A075053 -id:A075053 - cp's OEIS frontend

A076730 Maximum number of (distinct) primes that an n-digit number may shelter (i.e., primes contained among all digital substrings' permutations).

1, 4, 11, 31, 106, 402, 1953, 10542, 64905, 362451, 2970505

Offset: 1

Lekraj Beedassy, Nov 08 2002

See sequence A134596 for the least numbers of given length which yields these maxima over n-digit indices for A039993. - M. F. Hasler, Mar 11 2014

By definition this is a subsequence of A076497. The term a(10) was incorrectly given as 398100 = A075053(1123456789), which double-counts each prime using only one digit '1'. But a(10) = A039993(1123456789) = A076497(80) = 362451. The values given for a(9) and a(11) were also incorrect, the latter probably for the same reason, and for a(9) probably due to double-counting of primes with leading zeros. - M. F. Hasler and David A. Corneth, Oct 15 2019

			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).

M. Keith, Integers containing many embedded primes
W. Schneider, MATHEWS, Primeval Numbers

Cf. A072857, A076449, A076497, A134596 (largest n-digit primeval number).

Cf. A075053 (a variant of A039993), A134597 (= max A075053(1..10^n-1)).

a(n,m=0)=for(k=10^(n-1),10^n-1,A039993(k)>m&&m=A039993(k));m \\ M. F. Hasler, Mar 09 2014

Python
```
# see linked program in A076449
```

a(n) = A039993(A134596(n)) = max { A039993(m); m in A072857 and m < 10^n }. - M. F. Hasler, Mar 12 2014

a(n) = A076497(k) for k such that A072857(k) = A134596(n). - M. F. Hasler, Oct 15 2019

Link fixed by Charles R Greathouse IV, Aug 13 2009
a(6) from M. F. Hasler, Mar 09 2014
a(7)-a(11) from Robert G. Wilson v, Mar 11 2014
a(9)-a(11) corrected by M. F. Hasler, Oct 15 2019

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

Robert G. Wilson v, Mar 22 2024

Inspired by A076449.
a(n) >= A076449(n). As an example, a(727) is 3569887, but A076449(727) is 3567889, a difference of 1998. Notice that they possess identical digits.
a(n) = A076449(n) at n = 1, 3, 4, 5, 6, 7, 8, 10, 11, 14, 18, 19, 25, 26, 29, 33, 38, 40, 45, 46, ..., .
a(n) <> A076449(n) but they have identical digits at n = 12, 13, 17, 19, 20, 21, 24, 27, 31, 32, 34, 35, 36, 37, 39, ..., .
a(n) <> A076449(n) and they do not have identical digits at n = 2, 9, 15, 16, 22, 23, 28, 30, ..., .

			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.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1755 terms from Robert G. Wilson v)

Cf. A072857, A075053, A076449.

f = Compile[{{n, Integer}}, Floor@ Length[ Select[ Union[ FromDigits /@ Flatten[ Permutations /@ Subsets[ IntegerDigits@n], 1]], PrimeQ@# &]]]; p = 2; t[] := 0; While[p < 114500, a = f@p; If[ t[a] == 0, t[a] = p]; p = NextPrime@ p]; t /@ Range@ 100

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}.

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A076730 Maximum number of (distinct) primes that an n-digit number may shelter (i.e., primes contained among all digital substrings' permutations).

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A370322 Least prime p such that exactly n distinct primes can be formed using one or more of the digits of p.

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A173052 Partial sums of A072857.

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