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
Examples
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.
Links
- Jon E. Schoenfield, Table of n, a(n) for n = 1..10000
- G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 138140141.
Programs
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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 *) -
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
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