013 *)
Select[FromDigits[Flatten[IntegerDigits[#]]]&/@Partition[Range[600],5,1], PrimeQ] (* Harvey P. Dale, Nov 11 2014 *)

A210512 Primes formed by concatenating k, k and 3 for k >= 1.

Original entry on oeis.org

113, 223, 443, 773, 883, 10103, 11113, 14143, 25253, 26263, 28283, 32323, 35353, 41413, 50503, 61613, 68683, 71713, 77773, 80803, 83833, 85853, 88883, 97973, 1001003, 1011013, 1101103, 1131133, 1161163, 1181183, 1221223, 1241243, 1281283, 1331333, 1361363, 1391393

Offset: 1

Abhiram R Devesh, Jan 26 2013

This sequence is similar to A030458, A052089 and A210511.
k must not be a multiple of 3, otherwise the concatenation of k, k and 3 will also be a multiple of 3 and therefore not prime. This is a necessary but not sufficient condition.
Some of the terms can be found with this simple process: 5 - 3 = 2 = 1 + 1 giving 113; 7 - 3 = 4 = 2 + 2 giving 223; 11 - 3 = 8 = 4 + 4 giving 443; 17 - 3 = 14 = 7 + 7 giving 773; 19 - 3 = 16 = 8 + 8 giving 883. - J. M. Bergot, Jul 25 2022

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A030458, A052089, A210511.

[nn3: n in [1..140] | IsPrime(nn3) where nn3 is Seqint([3] cat Intseq(n) cat Intseq(n))]; // Bruno Berselli, Jan 30 2013

Select[Table[FromDigits[Flatten[{IntegerDigits[n], IntegerDigits[n], {3}}]], {n, 100}], PrimeQ] (* Alonso del Arte, Jan 27 2013 *)

import numpy as np
from functools import reduce
def factors(n):
    return reduce(list._add_, ([i, n//i] for i in range(1, int(n**0.5) +1) if n % i == 0))
for i in range(1, 1000):
    p1=int(str(i)+str(i)+"3")
    if len(factors(p1))<3:
        print(p1, end=',')

from sympy import isprime
def xf(n): return int(str(n)*2+'3')
def ok(n): return isprime(xf(n))
print(list(map(xf, filter(ok, range(1, 140))))) # Michael S. Branicky, May 21 2021

A210513 Primes formed by concatenating k, k, and 7.

Original entry on oeis.org

227, 337, 557, 887, 997, 11117, 24247, 26267, 27277, 29297, 30307, 32327, 39397, 48487, 51517, 54547, 60607, 62627, 65657, 68687, 69697, 72727, 74747, 78787, 81817, 87877, 89897, 90907, 92927, 93937, 95957, 101710177, 101910197, 103110317, 103410347, 103810387

Offset: 1

Abhiram R Devesh, Jan 26 2013

This sequence is similar to A030458, A052089, and A092994.
Base considered is 10.
Observations:
- k cannot be a multiple of 7.
- k cannot have a digital root 7 as the sum of the digits would be divisible by 3.
- There is no k between 100 and 1000 that can form a prime number of this form after 95957 the next prime is 101710177.
- k cannot have a digital root equal to 1 or 4, because then in the concatenation it contributes 2 or 8 to the digital root of the number, and that number is then divisible by 3.

			For k = 2, a(1) = 227.
For k = 3, a(2) = 337.
For k = 5, a(3) = 557.
For k = 8, a(4) = 887.
For k = 9, a(5) = 997.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A030458, A052089, A092994.

Select[Table[FromDigits[Flatten[{IntegerDigits[n], IntegerDigits[n], {7}}]], {n, 100}], PrimeQ] (* Alonso del Arte, Feb 01 2013 *)

import numpy as np
from functools import reduce
def factors(n):
    return reduce(list._add_, ([i, n//i] for i in range(1, int(n**0.5) + 1) if n % i == 0))
for i in range(1, 2000):
    p1=int(str(i)+str(i)+"7")
    if len(factors(p1))<3:
        print(p1, end=',')

from sympy import isprime
from itertools import count, islice
def agen(): yield from filter(isprime, (int(str(k)+str(k)+'7') for k in count(1)))
print(list(islice(agen(), 36))) # Michael S. Branicky, Jul 26 2022

a(34) and beyond from Michael S. Branicky, Jul 26 2022

A210514 Prime numbers generated by concatenating k, k, and 9.

