A069603 -id:A069603 - cp's OEIS frontend

A074338 a(1) = 2; a(n) is smallest number > a(n-1) such that the juxtaposition a(1)a(2)...a(n) is a prime.

2, 3, 9, 11, 13, 63, 71, 93, 187, 189, 201, 207, 243, 347, 369, 439, 473, 529, 611, 847, 1209, 1331, 1423, 1581, 1593, 1617, 1679, 1791, 2067, 2529, 2541, 2563, 2751, 3347, 3583, 3677, 3777, 4359, 4701, 4771, 5657, 6183, 6193, 6353, 6511, 6539, 6769, 6939

Offset: 1

Zak Seidov, Sep 23 2002

Cf. A033679, A069603, A074336, A074339, A074340, A074341, A074342, A074343, A074344, A074345, A074346.

a[1] = 2; a[n_] := a[n] = Block[{k = a[n - 1] + 1 + Mod[a[n - 1], 2], c = IntegerDigits @ Table[ a[i], {i, n - 1}]}, While[ !PrimeQ[ FromDigits @ Flatten @ Append[c, IntegerDigits[k]]], k += 2]; k]; Table[ a[n], {n, 48}] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Aug 05 2005

A111525 a(1) = 10; a(n) = smallest number such that the juxtaposition a(1)a(2)...a(n) is a prime.

Original entry on oeis.org

10, 1, 3, 3, 3, 29, 1, 3, 3, 11, 9, 7, 23, 61, 11, 3, 91, 137, 7, 11, 31, 93, 17, 9, 273, 51, 397, 9, 99, 41, 111, 129, 111, 801, 109, 131, 297, 37, 621, 21, 807, 143, 87, 57, 231, 187, 53, 169, 77, 613, 867, 41, 199, 773, 523, 227, 27, 499, 171, 329, 67, 483, 393, 179

Offset: 1

Amarnath Murthy, Jason Earls and Robert G. Wilson v, Aug 05 2005

Cf. A111524, A074346, A092528, A069603, A069605, A069606, A069607, A069608, A069609, A069610, A069611, A111525.

a[1] = 10; a[n_] := a[n] = Block[{k = 1, c = IntegerDigits @ Table[ a[i], {i, n - 1}]}, While[ !PrimeQ[ FromDigits @ Flatten @ Append[c, IntegerDigits[k]]], k += 2]; k]; Table[ a[n], {n, 63}]

A080155 a(1)=2; a(n) for n>1 is the smallest prime number > a(n-1) such that the concatenation of all previous terms is also prime.

Original entry on oeis.org

2, 3, 11, 31, 47, 229, 251, 577, 857, 859, 911, 1123, 1223, 1297, 1571, 2161, 2417, 2551, 2879, 3319, 5273, 6121, 6947, 7603, 8273, 12721, 12953, 13291, 15683, 16453, 17207, 18133, 20399, 23743, 23909, 25849, 28277, 28879, 35291, 35461, 36107, 43573

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 31 2003

See A073640 for the sequence involving concatenation of 2 successive terms, A080153 for 3 successive terms. Primeness is established using Maple's isprime() function, so later terms should be regarded as "probable".

			E.g. a(5)=47 since this is the smallest prime>a(4) which, when concatenated with the concatenation of a(1) to a(4) (=231131), also yields a prime, in this case 23113147.

T. D. Noe, Table of n, a(n) for n = 1..201

Cf. A073640, A080152, A080153, A080154, A080156.

Cf. A033679, A069603, A074338.

with(numtheory): pout := [2]: nout := [1]: for n from 2 to 5000 do: p := ithprime(n): d := parse(cat(seq(pout[i],i=1..nops(pout)),p)): if (isprime(d)) then pout := [op(pout),p]: nout := [op(nout),n]: fi: od: pout;

f[s_List] := Block[{p=NextPrime@s[[-1]], pp=FromDigits@Flatten[IntegerDigits/@s]}, While[!PrimeQ[pp*10^Floor[Log[10,p]+1]+p], p=NextPrime@p]; Append[s,p]]; Nest[f,{2},40]

For any n>1, a(n) is prime and a(n) > a(n-1). a(n) is the smallest prime for which a(1)//a(2)//...//a(n) is prime. // denotes concatenation.

