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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.
2
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
OFFSET
5,4
COMMENTS
Initially, except for n = 1..4, similar to A290174, but the two sequences differ from n = 28 onwards.
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
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Felix Fröhlich, Jun 03 2021
STATUS
approved