A030458 -id:A030458 - cp's OEIS frontend

A084551 Primes which are a concatenation of five consecutive numbers.

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

A236551 Primes formed from concatenation of PrimePi(n) and prime(n).

Original entry on oeis.org

2, 13, 311, 313, 419, 641, 643, 647, 653, 761, 983, 997, 9103, 11131, 11149, 12157, 12163, 14197, 15227, 15233, 18307, 18311, 18313, 20353, 20359, 21379, 21383, 21397, 22409, 23431, 24499, 25523, 25541, 26557, 29599, 30631, 30643, 30661, 30677, 31727, 33773

Offset: 1

K. D. Bajpai, Jan 28 2014

			pi(6) = 3: prime(6) = 13. Concatenation of 3 and 13 gives 313 which is prime and appears in the sequence.
pi(8) = 4: prime(6) = 19. Concatenation of 4 and 19 gives 419 which is prime and appears in the sequence.

K. D. Bajpai, Table of n, a(n) for n = 1..1461

Cf. A030458 (primes: concatenation of n and n+1), A084667 (primes: concatenation of n and prime(n)), A084669 (primes: concatenation of prime(n) and n).

with(StringTools): with(numtheory): KD := proc() local a,b,d; a:=pi(n); b:=ithprime(n); d:=parse(cat(a,b));  if isprime (d) then RETURN (d); fi;  end: seq(KD(), n=1..300);

Select[Table[FromDigits[Flatten[{IntegerDigits[PrimePi[n]], IntegerDigits[Prime[n]]}]], {n, 100}], PrimeQ] (* Alonso del Arte, Jan 28 2014 *)

A287244 Indices n for which concatenation of the numbers from n to n + 9 is prime.

Original entry on oeis.org

4, 20, 68, 88, 122, 140, 188, 200, 212, 220, 224, 244, 254, 374, 460, 490, 502, 518, 550, 568, 632, 688, 734, 814, 832, 884, 898, 908, 928, 982, 1022, 1108, 1472, 1490, 1498, 1514, 1544, 1580, 1894, 1954, 2074, 2230, 2294, 2384, 2492, 2524, 2722

Offset: 1

XU Pingya, May 22 2017

			a(1) = 4, since 45678910111213 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.
Eric Weisstein's World of Mathematics, Smarandache Sequences

Cf. A030458.

filter:= proc(n) isprime(parse(cat($n .. n+9))) end proc:
select(filter, 2*[$1..10^4]); # Robert Israel, Jan 08 2024

Position[Map[FromDigits@ Flatten@ IntegerDigits@ # &, Partition[Range@ 2800, 10, 1]], p_ /; PrimeQ@ p][[All, 1]] (* Michael De Vlieger, May 22 2017 *)

A287310 Primes that can be generated by the concatenation in base 8, in ascending order, of two consecutive integers read in base 10.

Original entry on oeis.org

19, 37, 521, 911, 1171, 1301, 1951, 2081, 2341, 2731, 2861, 3121, 3251, 3511, 32833, 35911, 37963, 43093, 44119, 46171, 53353, 56431, 57457, 59509, 61561, 68743, 71821, 77977, 85159, 87211, 88237, 90289, 95419, 99523, 100549, 114913, 117991, 123121, 124147, 126199

Offset: 1

Paolo P. Lava, May 24 2017

Primes of the form (1+8^k) m + 1 where m+1 < 8^k < 8(m+1). - Robert Israel, May 24 2017

			2 and 3 in base 8 are 2_8 and 3_8, and concat(2,3) = 23_8 in base 10 is 19;
8 and 9 in base 8 are 10_8 and 11_8 and concat(10,11) = 1011_8 in base 10 is 521.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A030458.

with(numtheory): P:= proc(q,h) local a,b,c,d,k,n; a:=convert(q+1,base,h); b:=convert(q,base,h); c:=[op(a),op(b)]; d:=0; for k from nops(c) by -1 to 1 do d:=h*d+c[k]; od; if isprime(d) then d; fi; end: seq(P(i,8),i=1..1000);

With[{b = 8}, Select[Map[FromDigits[Flatten@ IntegerDigits[#, b], b] &, Partition[Range@ 250, 2, 1]], PrimeQ]] (* Michael De Vlieger, May 25 2017 *)

A032663 Primes that are the smallest decimal concatenations of n with n + (0,1,2,3,...).

Original entry on oeis.org

11, 23, 13, 47, 37, 1217, 17, 29, 19, 211, 313, 617, 113, 619, 317, 419, 521, 421, 523, 827, 727, 223, 2143, 1033, 1741, 227, 127, 229, 331, 433, 131, 233, 739, 1447, 337, 439, 137, 239, 139, 241, 1151, 647, 547, 1861, 347, 449, 349, 653, 149, 251, 151, 859

Offset: 1

Patrick De Geest, May 15 1998

First terms of sequences '11', A030458, A032625-A032632,continued with displacements d > 9.

			6th term is prime 1217: 17 - 12 = displacement 5 (=6-1).

Cf. A032662.

Edited by Charles R Greathouse IV, Apr 30 2010

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

A134428 Primes formed by concatenating k with k+1, where k+1 is a prime.

Original entry on oeis.org

23, 67, 1213, 3637, 4243, 7879, 9697, 102103, 108109, 126127, 138139, 150151, 156157, 180181, 192193, 270271, 276277, 312313, 330331, 378379, 420421, 432433, 438439, 456457, 522523, 540541, 546547, 576577, 600601, 606607, 612613, 618619

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jan 18 2008

Subset of A030458 and A052087.

Paolo P. Lava, Table of n, a(n) for n = 1..2500

Cf. A030458, A052087.

P:=proc(n) local a;
a:=(ithprime(n)-1)*10^(ilog10(ithprime(n))+1)+ithprime(n);
if isprime(a) then a; fi; end: seq(P(i),i=1..10^3);

Select[Table[(p-1)*10^IntegerLength[p]+p,{p,Prime[Range[200]]}],PrimeQ] (* Harvey P. Dale, Aug 23 2019 *)

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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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A236551 Primes formed from concatenation of PrimePi(n) and prime(n).

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Maple

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A287244 Indices n for which concatenation of the numbers from n to n + 9 is prime.

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Maple

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A287310 Primes that can be generated by the concatenation in base 8, in ascending order, of two consecutive integers read in base 10.

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Maple

Mathematica

A032663 Primes that are the smallest decimal concatenations of n with n + (0,1,2,3,...).

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