cp's OEIS Frontend

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

%I A342049 #60 Mar 04 2021 03:20:22
%S A342049 89,5051,5657,6263,6869,8081,9091,9293,120121,186187,188189,200201,
%T A342049 216217,242243,246247,252253,278279,300301,308309,318319,338339,
%U A342049 342343,350351,362363,368369,390391,402403,410411,416417,426427,428429,440441,446447,450451,452453,470471,476477,482483
%N A342049 Primes formed by the concatenation of exactly two consecutive composite numbers.
%C A342049 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.
%C A342049 a(1) = 89 is also the smallest prime whose digits are composite (A051416).
%C A342049 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
%H A342049 G. L. Honaker, Jr. and Chris Caldwell, <a href="https://primes.utm.edu/curios/page.php?curio_id=646">Prime Curios! 89</a>.
%e A342049 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.
%o A342049 (PARI) isc(c) = (c>1) && ! isprime(c);
%o A342049 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
%o A342049 (PARI) 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
%o A342049 (PARI) 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
%o A342049 (Python)
%o A342049 from sympy import isprime
%o A342049 def agento(lim):
%o A342049   digs, pow10 = 1, 10
%o A342049   while True:
%o A342049     for c2 in range(max(pow10//10+1, 3), pow10, 2):
%o A342049       if not isprime(c2) and not isprime(c2-1):
%o A342049         c1c2 = (c2-1)*pow10+c2
%o A342049         if c1c2 > lim: return
%o A342049         if isprime(c1c2): yield c1c2
%o A342049     digs, pow10 = digs+1, pow10*10
%o A342049 print([an for an in agento(482483)]) # _Michael S. Branicky_, Feb 27 2021
%Y A342049 Cf. A002808, A087341.
%Y A342049 Subsequence of A030458 and A121608.
%K A342049 nonn,base
%O A342049 1,1
%A A342049 _Bernard Schott_, Feb 26 2021