Original entry on oeis.org

229, 449, 11119, 14149, 22229, 28289, 31319, 37379, 44449, 49499, 52529, 56569, 67679, 70709, 71719, 80809, 86869, 89899, 94949, 95959, 1061069, 1101109, 1131139, 1151159, 1161169, 1191199, 1241249, 1251259, 1331339, 1401409, 1431439, 1481489, 1571579, 1601609

Offset: 1

Abhiram R Devesh, Jan 26 2013

This series is similar to A030458 and A052089.

Base considered is 10.

			For k=2, a(1)= 229.
For k=4, a(2)= 449.
For k=11, a(3)= 11119.
For k=14, a(4)= 14149.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A030458, A052089.

Select[Table[FromDigits[Flatten[Join[IntegerDigits/@{n,n},{9}]]],{n,200}],PrimeQ] (* Harvey P. Dale, Apr 23 2015 *)

import numpy as np
from functools import reduce
def factors(n):
    return reduce(list._add_, ([i, n//i] for i in range(1, int(n**0.5) + 1) if n % i == 0))
for i in range(1, 2000):
    p1=int(str(i)+str(i)+"1")
    if len(factors(p1))<3:
        print(p1, end=',')

from sympy import isprime
from itertools import count, islice
def agen(): yield from filter(isprime, (int(str(k)+str(k)+'9') for k in count(1)))
print(list(islice(agen(), 34))) # Michael S. Branicky, Jul 26 2022

a(27) and beyond from Michael S. Branicky, Jul 26 2022

A341931 a(n) = smallest m > 0 such that the decimal concatenation n||n-1||n-2||...||m is prime, or -1 if no such prime exists.

Original entry on oeis.org

-1, -1, 2, 3, 3, 5, -1, 3, -1, -1, 7, 11, -1, 13, -1, -1, -1, 17, -1, 19, -1, -1, 19, 23, 23, 13, -1, 23, -1, 29, -1, 31, -1, -1, 33, -1, -1, 37, -1, -1, -1, 41, 41, 43, -1, -1, 3, 47, 17, -1, -1, 47, 37, 41, -1, 27, 47, -1, 57, 59, 47, 61, -1, -1, -1, -1, -1

Offset: 0

Chai Wah Wu, Feb 23 2021

a(n) <= A341701(n). a(82) = 1, are there any other n such that a(n) = 1?
Primes p such that a(p) < p: 7, 53, 73, 79, 89, 103, ...
n such that a(n) < A341701(n): 7, 10, 22, 46, 48, 53, 55, 73, ...
Similar argument as in A341716 shows that if n > 3 and a(n) >= 0, then a(n) is odd, n-a(n) !== 2 (mod 3) and n+a(n) !== 0 (mod 3).

			a(7) = 3 since 76543 is prime and 765432, 7654321 are not. a(10) = 7 since 10987 is prime.

Cf. A052088, A052089, A341701, A341702, A341715, A341716, A341717, A341932.

from sympy import isprime
def A341931(n):
    k, m, r = n, n-1, n if isprime(n) else -1
    while m > 0:
        k = int(str(k)+str(m))
        if isprime(k):
            r = m
        m -= 1
    return r

A341932 a(n) = largest k < n such that the decimal concatenation n||n-1||n-2||...||n-k is prime, or -1 if no such prime exists.

Original entry on oeis.org

-1, -1, 0, 0, 1, 0, -1, 4, -1, -1, 3, 0, -1, 0, -1, -1, -1, 0, -1, 0, -1, -1, 3, 0, 1, 12, -1, 4, -1, 0, -1, 0, -1, -1, 1, -1, -1, 0, -1, -1, -1, 0, 1, 0, -1, -1, 43, 0, 31, -1, -1, 4, 15, 12, -1, 28, 9, -1, 1, 0, 13, 0, -1, -1, -1, -1, -1, 0, 57, -1, 1, 0, -1

Offset: 0

Chai Wah Wu, Feb 23 2021

a(82) = 81, are there any other n such that a(n) = n-1?
Primes p such that a(p) > 0: 7, 53, 73, 79, 89, 103, ...
n such that a(n) > A341702(n): 7, 10, 22, 46, 48, 53, 55, 73, ...
Similar argument as in A341716 shows that if n > 3 and a(n) >= 0, then n-a(n) is odd, a(n) !== 2 (mod 3) and 2n-a(n) !== 0 (mod 3).

			a(22) = 3 since 22212019 is prime.

Cf. A052088, A052089, A341701, A341702, A341715, A341716, A341717, A341931.

from sympy import isprime
def A341932(n):
    k, m, r = n, n-1, 0 if isprime(n) else -1
    while m > 0:
        k = int(str(k)+str(m))
        if isprime(k):
            r = n-m
        m -= 1
    return r

a(n) = n-A341931(n) >= A341702(n).

