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

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

Author

Bernard Schott, Feb 26 2021

Keywords

• nonn
• base

Comments

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A002808, A087341.

Subsequence of A030458 and A121608.

Programs

PARI

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

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

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

Python

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