A030458 -id:A030458 - cp's OEIS frontend

A239992 Primes formed by concatenating a square number S with S+1, where S+1 is a prime number.

3637, 576577, 2433624337, 3240032401, 4410044101, 6969669697, 352836352837, 404496404497, 746496746497, 16641001664101, 17424001742401, 20050562005057, 20736002073601, 24523562452357, 26049962604997, 28022762802277, 34596003459601, 35494563549457, 42600964260097

Offset: 1

K. D. Bajpai, Mar 30 2014

It is observed that each term in the sequence ends either with digit 1 or 7.

			3637 is a prime formed by concatenaing 36 with 37 where 36= 6^2 and (36+1)= 37 is a prime.
576577 is a prime formed by concatenaing 576 with 577 where 576= 24^2 and (576+1)= 577 is a prime.

K. D. Bajpai, Table of n, a(n) for n = 1..1060

Cf. A000040, A030458.

with(StringTools):KD := proc() local a,b,s;s:=n*n;b:=s+1; if isprime(b) then a:= parse(cat(s,b )); if isprime(a) then RETURN (a); fi; fi;end: seq(KD(), n=1..5000);

A256176 Primes formed by concatenating n with n+1 and by concatenating n+2 with n+3.

Original entry on oeis.org

67, 89, 7879, 8081, 9091, 9293, 186187, 188189, 276277, 278279, 426427, 428429, 438439, 440441, 450451, 452453, 600601, 602603, 606607, 608609, 798799, 800801, 816817, 818819, 858859, 860861, 936937, 938939, 960961, 962963, 11401141, 11421143

Offset: 1

Bui Quang Tuan, Mar 18 2015

Subsequence of A030458.

First bisection: A156121.

			67, 89 are in the sequence because they are primes and 6, 7, 8, 9 are four consecutive integers.
7879, 8081 are in the sequence because they are primes and 78, 79, 80, 81 are four consecutive integers.
186187, 188189 are in the sequence because they are primes and 186, 187, 188, 189 are four consecutive integers.

Bui Quang Tuan, Table of n, a(n) for n = 1..100

f[n_] := FromDigits@ Flatten[IntegerDigits /@ Range[n, n + 1]]; {f@ #, f[# + 2]} & /@ Select[Range@ 1200, AllTrue[{f@ #, f[# + 2]}, PrimeQ] &] // Flatten (* Michael De Vlieger, Mar 18 2015 *)
fd[{a_,b_}]:=FromDigits[Join[IntegerDigits[a],IntegerDigits[b]]]; Select[ {fd[ Take[#,2]],fd[Take[#,-2]]}&/@Partition[Range[1500],4,1],AllTrue[ #,PrimeQ]&]//Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 17 2018 *)

lista(nn) = {for (n=1, nn, if (isprime(p=eval(concat(Str(n), Str(n+1)))) && isprime(q=eval(concat(Str(n+2), Str(n+3)))), print1(p, ", ", q, ", ")););} \\ Michel Marcus, Mar 18 2015

A287018 Primes that can be generated by the concatenation in base 2, in ascending order, of two consecutive integers read in base 10.

Original entry on oeis.org

11, 37, 137, 239, 661, 727, 859, 991, 2081, 2341, 2731, 2861, 3121, 3251, 3511, 9547, 10321, 10837, 11353, 13159, 13417, 13933, 14449, 15739, 34439, 40093, 43177, 43691, 45233, 46261, 60139, 61681, 63737, 135433, 138511, 139537, 144667, 146719, 151849, 154927

Offset: 1

Paolo P. Lava, May 18 2017

			2 and 3 in base 2 are 10 and 11 and concat(10,11) = 1011 is 11 in base 10.
4 and 5 in base 2 are 100 and 101 and concat(100,101) = 100101 is 37 in base 10.

Cf. A000040, A030458, A287019.

