A068166 -id:A068166 - cp's OEIS frontend

A068167 Define an increasing sequence as follows. Given the first term, called the seed (which need not share the property of the remaining terms), subsequent terms are obtained by inserting at least one digit in the previous term so as to obtain the smallest number with the specified property. This is the prime sequence with the seed a(1) = 2.

2, 23, 223, 1223, 10223, 102023, 1020023, 10200263, 102002603, 1020026303, 10200226303, 102002263031, 1020002263031, 10200022363031, 102000223263031, 1020000223263031, 10200002232630131, 102000022326301313, 1020000222326301313, 10200002223236301313

Offset: 1

Amarnath Murthy, Feb 25 2002

			The primes that can be obtained by inserting/placing a digit in a(2) = 23 are 223, 233, 239, 263, 283, 293, etc. a(3) = 223 is the smallest.

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

Cf. A068166, A030456.

a:= proc(n) option remember; local s, w, m;
      if n=1 then 2
    else w:=a(n-1); s:=""||w; m:=length(s);
         min(select(x->length(x)=m+1 and isprime(x),
         {seq(seq(parse(cat(seq(s[h], h=1..i), j,
         seq(s[h], h=i+1..m))), j=0..9), i=0..m)})[])
      fi
    end:
seq(a(n), n=1..23);  # Alois P. Heinz, Nov 07 2014

Corrected and extended by Robert Gerbicz, Sep 06 2002

A068169 Define an increasing sequence as follows. Given the first term called the seed (the seed need not have the property of the sequence.). Subsequent terms are defined as obtained by inserting/placing digits (at least one) in the previous term to obtain the smallest number with a given property. This is the growing prime sequence for the seed a(1) = 4.

Original entry on oeis.org

4, 41, 241, 2141, 21341, 213461, 2123461, 21123461, 211234561, 2112343561, 21123043561, 211230043561, 2112030043561, 21112030043561, 211120030043561, 2110120030043561, 21101020030043561, 211010200230043561, 2110102002300430561, 21010102002300430561

Offset: 1

Amarnath Murthy, Feb 25 2002

			The primes obtained by inserting/placing a digit in a(2) = 41 are 241, 419, 421 etc... a(3)= 241 is the smallest.

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

Cf. A068166, A068167, A068169.

Corrected and extended by Robert Gerbicz, Sep 06 2002

A068170 Define an increasing sequence as follows. Given the first term, called the seed (the seed need not have the property of the remaining terms of the sequence), subsequent terms are defined as obtained by inserting/placing digits (at least one) in the previous term to obtain the smallest number with a given property. This is the growing prime sequence for the seed a(1) = 5.

Original entry on oeis.org

5, 53, 353, 3253, 30253, 130253, 1300253, 10300253, 100300253, 1003002053, 10030020503, 100300200503, 1003002050503, 10013002050503, 100130002050503, 1001300002050503, 10013000020503503, 100013000020503503, 1000130000205035083, 10001300002015035083

Offset: 1

Amarnath Murthy, Feb 25 2002

			The primes obtained by inserting/placing a digit in a(2) = 53 are 353, 523, ...; a(3) = 353 is the smallest.

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

Cf. A068166, A068167, A068169.

Corrected and extended by Robert Gerbicz, Sep 06 2002

A068171 Define an increasing sequence as follows: Given the first term called the seed (the seed need not have the property of the sequence.), subsequent terms are defined as obtained by inserting/placing digits (at least one) in the previous term to obtain the smallest number with a given property. This is the growing prime sequence for the seed a(1) = 6.

Original entry on oeis.org

6, 61, 461, 3461, 33461, 332461, 3132461, 31320461, 313204061, 3130204061, 23130204061, 231302004061, 2131302004061, 21313020024061, 213130200240161, 2131230200240161, 12131230200240161, 121312302002401613, 1210312302002401613, 12103123020020401613, 121031230200203401613, 1210312300200203401613

Offset: 1

Amarnath Murthy, Feb 25 2002

			The primes obtained by inserting/placing a digit in a(2) = 61 are 461, 619, 641, etc...a(3) = 461 is the smallest.

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

Cf. A068166, A068167, A068169, A068170.

More terms from Robert Gerbicz, Sep 06 2002

Definition edited by Harvey P. Dale, Feb 28 2023

A068172 Define an increasing sequence as follows. Given the first term called the seed (the seed need not have the property of the sequence.). Subsequent terms are defined as obtained by inserting/placing digits (at least one) in the previous term to obtain the smallest number with a given property. This is the growing prime sequence for the seed a(1) = 7.

