A052087 -id:A052087 - cp's OEIS frontend

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

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 *)

A286706 Primes in A077309.

Original entry on oeis.org

23, 4567, 78910111213, 25262728293031323334353637383940414243444546474849

Offset: 1

XU Pingya, May 12 2017

That is, primes formed from concatenation of k numbers starting with k.
Subsequence of A052087.
a(5) must be larger than A077309(2500).
A077309(k) is prime for k = {2, 4, 7, 25, ...}. - Michael De Vlieger, May 13 2017

			The prime 4567 is a term since it is the concatenation of 4 decimal numbers beginning with the number 4. - _Michael De Vlieger_, May 13 2017

Eric Weisstein's World of Mathematics, Consecutive Number Sequences

Cf. A077309, A052087, A035333, A053067.

Select[Table[FromDigits@ Flatten@ Take[#, {n, n + (n - 1)}], {n, Ceiling[Length[#]/2]}], PrimeQ] &@ Array[IntegerDigits, 10^3] (* Michael De Vlieger, May 13 2017 *)

A317744 Prime numbers which result as a concatenation of a decimal number and its binary representation.

Original entry on oeis.org

11, 311, 5101, 131101, 2511001, 37100101, 51110011, 59111011, 731001001, 931011101, 971100001, 1191110111, 12910000001, 13110000011, 13710001001, 15310011001, 17310101101, 19311000001, 21311010101, 21511010111, 24711110111, 25511111111, 319100111111

Offset: 1

Philip Mizzi, Aug 05 2018

The decimal number plus its Hamming weight A000120 must not be divisible by 3. - M. F. Hasler, Apr 05 2024

			11 is in the sequence because the binary representation of 1 is 1 and the concatenation of 1 and 1 gives 11, which is prime.
931011101 is in the sequence because it is the concatenation of 93 and 1011101 (the binary representation of 93) and is prime.

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

Cf. A000040, A030458, A052087, A052088, A052089, A127421, A236551, A287300, A287310.

Select[Table[FromDigits[Join[IntegerDigits[n],IntegerDigits[n,2]]],{n,400}],PrimeQ] (* Harvey P. Dale, Jul 15 2020 *)

nb(n)=fromdigits(concat(n,binary(n)))
A317744_upto(N=666)=[p|n<-[1..N], is/*pseudo*/prime(p=nb(n))] \\ M. F. Hasler, Apr 05 2024

from sympy import isprime
def nbinn(n): return int(str(n)+bin(n)[2:])
def ok(n): return isprime(nbinn(n))
def aprefixupto(p): return [nbinn(k) for k in range(1, p+1, 2) if ok(k)]
print(aprefixupto(319)) # Michael S. Branicky, Dec 27 2020

A375481 Starting numbers for the terms in A035333.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 2, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 3, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 4, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 5, 57, 58

Offset: 1

Nicholas M. R. Frieler, Aug 17 2024

			a(19) = 12 since A035333(19) = 1213, the concatenation of 12 and 13.
a(20) =  1 since A035333(20) = 1234, the concatenation of 1, 2, 3 and 4.

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

For the terms generated by consecutive concatenations see A035333.For concatenations of various numbers of consecutive integers see A000027 (k=1), A127421 (k=2), A001703 (k=3), A279204 (k=4).For primes that are the concatenation of two or more consecutive integers see A052087.

ConsecutiveNumber[num_, numberOfNumbers_] := Module[{},
  If[numberOfNumbers == 1, Return[num]];
  FromDigits@(IntegerDigits[num]~Join~
     IntegerDigits@ConsecutiveNumber[num + 1, numberOfNumbers - 1])
 ]
ConsecutiveNumberDigits[maxDigits_] := Module[{numList = {}, c, d},
  Do[
   numList =
     numList~Join~(Association[# -> ConsecutiveNumber[#, c]] & /@
        Range[Power[10, d - 1], Power[10, d] - 1]);,
   {d, 1, Floor[maxDigits/2]}, {c, 2, Floor[maxDigits/d]}
   ];
  SortBy[Select[numList, Values[#][[1]] < Power[10, maxDigits] &],
   Values[#] &]
 ]
Keys[ConsecutiveNumberDigits[8]]//Flatten (* number in this line corresponds to the maximum number of digits of the concatenated terms *)

import heapq
from itertools import islice
def agen(): # generator of terms
    c = 12
    h = [(c, 1, 2)]
    nextcount = 3
    while True:
        (v, s, l) = heapq.heappop(h)
        yield s
        if v >= c:
            c = int(str(c) + str(nextcount))
            heapq.heappush(h, (c, 1, nextcount))
            nextcount += 1
        l += 1; v = int(str(v)[len(str(s)):] + str(l)); s += 1
        heapq.heappush(h, (v, s, l))
print(list(islice(agen(), 70))) # Michael S. Branicky, Aug 18 2024

cp's OEIS Frontend

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

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

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A286706 Primes in A077309.

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A317744 Prime numbers which result as a concatenation of a decimal number and its binary representation.

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A375481 Starting numbers for the terms in A035333.

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