Subsequence of primes of A087737.

with(numtheory): P:= proc(q) local a,b,c,n; a:=convert(q+1,binary,decimal); b:=convert(q,binary,decimal); c:=convert(b*10^(ilog10(a)+1)+a,decimal,binary); if isprime(c) then c; fi; end: seq(P(i),i=1..1000);

Select[Map[FromDigits[Apply[Join, IntegerDigits[#, 2]], 2] &, Partition[Range@ 320, 2, 1]], PrimeQ] (* Michael De Vlieger, May 18 2017 *)

lista(nn) = {for (n=1, nn, if (isprime(p=fromdigits(Vec(concat(binary(n), binary(n+1))), 2)), print1(p, ", ")));} \\ Michel Marcus, May 20 2017

A287300 Primes that can be generated by the concatenation in base 3, in ascending order, of two consecutive integers read in base 10.

Original entry on oeis.org

5, 31, 41, 61, 71, 281, 337, 421, 449, 617, 673, 701, 2297, 2543, 2707, 2789, 2953, 3527, 3691, 4019, 5003, 5167, 5413, 5659, 5741, 5987, 6151, 6397, 21961, 22937, 23669, 24889, 25621, 26597, 27329, 27817, 28549, 28793, 30013, 31477, 31721, 32941, 34649, 35381

Offset: 1

Paolo P. Lava, May 23 2017

			1 and 2 in base 3 are 1 and 2 and concat(1,2) = 12 in base 10 is 5;
3 and 4 in base 3 are 10 and 11 and concat(10,11) = 1011 in base 10 is 31.

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,3),i=1..1000);

With[{b = 3}, Select[Map[FromDigits[Flatten@ IntegerDigits[#, b], b] &, Partition[Range@ 150, 2, 1]], PrimeQ]] (* Michael De Vlieger, May 23 2017 *)

A309851 Primes formed by concatenating n and 2n-1.

Original entry on oeis.org

11, 23, 47, 59, 1019, 1223, 1427, 1733, 2039, 2141, 2243, 2447, 2549, 2753, 2957, 3467, 3671, 4079, 4283, 4691, 4793, 5099, 52103, 55109, 61121, 65129, 70139, 75149, 77153, 82163, 86171, 95189, 102203, 104207, 112223, 119237, 124247, 130259, 132263, 137273, 145289, 146291, 147293, 149297, 150299, 160319

Offset: 1

A.H.M. Smeets, Aug 20 2019

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000040, A052089 (n and n-1), A030458 (n and n+1), A309808 (n and 2n+1).

[a:m in [1..170]|IsPrime(a) where a is 10^(#Intseq(2*m-1))*m+2*m-1]; // Marius A. Burtea, Oct 09 2019

f[n_] := FromDigits @ (Join @@ IntegerDigits[{n, 2n-1}]); Select[f /@ Range[160], PrimeQ]  (* Amiram Eldar, Sep 24 2019 *)

from sympy import isprime
A309851_list = [m for m in (int(str(n)+str(2*n-1)) for n in range(1,10**5)) if isprime(m)] # Chai Wah Wu, Oct 23 2019

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

A342049 Primes formed by the concatenation of exactly two consecutive composite numbers.

Original entry on oeis.org

89, 5051, 5657, 6263, 6869, 8081, 9091, 9293, 120121, 186187, 188189, 200201, 216217, 242243, 246247, 252253, 278279, 300301, 308309, 318319, 338339, 342343, 350351, 362363, 368369, 390391, 402403, 410411, 416417, 426427, 428429, 440441, 446447, 450451, 452453, 470471, 476477, 482483

Offset: 1

Bernard Schott, Feb 26 2021

When a prime is obtained by the concatenation of exactly two consecutive composite numbers, the first one always ends with 0, 2, 6, 8 while the second one ends respectively with 1, 3, 7, 9.
a(1) = 89 is also the smallest prime whose digits are composite (A051416).
a(n) has an even number of digits. If it would have an odd number of digits then it is like 99..99100..00 but that is composite. - David A. Corneth, Feb 27 2021

			If (2,q) is the smallest term formed by the concatenation of 2 consecutive composite numbers with each q digits: (2,1) = a(1) = 89, (2,2) = a(2) = 5051, (2,3) = a(9) = 120121, (2,4) = 10021003, (2,5) = 1001010011, (2,6) = 100010100011.

