A356273 a(n) is the position of the least prime in the ordered set of numbers obtained by inserting/placing any digit anywhere in the digits of n (except a zero before 1st digit), or 0 if there is no prime in that set.

2, 5, 1, 5, 8, 7, 1, 11, 1, 2, 1, 10, 1, 14, 7, 10, 1, 10, 1, 0, 4, 7, 4, 7, 8, 11, 1, 11, 4, 10, 1, 0, 2, 14, 11, 16, 1, 14, 1, 5, 2, 7, 8, 11, 16, 11, 3, 19, 1, 8, 1, 8, 3, 10, 17, 14, 1, 20, 3, 7, 4, 0, 1, 11, 14, 13, 1, 17, 2, 8, 2, 16, 1, 14, 13, 14, 2, 22, 1, 17

Offset

1,1

Comments

It appears that a(n) = 0 for n in A124665.

891, a term of A124665 and with a(891) = 9, is the first counterexample. - Michael S. Branicky, Aug 01 2022

Links

Michel Marcus, Table of n, a(n) for n = 1..10000

Example

For n=1, the resulting set is (10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 31, 41, 51, 61, 71, 81, 91) where the least prime 11 is at position 2, so a(1) = 2.
For n=2, the resulting set is (12, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 32, 42, 52, 62, 72, 82, 92) where the least prime 23 is at position 5, so a(2) = 5.

Mathematica

Table[Function[w, If[IntegerQ[#], #, 0] &@ FirstPosition[Rest@ Union@ Flatten@ Table[FromDigits@ Insert[w, d, k], {k, Length[w] + 1, 1, -1}, {d, 0, 9}], _?PrimeQ][[1]]][IntegerDigits[n]], {n, 80}] (* Michael De Vlieger, Aug 01 2022 *)

Prog

(PARI) get(d, rd, n, k) = {if (n==0, return(fromdigits(concat(d, k)))); if (n==#d, return(fromdigits(concat(k, d)))); my(v = concat(Vec(d, #d-n), k)); v = concat(v, Vecrev(Vec(rd, n))); fromdigits(v); }
a(n) = {my(d=digits(n), rd = Vecrev(d), list = List(), p); for (n=0, #d, my(kstart = if (n==#d, 1, 0)); for (k=kstart, 9, listput(list, get(d, rd, n, k)); ); ); my(svec = Set(Vec(list))); for (k=1, #svec, if (isprime(svec[k]), return(k)); ); }
(Python)
from sympy import isprime
def a(n):
    s = str(n)
    out = set(s[:i]+c+s[i:] for i in range(len(s)+1) for c in "0123456789")
    out = sorted(int(k) for k in out if k[0] != "0")
    ptest = (i for i, k in enumerate(sorted(out), 1) if isprime(int(k)))
    return next(ptest, 0)
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Aug 01 2022

Crossrefs

Related to the process in A068166, A068167, A068169, A068170, A068171, A068172, A068173, and A068174.

Cf. A124665.

Sequence in context: A289848 A258020 A021803 * A077382 A046527 A008343

Adjacent sequences: A356270 A356271 A356272 * A356274 A356275 A356276

Keyword

nonn,base

Author

Michel Marcus, Aug 01 2022

Status

approved