A256481 -id:A256481 - cp's OEIS frontend

A256480 Smallest prime obtained by appending n to a nonzero number with identical digits or 0 if no such prime exists.

0, 11, 0, 13, 0, 0, 0, 17, 0, 19, 0, 211, 0, 113, 0, 0, 0, 317, 0, 419, 0, 421, 0, 223, 0, 0, 0, 127, 0, 229, 0, 131, 0, 233, 0, 0, 0, 137, 0, 139, 0, 241, 0, 443, 0, 0, 0, 347, 0, 149, 0, 151, 0, 353, 0, 0, 0, 157, 0, 359, 0, 461, 0, 163, 0, 0, 0, 167, 0, 269

Offset: 0

Chai Wah Wu, Mar 31 2015

a(n) = 0 if n is even or a multiple of 5. Conjecture: all other terms are nonzero. Conjecture verified for n <= 10^7.

"Appending" means "on the right".

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Chai Wah Wu, On a conjecture regarding primality of numbers constructed from prepending and appending identical digits, arXiv:1503.08883 [math.NT], 2015.

Cf. A090287, A256481.

from gmpy2 import digits, mpz, is_prime
def A256480(n,limit=2000):
    sn = str(n)
    if not (n % 2 and n % 5):
        return 0
    for i in range(1,limit+1):
        for j in range(1,10):
            si = digits(j,10)*i
            p = mpz(si+sn)
            if is_prime(p):
                return int(p)
    else:
        return 'search limit reached.'

A346979 Count of the prime decimal descendants of n.

Original entry on oeis.org

83, 63, 23, 22, 23, 11, 29, 23, 3, 4, 54, 1, 9, 14, 6, 7, 3, 4, 7, 40, 0, 4, 19, 15, 8, 7, 10, 14, 5, 6, 2, 7, 0, 16, 9, 11, 12, 13, 4, 1, 34, 1, 8, 14, 5, 1, 13, 5, 5, 16, 6, 0, 9, 0, 24, 4, 6, 19, 2, 9, 25, 16, 0, 7, 4, 4, 3, 11, 2, 7, 7, 4, 1, 15, 2, 8, 8

Offset: 0

Ya-Ping Lu, Aug 09 2021

The number of direct decimal descendants (i.e., decimal children) of n is A038800(n). The number of prime decimal descendants of the n-th prime is A214342(p_n). a(n) is the number of prime decimal descendants of n, which include the prime decimal children of n, the prime decimal children of the prime decimal children of n, and so on.
a(0) = Sum_{m=1..4} (A214342(m) + 1); a(1) = Sum_{m=5..8} (A214342(m) + 1).
a(A032352(m)) = 0; a(A119289(m)) = 0.
A214342 is a subset, as A214342(m) = a(prime(m)).
Conjecture 1: a(n) <= 83. Conjecture 2: lim_{n->oo} (n0/n) = 1, where n0 is the number of zero terms, a(k) = 0, for k <= n.

			a(4) = 23. The 23 prime decimal descendants of 4 are shown in the tree below.
       _____ 4__________________________
      /      |                          \
     41   ___43______________            47
    /    /   |               \             \
  419  431  433               439          479
            / \              /   \        /   \
        4337  4339         4391  4397   4793  4799
             /  |  \        |     |     /  \
        43391 43397 43399 43913 43973 47933 47939
                            |
                         439133
                            |
                        4391339

Cf. A030665, A032352, A038800, A119289, A122072, A214342, A218255, A256481.

Table[Length@Rest@Flatten[FixedPointList[(b=#;Select[Flatten[(a=#;FromDigits/@(Join[IntegerDigits@a,{#}]&/@If[b=={0},Range@9,{1,3,7,9}]))&/@b],PrimeQ])&,{n}]],{n,0,76}] (* Giorgos Kalogeropoulos, Aug 16 2021 *)

from sympy import isprime
def p_count(k):
    global ct; d = [2, 3, 5, 7] if k == 0 else [1, 3, 7, 9]
    for i in range(4):
        m = 10*k + d[i]
        if isprime(m): ct += 1; p_count(m)
    return ct
for n in range(100):
    ct = 0; print(p_count(n))

cp's OEIS Frontend

A256480 Smallest prime obtained by appending n to a nonzero number with identical digits or 0 if no such prime exists.

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