G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 89.

Cf. A002808, A087341.

Subsequence of A030458 and A121608.

isc(c) = (c>1) && ! isprime(c);
isok(p) = {if (isprime(p), my(d=digits(p)); for (i=1, #d-1, my(b = fromdigits(vector(i, k, d[k]))); if (d[i+1], my(c = fromdigits(vector(#d-i, k, d[k+i]))); if (isc(b) && isc(c) && ((primepi(c) - primepi(b)) == c-b-1), return (1)); ); ); ); } \\ Michel Marcus, Feb 27 2021

first(n) = { pc = 4; my(res = vector(n)); t = 0; forcomposite(c = 6, oo, nc = pc * 10^#digits(c) + c; if(isprime(nc), t++; res[t] = nc; if(t >= n, return(res) ) ); pc = c; ) } \\ David A. Corneth, Feb 27 2021

is(n) = { my(d = digits(n)); if(#d % 2 == 1, return(0) ); fc = fromdigits(vector(#d \ 2, i, d[i])); lc = fromdigits(vector(#d \ 2, i, d[i+#d\2])); lc - fc == 1 && !isprime(fc) && !isprime(lc) && nextprime(fc)==nextprime(lc) && isprime(n) } \\ David A. Corneth, Feb 27 2021

from sympy import isprime
def agento(lim):
  digs, pow10 = 1, 10
  while True:
    for c2 in range(max(pow10//10+1, 3), pow10, 2):
      if not isprime(c2) and not isprime(c2-1):
        c1c2 = (c2-1)*pow10+c2
        if c1c2 > lim: return
        if isprime(c1c2): yield c1c2
    digs, pow10 = digs+1, pow10*10
print([an for an in agento(482483)]) # Michael S. Branicky, Feb 27 2021

A372207 a(n) is the smallest prime number that is obtained by concatenating 2 consecutive n-digit integers.

Original entry on oeis.org

23, 1213, 102101, 10021001, 1000810007, 100010100011, 10000241000023, 1000004210000041, 100000002100000003, 10000000081000000009, 1000000000810000000007, 100000000002100000000001, 10000000000021000000000001, 1000000000001010000000000011, 100000000000002100000000000003

Offset: 1

Gonzalo Martínez, May 19 2024

It is possible to form primes by concatenating an integer with its successor (A030458), as well as by concatenating an integer with its predecessor (A052089).

			a(3) = 102101, since when concatenating 3-digit numbers, the first three numbers obtained, in increasing order, are 100101, 101100, 101102, which are composite, while the next number in the list is 102101, which is prime.

Michael S. Branicky, Table of n, a(n) for n = 1..500

Cf. A030458, A052089.

from sympy import isprime
def a(n):
    if n == 1: return 23
    lb, ub = 10**(n-1), 10**n
    for k in range(lb, ub, 2):
        base = k*ub
        for inc in [k-1, k+1]:
            if inc >= lb and isprime(t:=base+inc):
                return t
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, May 19 2024

a(12) and beyond from Michael S. Branicky, May 19 2024

A275238 a(n) = n*(10^floor(log_10(n)+1) + 1) + (-1)^n.

