A039993 -id:A039993 - cp's OEIS frontend

A134597 a(n) gives the maximal value of A075053(m) for any n-digit number m.

1, 4, 11, 31, 106, 402, 1953

Offset: 1

N. J. A. Sloane, Jan 25 2008

In A075053(m), the primes obtained as permutations of digits of m are counted several times if they can be obtained in several different ways. See sequence A076730 which uses A039993 instead, i.e., counting only different primes. - M. F. Hasler, Mar 11 2014

The original data given for n = 3, 4, 5 was erroneously A007526(n). - Up to n = 6, a(n) = A076730(n), but the two will differ not later than for n = 10, where A134596(10) = 1123456789 gives a(10) >= 398100 = A075053(1123456789) > A039993(1123456789) = 362451 = A076730(10). The difference arises because each prime containing a single '1' will be counted twice by A075053, but only once by A039993. - M. F. Hasler, Oct 14 2019

			From _M. F. Hasler_, Oct 14 2019: (Start)
a(2) = 4 = A075053(37), because from 37 one can obtain the primes {3, 7, 37, 73}, and there is obviously no 2-digit number which could give more primes.
a(3) = 11 = A075053(137), because from 137 one can obtain the primes {3, 7, 13, 17, 31, 37, 71, 73, 137, 173, 317}, and no 3-digit number yields more.
a(4) = 31 = A075053(1379), because from 1379 one can obtain the 31 primes {3, 7, 13, 17, 19, 31, 37, 71, 73, 79, 97, 137, 139, 173, 179, 193, 197, 317, 379, 397, 719, 739, 937, 971, 1973, 3719, 3917, 7193, 9137, 9173, 9371}, and no 4-digit number yields more.
a(5) = 106 = A075053(13679). a(6) = 402 = A075053(123479).
a(7) = 1953 = A075053(1234679). (End)

M. Keith, Integers containing many embedded primes

Cf. A075053, A039993, A134596.

Cf. A239196 for record indices of A075053, A239197 for associated record values.

A134597(n)={my(m=0);forvec(D=vector(n,i,[0,9]), vecsum(D)%3||next;m=max(A075053(fromdigits(D),D),m),1);m} \\ M. F. Hasler, Oct 14 2019

a(n) <= A007526(n), with equality iff n <= 2. [Keith]

a(n) = max { A075053(m); 10^(n-1) <= m < 10^n } >= A076730(n) = max { A039993(m); 10^(n-1) <= m < 10^n }. - M. F. Hasler, Mar 11 2014

Link fixed by Charles R Greathouse IV, Aug 13 2009
Definition corrected by M. F. Hasler, Mar 11 2014
Data corrected and extended by M. F. Hasler, Oct 14 2019

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. F. Hasler, Mar 12 2014

This and A239196 are the analogs (related to A075053) of A076497 and A072857 (primeval numbers), related to A039993.

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

a(n)=A075053(A239196(n)).

A039992 Number of distinct primes embedded in prime p(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 1, 3, 2, 3, 4, 1, 2, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 3, 2, 4, 5, 2, 7, 6, 7, 11, 6, 6, 3, 7, 7, 8, 11, 10, 3, 4, 6, 10, 4, 3, 4, 3, 3, 4, 6, 4, 4, 4, 4, 3, 6, 4, 3, 6, 6, 5, 7, 5, 11, 5, 7, 8, 4, 4, 7, 7, 7, 10, 3, 6, 10, 2, 1, 6, 4, 6, 3, 4, 3, 1, 5, 4, 4, 5, 6, 3, 6, 1, 4, 3, 4, 6, 3, 5

Offset: 1

David W. Wilson

a(n) counts permuted subsequences of digits of p(n) which denote primes.

We put all the digits of prime(n) into a bag and ask how many distinct primes can be formed using some or all of these digits.

			a(35)=6 because from the digits of p(35)=149, six numbers can be formed, 19, 41, 149, 419, 491 & 941, which are primes.

a(n) = A045719(n)+1 = A039993(p(n)) A101988 gives another version.

Needs["DiscreteMath`Combinatorica`"]; f[n_] := Length[ Union[ Select[ FromDigits /@ Flatten[ Permutations /@ Subsets[ IntegerDigits[ Prime[n]]], 1], PrimeQ]]]; Table[f[n], {n, 102}] (* Ray Chandler and Robert G. Wilson v, Feb 25 2005 *)

A134649 Minimal n-primeval numbers: a(n) = smallest number M such that all primes with <= n digits are embedded in a permuted subsequence of M's digits.

