A076730
Maximum number of (distinct) primes that an n-digit number may shelter (i.e., primes contained among all digital substrings' permutations).
Original entry on oeis.org
1, 4, 11, 31, 106, 402, 1953, 10542, 64905, 362451, 2970505
Offset: 1
We have a(3)=11, since among numbers 100 through 999, the smallest ones having 5, 6, 7, 8, 10, 11 embedded primes are respectively 107, 127, 113, 167, 179, 137 (the last of these being the first reaching the maximum number of 11 embedded primes, viz. 3, 7, 13, 17, 31, 37, 71, 73, 137, 173, 317).
A239197
The record values A075053 associated to the records (indices) listed in A239196.
Original entry on oeis.org
0, 1, 3, 4, 5, 9, 11, 17, 19, 21, 23, 25, 26, 29, 31, 32, 33, 44, 48, 52, 66, 89, 96, 106, 117, 164, 211, 236, 248, 311, 349
Offset: 1
-
m=-1; for(k=1, 9e9, A075053(k)>m&&print1(m=A075053(k),",")) \\ Not very efficient; from 199, 1999, 19999 etc one can jump to the next larger power of 10. - M. F. Hasler, Mar 12 2014
A307623
Numbers that set a record for the number of distinct composite numbers that can be obtained by permuting some subset of their digits.
Original entry on oeis.org
1, 4, 12, 18, 46, 102, 108, 124, 126, 148, 246, 468, 1002, 1008, 1014, 1022, 1023, 1025, 1026, 1068, 1234, 1236, 1245, 1248, 1268, 1458, 2456, 2468, 10023, 10025, 10026, 10068, 10124, 10125, 10146, 10224, 10234, 10236, 10245, 10248, 10458, 12345, 12348, 12369
Offset: 1
108 is in this sequence because the number of composite numbers which can be obtained by permuting some or all of digits of 108 is larger than the number of composite numbers obtainable in the same way for any smaller integer. With 108, you can form 9 composite numbers: 8, 10, 18, 80, 81, 108, 180, 801, 810. It's impossible to form n >= 9 composite numbers in the same way with any integer < 108.
Cf.
A072857 (the same with primes instead of composite numbers) and
A307624.
-
f[n_] := Length[Union[ Select[FromDigits /@ Flatten[Permutations /@ Subsets[IntegerDigits[n]], 1], CompositeQ]]];
d=-1; res={};Do[b=f[n];If[b>d,AppendTo[res,n];d=b],{n,10000}];res
A135377
Smallest n-primeval prime, i.e., minimal prime number containing all A006880(n) primes < 10^n embedded in it as permutations of some of its substrings.
Original entry on oeis.org
2357, 1123465789, 10112233445566788997, 100111222333444555666777998889, 1000111222233334444555666777798889899, 100001111222233333444445555566666777778888999989
Offset: 1
Mike Keith's website uses a shorthand notation for these numbers. The 4-primeval prime 100111222333444555666777998889 is written in this notation as (1) 2 3 3 3 3 3 3 3 0 998889. The (1) represents the leading 1 digit (which will always be present). The next number says how many consecutive 0's follow the leading 1 and the next says how many consecutive 1's follow that and so on up to the number of consecutive 8's. The final grouping explicitly shows how the last group of 8's and 9's are arranged.
The following are the n-primeval primes as found by _Jérôme STORTI_ in this notation:
5 (1) 3 3 4 4 4 3 3 4 0 98889899
6 (1) 4 4 4 5 5 5 5 5 4 999989
7 (1) 5 5 5 6 5 5 5 6 3 98899999
8 (1) 5 6 7 7 6 7 7 7 6 98999999
9 (1) 7 7 8 8 8 7 8 8 6 9999989899
10 (1) 8 8 8 9 9 9 9 9 7 9999899999
11 (1) 8 9 10 10 10 9 10 10 6 9889989999999
12 (1) 10 10 10 11 11 11 10 11 9 9998999999899
13 (1) 10 11 11 12 11 12 11 12 9 99899999999899
14 (1) 11 13 13 13 12 12 12 13 11 989999989999999
15 (1) 12 13 14 14 13 14 13 14 12 9999999988999999
16 (1) 13 14 14 15 14 14 14 15 12 99999999999999889
17 (1) 14 15 15 16 15 15 15 16 14 998999999999998999
18 (1) 16 17 17 17 16 17 17 17 14 9989999999999899999
19 (1) 17 18 17 18 17 17 17 18 15 988999999899999999999
20 (1) 17 19 18 19 19 18 19 19 16 999999998999999999989
21 (1) 18 19 19 20 19 19 20 20 17 9899999999999999998999
22 (1) 18 20 20 21 20 21 21 21 18 99998999999999999998999
23 (1) 21 23 21 22 21 21 22 22 19 999999889999999999999999
24 (1) 20 22 22 23 22 22 22 23 21 999999999999999989999999
25 (1) 23 23 23 24 23 23 23 24 22 9999999999999999998999999
26 (1) 23 24 24 25 25 25 24 25 22 999999999999999999899999989
27 (1) 24 25 25 26 25 25 25 26 23 9999999998999999999999998999
28 (1) 25 26 27 27 27 26 27 27 25 9999899999999999999999999999
29 (1) 25 27 27 28 27 27 27 28 25 999999989999999999999999999989
30 (1) 26 29 28 29 29 28 28 29 27 999999999999998999999999999999
31 (1) 28 29 29 30 29 29 29 30 27 99999889999999999999999999999999
a(2) = 1123465789 because this is the smallest prime out of which each of the first 25 primes below 10^2, viz. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 can be formed using its digits.
A370322
Least prime p such that exactly n distinct primes can be formed using one or more of the digits of p.
Original entry on oeis.org
2, 29, 13, 37, 107, 127, 113, 167, 1033, 179, 137, 1063, 1217, 1013, 1399, 1249, 1163, 1123, 1307, 1193, 1097, 10477, 11351, 1439, 1279, 1237, 3947, 11353, 1367, 10343, 1973, 10271, 10079, 10831, 10321, 10243, 10253, 10247, 13093, 10267, 10163, 10429, 12487, 11437, 10357, 10337
Offset: 1
a(0) would be 1, but 1 is not a prime (A075053);
a(1) is 2, the first prime;
a(2) is 29 since {2 & 29} are primes but {9 & 92} are not;
a(3) is 13 since {3, 13 & 31} are primes, but 1 is not;
a(4) is 37 since all the permutations are prime, i.e.: {3, 7, 37 & 73};
a(5) is 107 since {7, 17, 71, 107 & 701} are primes; etc.
Comments