A132903 -id:A132903 - cp's OEIS frontend

A030469 Primes which are concatenations of three consecutive primes.

5711, 111317, 171923, 313741, 414347, 8997101, 229233239, 239241251, 263269271, 307311313, 313317331, 317331337, 353359367, 359367373, 383389397, 389397401, 401409419, 409419421, 439443449, 449457461

Offset: 1

Patrick De Geest

a(n) = "p(k) p(k+1) p(k+2)" where p(k) is k-th prime

It is conjectured that sequence is infinite. - from Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 09 2009

			(1) 5=p(3), 7=p(4), 11=p(5) gives a(1).
(2) 7=p(4), 11=p(5), 13=p(6), but 71113 = 7 x 10159

Richard E. Crandall, Carl Pomerance: Prime Numbers, Springer 2005 - from Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 09 2009
John Derbyshire: Prime obsession, Joseph Henry Press, Washington, DC 2003 - from Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 09 2009
Marcus du Sautoy: Die Musik der Primzahlen. Auf den Spuren des groessten Raetsels der Mathematik, Beck, Muenchen 2004

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A030461, A167517, A132903, A068655, A030997, A030473, A086041, A099727.

Select[Table[FromDigits[Flatten[IntegerDigits/@{Prime[n],Prime[n+1],Prime[n+2]}]],{n,11000}],PrimeQ] (* Zak Seidov, Oct 16 2009 *)
concat[{a_,b_,c_}]:=FromDigits[Flatten[IntegerDigits/@{a,b,c}]]; Select[ concat/@ Partition[ Prime[ Range[200]],3,1],PrimeQ] (* Harvey P. Dale, Sep 06 2017 *)

for(i=1,999, isprime(p=eval(Str(prime(i),prime(i+1),prime(i+2)))) & print1(p," ")) \\ M. F. Hasler, Nov 10 2009

A132903 INTERSECT A000040. - R. J. Mathar, Nov 11 2009

A383114 Triprimes that are concatenations of three consecutive primes.

Original entry on oeis.org

357, 131719, 434753, 475359, 717379, 97101103, 101103107, 103107109, 109113127, 139149151, 151157163, 157163167, 163167173, 167173179, 173179181, 193197199, 197199211, 199211223, 227229233, 241251257, 281283293, 293307311, 331337347, 347349353, 367373379, 397401409, 461463467, 487491499

Offset: 1

Will Gosnell and Robert Israel, May 23 2025

Numbers that are the concatenation (in increasing order) of three consecutive primes and are the product of three primes, counted with multiplicity.

			a(3) = 434753 is a term because it is the concatenation of the consecutive primes 43, 47 and 53 and 434753 = 11^2 * 3593 is the product of three primes, counted with multiplicity.

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

Intersection of A014612 and A132903.

cat3:= proc(a,b,c)
  (a*10^(1+ilog10(b))+b)*10^(1+ilog10(c))+c
end proc:
select(t ->numtheory:-bigomega(t)=3, [seq(cat3(ithprime(i),ithprime(i+1),ithprime(i+2)),i=1..100)]);

p3[p_]:=FromDigits[Join[IntegerDigits[Prime[p]],IntegerDigits[Prime[p+1]],IntegerDigits[Prime[p+2]]]];Select[Array[p3,100],PrimeOmega[#]==3&] (* James C. McMahon, Jun 09 2025 *)

A167517 Emirps (A007500) which are concatenation of three consecutive primes (A030469).

Original entry on oeis.org

353359367, 193319491951, 334733593361, 344934573461, 346734693491, 732173317333, 902990419043, 104591046310477, 133091331313327, 141591417314177, 146571466914683, 150131501715031, 154431545115461

Offset: 1

Jonathan vos Post and M. F. Hasler, Nov 10 2009

A subsequence of A007500, A030469, A132903.

for(i=1,9999, isprime(eval(p=Str(prime(i),prime(i+1),prime(i+2)))) & isprime(eval(concat(vecextract(Vec(p),"-1..1"))))& print1(p,", "))

A167517 = A007500 n A030469 = A007500 n A132903 (where "n" means intersection).