Original entry on oeis.org

2357, 1123456789, 1012233445566778899, 10011222333444555666777888999, 1000111222233334444555666777788889999, 100001111222233333444445555566666777778888899999, 100000111112222233333344444555556666677777788888999999

Offset: 1

N. J. A. Sloane, Jan 25 2008

M. Keith, Integers containing many embedded primes

Cf. A039993, A135377.

Extended by Charles R Greathouse IV, Aug 13 2009

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

Lekraj Beedassy, Dec 09 2007

a(1) - a(4) were computed by Mike Keith in 2008 and a(4) - a(31) by Jérôme STORTI in 2002.

			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.

M. Keith, Integers containing many embedded primes

Cf. A072857, A039993, A134649.

Link fixed by Charles R Greathouse IV, Aug 13 2009

A229908 The least prime that has at least n embedded primes, reading forward and backward.

Original entry on oeis.org

2, 13, 13, 37, 113, 113, 137, 1237, 1373, 1733, 1733, 11317, 11731, 12713, 19973, 91733, 113173, 113371, 113371, 173347, 991733, 1123379, 1134673, 1137991, 1237199, 2333719, 7433719, 11133719, 11399173, 11399173, 11791733, 37914713, 97433719, 113217397, 113217397, 113337199, 113337199, 113337199, 1113371999, 1113371999, 1113991733, 1139917321, 1139917321, 1139917333

Offset: 1

Carlos Rivera, Dec 19 2013

Results of my search up to 2^32.

			Example: a(8) = 1237 because there are the following eight embedded primes: 1237, 2, 23, 3, 37, 7 and 73, 7321.

Giovanni Resta, Table of n, a(n) for n = 1..65 (terms < 3.5*10^12)
C. Rivera, Puzzle 724. Extending the OEIS sequence A229908

Cf. A039993, A039997.

Missing a(19) from Giovanni Resta, Jan 25 2014

A173052 Partial sums of A072857.

Original entry on oeis.org

1, 3, 16, 53, 160, 273, 410, 1423, 2460, 3539, 4776, 6143, 7522, 17601, 27724, 37860, 47999, 58236, 68515, 78882, 89261, 101640, 115319, 215598, 315977, 417214, 519561, 621940, 725619, 849098, 1850335, 2852682, 3855061, 4858740, 5871089

Offset: 1

Jonathan Vos Post, Feb 08 2010

Partial sums of primeval numbers. Primeval number: a prime which "contains" more primes in it than any preceding number. Here "contains" means may be constructed from a subset of its digits. E.g., 1379 contains 3, 7, 13, 17, 19, 31, 37, 71, 73, 79, 97, 137, 139, 173, 179, 193, 197, 317, 379, 397, 719, 739, 937, 971, 1973, 3719, 3917, 7193, 9137, 9173 and 9371. The subsequence of prime partial sums of primeval numbers begins: 3, 53, 1423, 3539, 6143, 89261, 115319, 315977. What is the smallest primeval prime partial sums of primeval numbers, i.e. the intersection of this sequence with A119535?

			a(36) = 1 + 2 + 13 + 37 + 107 + 113 + 137 + 1013 + 1037 + 1079 + 1237 + 1367 + 1379 + 10079 + 10123 + 10136 + 10139 + 10237 + 10279 + 10367 + 10379 + 12379 + 13679 + 100279 + 100379 + 101237 + 102347 + 102379 + 103679 + 123479 + 1001237 + 1002347 + 1002379 + 1003679 + 1012349 + 1012379.

Chris K. Caldwell AND G. L. Honaker, Jr., A BRIEF PRIME CURIOS! GLOSSARY.

Cf. A000040, A072857, A039993, A075053, A076497, A076449, A119535 (prime subsequence).

a(n) = SUM[i=1..n] A072857(i) = SUM[i=1..n] {numbers that set a record for the number of distinct primes that can be obtained by permuting some subset of their digits}.

cp's OEIS Frontend

A134597 a(n) gives the maximal value of A075053(m) for any n-digit number m.

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A239197 The record values A075053 associated to the records (indices) listed in A239196.

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A039992 Number of distinct primes embedded in prime p(n).

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Mathematica

A134649 Minimal n-primeval numbers: a(n) = smallest number M such that all primes with <= n digits are embedded in a permuted subsequence of M's digits.

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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.

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A229908 The least prime that has at least n embedded primes, reading forward and backward.

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A173052 Partial sums of A072857.

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