A068173 -id:A068173 - cp's OEIS frontend

A068174 Define an increasing sequence as follows. Start with an initial term, the seed (which need not have the property of the sequence); subsequent terms are obtained by inserting/placing at least one digit in the previous term to obtain the smallest number with the given property. This is the prime sequence with the seed a(1) = 9.

9, 19, 109, 1009, 10009, 100019, 1000159, 10001569, 100001569, 1000015069, 10000135069, 100001350649, 1000013500649, 10000130500649, 100001303500649, 1000013032500649, 10000103032500649, 100001030325003649, 1000010130325003649, 10000101303250036493

Offset: 1

Amarnath Murthy, Feb 25 2002

			The primes obtained by inserting/placing a digit in a(2) = 19 are 109, 139, 149, 179, 199 etc. and a(3) = 109 is the smallest.

Alois P. Heinz, Table of n, a(n) for n = 1..300

Cf. A068166, A068167, A068169, A068170, A068171, A068172, A068173.

f[n_] := Block[{b = PadLeft[ IntegerDigits[n], Floor[ Log[10, n] + 1]], k = 0}, While[ !PrimeQ[ FromDigits[ Insert[b, k, -2]]], k++ ]; FromDigits[ Insert[b, k, -2]]]; NestList[ f, 9, 18]

Edited by N. J. A. Sloane and Robert G. Wilson v, May 08 2002

Corrected and extended by Robert Gerbicz, Sep 06 2002

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.

Original entry on oeis.org

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

Michel Marcus, Aug 01 2022

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

			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.

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

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

Cf. A124665.

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 *)

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)););}

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

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