A244007 Semiprimes which are concatenation of three consecutive primes.

Original entry on oeis.org

235, 71113, 192329, 232931, 293137, 535961, 616771, 677173, 737983, 798389, 838997, 107109113, 137139149, 149151157, 181191193, 191193197, 211223227, 223227229, 233239241, 257263269, 269271277, 277281283, 337347349, 349353359, 373379383, 421431433, 431433439

Offset: 1

K. D. Bajpai, Jun 17 2014

The semiprimes in A132903.

			235 is in the sequence because concatenation of [2, 3, 5] = 235 = 5 * 47, which is semiprime.
71113 is in the sequence because concatenation of [7, 11, 13] = 71113 = 7 * 10159, which is semiprime.
111317 is not in the sequence because, though 111317 is concatenation of three consecutive primes [11, 13, 17], but it is not semiprime.

Cf. A000040, A001358, A030469, A132903, A244057.

with(numtheory): with(StringTools): A244007:= proc() local a,b,c,k,m; a:=ithprime(n); b:=ithprime(n+1); c:=ithprime(n+2);m:=parse(cat(a,b,c)); k:=bigomega(m); if (k)=2 then RETURN (m); fi; end: seq(A244007 (), n=1..100);

A244007 = {}; Do[t = FromDigits[Flatten[IntegerDigits /@ {Prime[n], Prime[n + 1], Prime[n + 2]}]]; If  [PrimeOmega[t] == 2,  AppendTo[A244007, t]], {n, 100}]; A244007

A244163 Primes which are the concatenation of three consecutive primes p, q, r while the sum (p + q + r) yields another prime.

Original entry on oeis.org

5711, 111317, 171923, 313741, 414347, 229233239, 389397401, 401409419, 409419421, 449457461, 701709719, 773787797, 787797809, 797809811, 140914231427, 157915831597, 163716571663, 202920392053, 212921312137, 252125312539, 259125932609, 263326472657, 268926932699

Offset: 1

K. D. Bajpai, Jun 21 2014

Subsequence of A030469.

The first five terms of this sequence resemble exactly those of A030469.

			5711 is in the sequence because the concatenation of [5, 7, 11] = 5711 which is prime. The sum [5 + 7 + 11] = 23 is another prime.
111317 is in the sequence because the concatenation of [11, 13, 17] = 111317 which is prime. The sum [11 + 13 + 17] = 41 is another prime.

K. D. Bajpai, Table of n, a(n) for n = 1..2250

Cf. A000040, A030469, A034961, A034962, A132903.

A244163:= proc() local a,b,c,k,m; a:=ithprime(n); b:=ithprime(n+1); c:=ithprime(n+2); m:=a+b+c; k:=parse(cat(a,b,c)); if isprime(k) and isprime(m) then RETURN (k); fi; end: seq(A244163 (), n=1..500);

prQ[{a_,b_,c_}]:=Module[{p=FromDigits[Flatten[IntegerDigits/@ {a,b,c}]]}, If[ AllTrue[ {p,a+b+c},PrimeQ],p,Nothing]]; prQ/@Partition[ Prime[ Range[ 500]],3,1] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 05 2021 *)

A244186 Primes which are the concatenation of five consecutive primes p, q, r, s, t while the sum (p + q + r + s + t) is another prime.

Original entry on oeis.org

711131719, 5359616771, 6771737983, 149151157163167, 401409419421431, 479487491499503, 757761769773787, 14091423142714291433, 18111823183118471861, 21132129213121372141, 26892693269927072711, 27192729273127412749, 36133617362336313637, 37613767376937793793

Offset: 1

K. D. Bajpai, Jun 21 2014

Subsequence of A086041.