Original entry on oeis.org

7, 17, 107, 1087, 10487, 104087, 1024087, 10024087, 100024087, 1000124087, 10001240087, 100012400837, 1000124008327, 10000124008327, 100001124008327, 1000011224008327, 10000110224008327, 100001100224008327, 1000010100224008327, 10000101002240083271, 100001010022400283271, 1000010100221400283271

Offset: 1

Amarnath Murthy, Feb 25 2002

			The primes obtained by inserting/placing a digit in a(2) = 17 are 107,127, 137, etc...a(3) = 107 is the smallest.

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

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

More terms from Robert Gerbicz, Sep 06 2002

A068173 Define an increasing sequence as follows. Given the first term called the seed (the seed need not have the property of the sequence.). Subsequent terms are defined as obtained by inserting/placing digits (at least one) in the previous term to obtain the smallest number with a given property. This is the growing prime sequence for the seed a(1) = 8.

Original entry on oeis.org

8, 83, 283, 1283, 12583, 112583, 1102583, 11002583, 110025803, 1010025803, 10100258303, 101002258303, 1010022508303, 10100225080303, 101002250803093, 1010022508030793, 10100225080303793, 101002250803030793, 1010022508030305793, 10100224508030305793

Offset: 1

Amarnath Murthy, Feb 25 2002

			The primes obtained by inserting/placing a digit in a(2) = 89 are 389, 809, etc...a(3) = 389 is the smallest.

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

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

Corrected and extended by Robert Gerbicz, Sep 06 2002

A249447 Least n-digit prime whose digit sum is also prime.

Original entry on oeis.org

2, 11, 101, 1013, 10037, 100019, 1000033, 10000019, 100000037, 1000000033, 10000000019, 100000000019, 1000000000039, 10000000000037, 100000000000031, 1000000000000037, 10000000000000079, 100000000000000013, 1000000000000000031, 10000000000000000051, 100000000000000000039

Offset: 1

Paolo P. Lava, Oct 29 2014

Subsequence of A046704 (primes with digits sum being prime).

Some terms of this sequence are also in A003617, the least n-digit primes. - Michel Marcus, Oct 30 2014

			a(1) = 2 because it is the least prime with just one digit.
a(2) = 11 because it is the least prime with 2 digits whose sum, 1 + 1 = 2, is a prime.
Again, a(7) = 1000033 because it is the least prime with 7 digits whose sum is a prime: 1 + 0 + 0 + 0 + 0 + 3 + 3 = 7.

Paolo P. Lava, Table of n, a(n) for n = 1..100

Cf. A003617, A046704, A068166, A249448.

P:=proc(q) local a,b,k,n; for k from 0 to q do
for n from 10^k to 10^(k+1)-1 do if isprime(n) then a:=n; b:=0;
while a>0 do b:=b+(a mod 10); a:=trunc(a/10); od;
if isprime(b) then print(n); break; fi; fi;
od; od; end: P(10^3);

a(n) = {p = nextprime(10^(n-1)); while (!isprime(sumdigits(p)), p = nextprime(p+1)); p;} \\ Michel Marcus, Oct 29 2014

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.

Original entry on oeis.org

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

A068168 Define an increasing sequence as follows. Given the first term called the seed (the seed need not have the property of the sequence.). Subsequent terms are defined as obtained by inserting/placing digits (at least one) in the previous term to obtain the smallest number with a given property. This is the growing prime sequence for the seed a(1) = 3.

Original entry on oeis.org

3, 13, 103, 1013, 10103, 100103, 1001003, 10010023, 100010023, 1000100239, 10001000239, 100010002039, 1000100020319, 10001000200319, 100001000200319, 1000010002000319, 10000100002000319, 100001000020003109, 1000010000200031039, 10000100002000310329

Offset: 1

Amarnath Murthy, Feb 25 2002

a(5) onwards the sequence is A068166.

			The primes obtained by inserting/placing a digit in a(2) = 13 are 113,131,313 etc... a(3)= 113 is the smallest.

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

Cf. A068166, A068167.

a:= proc(n) option remember; local s, w, m;
      if n=1 then 3
    else w:=a(n-1); s:=""||w; m:=length(s);
         min(select(x->length(x)=m+1 and isprime(x),
         {seq(seq(parse(cat(seq(s[h], h=1..i), j,
         seq(s[h], h=i+1..m))), j=0..9), i=0..m)})[])
      fi
    end:
seq(a(n), n=1..23);  # Alois P. Heinz, Nov 07 2014

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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A249447 Least n-digit prime whose digit sum is also prime.

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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.

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