A342049 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

LINKS

Table of n, a(n) for n=1..38.

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

EXAMPLE

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.

PROG

(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