Numbers: Concatenation of 5 consecutive primes at A132905.

			711131719 is in the sequence because the concatenation of [7, 11, 13, 17, 19] = 711131719 which is prime. The sum [7 + 11 + 13 + 17 + 19] = 67 is another prime.
5359616771 is in the sequence because the concatenation of [53, 59, 61, 67, 71] = 5359616771 which is prime. The sum [53 + 59 + 61 + 67 + 71] = 311 is another prime.

K. D. Bajpai, Table of n, a(n) for n = 1..1703

Cf. A000040, A086041, A034961, A034962, A132903, A244163.

FromDigits[Flatten[IntegerDigits/@#]]&/@Select[Partition[Prime[Range[ 1000]],5,1],AllTrue[{Total[#],FromDigits[Flatten[ IntegerDigits/@ #]]}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 24 2014 *)

A385536 Integers k formed by the concatenation of 3 consecutive prime numbers such that k is divisible by at least one of those 3 primes.

Original entry on oeis.org

235, 357, 71113, 374143, 616771, 167173179, 143914471451, 909071909089909091, 104651162787110465116278991046511627907

Offset: 1

Gonzalo Martínez, Jul 02 2025

Subsequence of A132903.
If p is A255669, as it divides the concatenation of the next two primes, then p divides the concatenation of p with the next two primes. Thus, the first 4 terms of A255669 give rise to a(2), a(3), a(5) and a(6). In this sequence the number formed by concatenating 3 consecutive primes is allowed to be divisible by at least one of those 3, which generates more possibilities.
a(9) has p > 10^11 and thus >= 36 digits. - Michael S. Branicky, Jul 02 2025
a(9) has p <=  A258182(11) - 36 = 1046511627871.
a(10) has p <= A258182(22) - 226.

			235 is a term, since it is the concatenation of the consecutive primes 2, 3, 5 and is divisible by 5.
2, 3 and 5: 235 = 5*47
3, 5, and 7: 357 = 3*119 = 7*51
7, 11 and 13: 71113 = 7*10159
37, 41 and 43: 374143 = 43*8701
61, 67 and 71: 616771 = 61*10111
167, 173 and 179: 167173179 = 167*1001037
1439, 1447 and 1451: 143914471451 = 1447*99457133
909071, 909089 and 909091: 909071909089909091 = 909091*999979000001

Cf. A132903, A255669, A258182.

tcat:= proc(a,b,c)
   c + (b + 10^(1+ilog10(b))*a)*10^(1+ilog10(c))
end proc:
R:= NULL: count:= 0:
q:= 2: r:= 3:
while count < 8 do
  p:= q; q:= r; r:= nextprime(r);
  x:= tcat(p,q,r);
  if igcd(x,p*q*r)>1 then
    R:= R,x; count:= count+1;
  fi
od:
R; # Robert Israel, Jul 06 2025

cat[s_] := FromDigits[Flatten[IntegerDigits[s]]]; q[s_] := AnyTrue[s, Divisible[cat[s], #] &]; cat /@ Select[Partition[Prime[Range[72000]], 3, 1], q] (* Amiram Eldar, Jul 03 2025 *)

a(9) from Michael S. Branicky, Jul 03 2025

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A030469 Primes which are concatenations of three consecutive primes.

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A383114 Triprimes that are concatenations of three consecutive primes.

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A167517 Emirps (A007500) which are concatenation of three consecutive primes (A030469).

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A244007 Semiprimes which are concatenation of three consecutive primes.

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A244163 Primes which are the concatenation of three consecutive primes p, q, r while the sum (p + q + r) yields another prime.

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A244186 Primes which are the concatenation of five consecutive primes p, q, r, s, t while the sum (p + q + r + s + t) is another prime.

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A385536 Integers k formed by the concatenation of 3 consecutive prime numbers such that k is divisible by at least one of those 3 primes.

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