A342214 Primes formed by the concatenation of exactly three successive composite numbers.

138140141, 180182183, 240242243, 250252253, 330332333, 400402403, 408410411, 478480481, 546548549, 570572573, 600602603, 646648649, 660662663, 676678679, 768770771, 838840841, 876878879, 928930931, 940942943, 970972973, 996998999, 100810101011, 109610981099

Offset

1,1

Comments

The primes that are obtained by the concatenation of exactly three successive composite numbers are always of the form c||c+2||c+3, with c+1 prime and c+3 odd <> 5, hence c must necessary ends with 0, 6, 8 (see examples).

No such primes can be obtained with the two other possible configurations of 3 successive composite numbers: c||c+1||c+2 or c||c+1||c+3.

The number of digits in each term is a multiple of 3. If a term existed for which this were not true, then c would necessarily be of the form 10^k - 2 (A099150), but then c+1 = 10^k - 1 would not be prime.

Links

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 138140141.

Example

a(1) = 138140141 because 138, 140, 141 are 3 successive composite numbers, then concat(138, 140, 141) = 138140141 is prime and is the least prime with this property (see link Prime Curios!).
The smallest such primes whose first composite ends respectively with 0, 6, 8 are: a(2) = 180182183, a(9) = 546548549, a(1) = 138140141.
If (3,q) is the smallest term formed by the concatenation of 3 successive composite numbers with each q digits: (3,3) = a(1) = 138140141, (3,4) = a(22) = 100810101011.

Mathematica

nextc[n_] := Module[{k = n + 1}, While[PrimeQ[k], k++]; k]; seq = {}; n1 = 4; n2 = nextc[n1]; Do[n3 = nextc[n2]; c = FromDigits @ Flatten @ Join[IntegerDigits /@ {n1, n2, n3}]; If[PrimeQ[c], AppendTo[seq, c]]; n1 = n2; n2 = n3, {1000}]; seq (* Amiram Eldar, Mar 05 2021 *)

Prog

(PARI) lista(nn) = {my(ca=4, cb=6); forcomposite(c=7, nn, if (isprime(x=eval(concat(Str(ca), concat(Str(cb), Str(c))))), print1(x, ", ")); ca = cb; cb = c; ); } \\ Michel Marcus, Mar 05 2021
(Python)
from sympy import isprime
def aupto(limit):
  c, t, alst = 6, 689, []
  while t < limit:
    t = int("".join(map(str, [c, c+2, c+3])))
    if isprime(c+1) and not isprime(c+3) and isprime(t): alst.append(t)
    c += [6, 4, 2, 2, 2][(c%10)//2]
  return alst
print(aupto(109610981099)) # Michael S. Branicky, Mar 05 2021

Crossrefs

Cf. A087341, A281684, A342049 (similar, with 2 consecutive composites).

Sequence in context: A105297 A298729 A203505 * A251505 A034642 A109093

Adjacent sequences: A342211 A342212 A342213 * A342215 A342216 A342217

Keyword

nonn,base

Author

Bernard Schott, Mar 05 2021

Status

approved