Original entry on oeis.org

1, 10, 23, 32, 45, 54, 67, 76, 89, 98, 1011, 1110, 1213, 1312, 1415, 1514, 1617, 1716, 1819, 1918, 2021, 2120, 2223, 2322, 2425, 2524, 2627, 2726, 2829, 2928, 3031, 3130, 3233, 3332, 3435, 3534, 3637, 3736, 3839, 3938, 4041, 4140, 4243, 4342, 4445, 4544, 4647, 4746, 4849, 4948, 5051, 5150, 5253, 5352, 5455, 5554

Offset: 0

Ilya Gutkovskiy, Jul 21 2016

Concatenation of n with n+(-1)^n (A004442).
Subsequence of A248378.
Primes in this sequence: 23, 67, 89, 1213, 3637, 4243, 5051, 5657, 6263, 6869, 7879, 8081, 9091, 9293, 9697, 102103, ... (A030458).
Numbers n such that a(n) is prime: 2, 6, 8, 12, 36, 42, 50, 56, 62, 68, 78, 80, 90, 92, 96, 102, 108, 120, 126, 138, ... (A030457).

			a(0) =  0 + 1 = 1;
a(1) = 11 - 1 = 10;
a(2) = 22 + 1 = 23;
a(3) = 33 - 1 = 32;
a(4) = 44 + 1 = 45;
a(5) = 55 - 1 = 54, etc.
or
a(0) =  1 -> concatenation of 0 with 0 + (-1)^0 = 1;
a(1) = 10 -> concatenation of 1 with 1 + (-1)^1 = 0;
a(2) = 23 -> concatenation of 2 with 2 + (-1)^2 = 3;
a(3) = 32 -> concatenation of 3 with 3 + (-1)^3 = 2;
a(4) = 45 -> concatenation of 4 with 4 + (-1)^4 = 5;
a(5) = 54 -> concatenation of 5 with 5 + (-1)^5 = 4, etc.
........................................................
a(2k) = 1, 23, 45, 67, 89, 1011, 1213, 1415, 1617, 1819, ...

Cf. A001704, A002285, A004442, A020338, A030457, A030458, A030656, A033999, A064834, A248378.

Table[n (10^Floor[Log[10, n] + 1] + 1) + (-1)^n, {n, 0, 55}]

a(n) = if(n, n*(10^(logint(n,10)+1) + 1) + (-1)^n, 1) \\ Charles R Greathouse IV, Jul 21 2016

a(n) = A020338(n) + A033999(n).
a(2k) = A030656(k).
A064834(a(n)) > 0, for n > 0.
a(n) ~ 10*n*10^floor(c*log(n)), where c = 1/log(10) = 0.4342944819... = A002285.

A287302 Primes that can be generated by the concatenation in base 4, in ascending order, of two consecutive integers read in base 10.

Original entry on oeis.org

11, 103, 137, 239, 1171, 1301, 1951, 2081, 2341, 2731, 2861, 3121, 3251, 3511, 16963, 17477, 20047, 21589, 23131, 26729, 30841, 34439, 40093, 43177, 43691, 45233, 46261, 60139, 61681, 63737, 270601, 272651, 278801, 291101, 295201, 297251, 315701, 321851, 325951

Offset: 1

Paolo P. Lava, May 23 2017

			2 and 3 in base 4 are 2 and 3 and concat(2,3) = 23 in base 10 is 11;
6 and 7 in base 4 are 12 and 13 and concat(12,13) = 1213 in base 10 is 103.

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,4),i=1..1000);

With[{b = 4}, Select[Map[FromDigits[Flatten@ IntegerDigits[#, b], b] &, Partition[Range@ 320, 2, 1]], PrimeQ]] (* Michael De Vlieger, May 23 2017 *)

cp's OEIS Frontend

A239992 Primes formed by concatenating a square number S with S+1, where S+1 is a prime number.

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A256176 Primes formed by concatenating n with n+1 and by concatenating n+2 with n+3.

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A287018 Primes that can be generated by the concatenation in base 2, in ascending order, of two consecutive integers read in base 10.

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A287300 Primes that can be generated by the concatenation in base 3, in ascending order, of two consecutive integers read in base 10.

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Maple

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A309851 Primes formed by concatenating n and 2n-1.

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

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A342049 Primes formed by the concatenation of exactly two consecutive composite numbers.

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A372207 a(n) is the smallest prime number that is obtained by concatenating 2 consecutive n-digit integers.

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