A344937 - OEIS

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A344937

a(n) is the largest k such that when strings of zeros of lengths t = 1..k are inserted between every pair of adjacent digits of prime(n), the resulting numbers are all primes.

1, 1, 1, 3, 0, 0, 0, 1, 2, 0, 0, 2, 2, 1, 2, 4, 0, 1, 0, 2, 4, 0, 0, 1, 1, 2, 0, 3, 0, 0, 0, 1, 0, 0, 2, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0

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

OFFSET

5,4

COMMENTS

Initially, except for n = 1..4, similar to A290174, but the two sequences differ from n = 28 onwards.

LINKS

Table of n, a(n) for n=5..91.

EXAMPLE

For n = 8: prime(8) = 19 and the numbers 109, 1009 and 10009 are all prime, while 100009 is not. Thus it is possible to insert strings of zeros of lengths 1, 2 and 3 between all adjacent digits of 19 such that the resulting number is prime. Since 3 is the largest length of such a string in case of 19, a(8) = 3.

MATHEMATICA

Table[k=0; While[PrimeQ@FromDigits@Flatten@Riffle[IntegerDigits@Prime@n, {Table[0, k]}], k++]; k-1, {n, 5, 100}] (* Giorgos Kalogeropoulos, Jun 03 2021 *)

PROG

(PARI) eva(n) = subst(Pol(n), x, 10)

insert_zeros(num, len) = my(d=digits(num), v=[]); for(k=1, #d-1, v=concat(v, concat([d[k]], vector(len)))); v=concat(v, d[#d]); eva(v)

a(n) = my(p=prime(n), ip=p); for(k=1, oo, ip=insert_zeros(p, k); if(!ispseudoprime(ip), return(k-1)))

(Python)

from sympy import isprime, prime

def insert_zeros(n, k): return int(("0"*k).join(list(str(n))))

def a(n):

pn, k = prime(n), 1

while isprime(insert_zeros(pn, k)): k += 1

return k - 1

print([a(n) for n in range(5, 92)]) # Michael S. Branicky, Jun 03 2021