A085720 Start of a run of 7 successive numbers which when concatenated form a prime.

Original entry on oeis.org

7, 37, 157, 185, 187, 271, 301, 355, 475, 485, 523, 533, 577, 611, 653, 661, 667, 731, 733, 755, 761, 791, 853, 911, 913, 937, 983, 1085, 1111, 1187, 1205, 1253, 1397, 1417, 1585, 1631, 1655, 1685, 1697, 1711, 1723, 1841, 1907, 1975, 2035, 2077, 2105, 2185

Offset: 1

Zak Seidov, Jun 27 2003

Concatenation of three and six successive numbers are always composite.

Primes as concatenation of two, four and five successive numbers are in A030458, A030471, A052087, A052088, A052089.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A030458, A030471, A052087, A052088, A052089.

f[n_] := FromDigits[ Flatten[ Table[ IntegerDigits[i], {i, n, n + 6}]]]; Select[ Range[2190], PrimeQ[ f[ # ]] & ]
Select[Range[2500],PrimeQ[FromDigits[Flatten[IntegerDigits/@Range[#,#+6]]]]&] (* Harvey P. Dale, Aug 15 2021 *)

Edited by Robert G. Wilson v, Jun 28 2003

Edited by Charles R Greathouse IV, Apr 24 2010

A104323 Primes which are the reverse concatenation of four consecutive numbers.

Original entry on oeis.org

10987, 22212019, 86858483, 94939291, 100999897, 106105104103, 112111110109, 122121120119, 124123122121, 146145144143, 184183182181, 226225224223, 232231230229, 244243242241, 274273272271, 332331330329, 362361360359

Offset: 1

Shyam Sunder Gupta, Apr 17 2005

			The first term is 10987 which is a prime and is the reverse concatenation of 7,8,9 and 10 which are four consecutive numbers.

Zak Seidov, Table of n, a(n) for n = 1..328

Cf. A052089.

Select[Table[FromDigits[Flatten[IntegerDigits/@Range[n,n-3,-1]]],{n,4,400}],PrimeQ] (* Harvey P. Dale, Aug 02 2021 *)

forstep(n=10,400,2,isprime(t=eval(Str(n,n-1,n-2,n-3)))&print1(t",")) \\ Zak Seidov, May 08 2013

A279213 Primes formed by concatenating n with n-3.

Original entry on oeis.org

41, 107, 1613, 2017, 3229, 4441, 4643, 5653, 7673, 9491, 106103, 116113, 124121, 130127, 136133, 170167, 172169, 182179, 184181, 196193, 206203, 212209, 214211, 220217, 224221, 230227, 272269, 274271, 280277, 302299, 304301, 320317, 322319, 326323, 334331

Offset: 1

Vincenzo Librandi, Dec 08 2016

			For n = 16, n-3 = 13. Concatenating 16 and 13 gives 1613 which is a prime. So, 1613 is in the sequence. - _Indranil Ghosh_, Jan 23 2017

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A052089, A104332.

[m: n in [4..400 by 2] | IsPrime(m) where m is Seqint(Intseq(n-3) cat Intseq(n))];

Select[Table[FromDigits[Join[Flatten[IntegerDigits[{n, n -3}]]]], {n, 400}], PrimeQ]

terms(n) = my(i=0, k=3); while(i < n, my(x=eval(Str(k, k-3))); if(ispseudoprime(x), print1(x, ", "); i++); k++)
/* Print initial 35 terms as follows: */
terms(35) \\ Felix Fröhlich, Jan 23 2017

from sympy import isprime
i=4
j=1
while j<=10000:
    if isprime(int(str(i)+str(i-3)))==True:
        print(str(j)+" "+str(i)+str(i-3))
        j+=1
    i+=1 # Indranil Ghosh, Jan 23 2017

cp's OEIS Frontend

A341702 a(n) is the smallest k < n such that the decimal concatenation n||n-1||n-2||...||n-k is prime, or -1 if no such prime exists.

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A084551 Primes which are a concatenation of five consecutive numbers.

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A210512 Primes formed by concatenating k, k and 3 for k >= 1.

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A210513 Primes formed by concatenating k, k, and 7.

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A210514 Prime numbers generated by concatenating k, k, and 9.

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A341931 a(n) = smallest m > 0 such that the decimal concatenation n||n-1||n-2||...||m is prime, or -1 if no such prime exists.

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A341932 a(n) = largest k < n such that the decimal concatenation n||n-1||n-2||...||n-k is prime, or -1 if no such prime exists.

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A085720 Start of a run of 7 successive numbers which when concatenated form a prime.