A358344 a(1) = 0; a(n) = the smallest number such that the concatenation a(1)a(2)...a(n) is prime in the smallest allowed base; sequence terminates at index m if a(1)a(2)...a(m)k is composite in the smallest allowed base for all k.

Original entry on oeis.org

0, 2, 1, 2, 2, 3, 1, 5, 9, 7, 21, 5, 31, 49, 39, 104, 2, 34, 44, 74, 22, 64, 16, 107, 549, 81, 207, 273, 87, 497, 27, 556, 42, 150, 32, 44, 144, 340, 28, 198, 677, 13, 61, 209, 377, 893, 329, 391, 49, 83, 425, 197, 1017, 205, 191, 163, 1131, 291, 281, 295, 389

Offset: 1

Samuel Harkness, Nov 11 2022

For all n > 1, a(n) > 0.
For all n > 3, if a(n) is even or odd, then until a new number a(n+k) > a(n), all a(n+k) must also be even or odd, respectively.
"Smallest allowed base" is max{a(1), a(2), ..., a(n)} + 1. E.g., a(3) uses base 3 because max{0, 2, 1} + 1 = 3.
Treat a(n) >= 10 as one "digit". E.g., if three consecutive terms were 8, 12, 4, treat the concatenation as "8C4" to be read in base 13 instead of "8124."
It is unknown whether this sequence has infinite terms. There exist initial values which, using the method described in the definition, reach a point that guarantees no new primes. E.g., Michael S. Branicky showed for a(1) = 13, after 4 terms {13, 9, 3, 5} the tested number for a fifth term "k" is not prime for 0 <= k < 13, and the equation for the tested number once k >= 13 is 13*(k+1)^4 + 9*(k+1)^3 + 3*(k+1)^2 + 5(k+1) + k = (k+2)(13*k^3 + 35*k^2 + 38*k + 15), thus never prime. Because there exist initial values which guarantee no new terms after various lengths, any initial value may eventually reach such a point.

			For a(6): The concatenation a(1)a(2)a(3)a(4)a(5) gives 2122. The smallest base in which 2122 can be read is max{2, 1, 2, 2} + 1 = 3, so test 21220_3 = 213 (nonprime), 21221_3 = 214 (nonprime), 21222_3 = 215 (nonprime). Now, 21223 is the next candidate; note that the new smallest allowed base is max{2, 1, 2, 2, 3} + 1 = 4, so test 21223_4 = 619 (prime). Thus, a(6) = 3.

Samuel Harkness, Table of n, a(n) for n = 1..1309

Cf. A023107, A024770, A069603.

V = {0}; While[Length[V] <= 60, c = 0; d = 0; b = Max[V] + 1; CCC = 0; While[c == 0, X = V; X = Append[X, d]; CCC = 0; For[i = 1, i <= Length[X], i++, CCC += Part[X, i]*b^(Length[X] - i)]; If[PrimeQ[CCC], c = 1]; d++; If[d == b, b++]]; V = X]; Print[V]

from sympy import isprime
from itertools import count, islice
def fd(d, b): return sum(di*b**i for i, di in enumerate(d[::-1]))
def anext(alst):
    b = max(alst)
    return next(k for k in count(1) if isprime(fd(alst+[k], max(b, k)+1)))
def agen():
    alst = [0]
    while True: yield alst[-1]; alst.append(anext(alst))
print(list(islice(agen(), 61))) # Michael S. Branicky, Nov 11 2022

Escape clause added by Jianing Song, Nov 28 2022

cp's OEIS Frontend

A074338 a(1) = 2; a(n) is smallest number > a(n-1) such that the juxtaposition a(1)a(2)...a(n) is a prime.

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A111525 a(1) = 10; a(n) = smallest number such that the juxtaposition a(1)a(2)...a(n) is a prime.

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A080155 a(1)=2; a(n) for n>1 is the smallest prime number > a(n-1) such that the concatenation of all previous terms is also prime.

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A358344 a(1) = 0; a(n) = the smallest number such that the concatenation a(1)a(2)...a(n) is prime in the smallest allowed base; sequence terminates at index m if a(1)a(2)...a(m)k is composite in the smallest allowed base for all k.

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