A099756 -id:A099756 - cp's OEIS frontend

A100369 Largest primes arising in A099756 which were built up from n distinct digits. This sequence differs from A007810 because more than one copy of each digit is permitted.

11, 787, 22259, 70879, 607889, 4456789, 40456789, 304456879, 1123465789, 10123457689

Offset: 1

Labos Elemer, Nov 29 2004

These primes are "largest among earliest primes" at fixed number of distinct digits chosen from A099756. Their position in A099756 are: {1,27,90,198,440,774,858,930,949,950}.

			n=3: a[3]=22259, built up from the 3-subset = {2,5,9} of decimal digits.
It appears in A099756 as the 90th term. It is by definition of A099756 is
the smallest prime that can be constructed from {2,5,9} and at the same time
it is the largest prime if running through all 3-subsets of decimal digits.
All terms includes at least 2 copies of some digit.
Differs from A007810[3]=983.

Cf. A099651, A099653, A099654, A007809, A007810, A099756.

Edited by Charles R Greathouse IV, Aug 03 2010

A100370 Primes in A099756, sorted.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 43, 47, 53, 59, 61, 67, 79, 83, 89, 101, 103, 107, 109, 127, 137, 139, 149, 151, 157, 163, 167, 179, 181, 211, 227, 239, 241, 251, 257, 263, 269, 281, 283, 307, 347, 349, 359, 367, 379, 389, 401, 409, 431, 449, 457, 461

Offset: 1

Labos Elemer, Nov 30 2004

Inspired by A099756.

			Positions of "minimal terms" (see A007809) inside A099756 and here, in A100370, are {2,8,31,138,320,574,779,900,942,950} or {1,6,23,84,250,494,721,873,934,950} respectively.
This is because the orders of A099756 and A100370 are based on different criteria.

Robert G. Wilson v, Table of n, a(n) for n = 1..950

Cf. A007809, A007810, A099651, A099653, A099654, A099756, A100369.

< 0 &][[-2]]*10^(Length[ss[[n]]] -  If[ Mod[ FromDigits@ ss[[n]], 3] == 0, 0, 1]) - 1]}, While[ Union@ IntegerDigits@ p != id, p = NextPrime@ p]; p]; f[3] = 3; Sort@ Array[f, 950]

A099651 For each of the C(10,2) = 45 pairs of decimal digits, record the smallest prime containing only these digits (if one exists); sort.

Original entry on oeis.org

13, 17, 19, 23, 29, 37, 41, 43, 47, 53, 59, 61, 67, 79, 83, 89, 101, 151, 181, 211, 227, 449, 557, 787

Offset: 1

Labos Elemer, Nov 11 2004

The sequence consists of 24 terms, of which 16 cases < 100.

From 45 combinations of 10 decimal digits only 24 can be prime. All least cases are here.

			Primes with digits of 8 and 9 are in A020472:{89,8999,89899,89989..}. The smallest = 89 is here.
The 24 digit pairs sorted least to greatest that can be prime are {01, 12, 13, 14, 15, 16, 17, 18, 19, 23, 27, 29, 34, 35, 37, 38, 47, 49, 57, 59, 67, 78, 79, 89}. - _Michael De Vlieger_, Mar 02 2017

Cf. A020449-A020472, A099756.

Sort@ Map[Module[{k = 1}, While[! SameQ[Union@ IntegerDigits@ Prime@ k, #], k++]; Prime@ k] &, Function[r, {{0, 1}}~Join~DeleteCases[Union@ Map[Sort, Tuples[Range@ 9, 2]], w_ /; Or[Times @@ Boole@ Map[EvenQ, w] > 0, SameQ @@ w, Times @@ Boole@ Map[Mod[#, 3] == 0 &, w] > 0, SubsetQ[r, w], w == {5, 6}]]]@ Select[Range[2, 9], PowerMod[10, #, #] == 0 &]] (* Michael De Vlieger, Mar 02 2017 *)

A099654 a(n) is the number of n-subsets [n=1,2,...,10] of the 10 decimal digits from which no prime numbers can be constructed. See also A099653.

Original entry on oeis.org

5, 21, 24, 16, 6, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Labos Elemer, Nov 15 2004

Number of "antiprime-digit-subclasses".

Subsets were selected from {0, 2, 4, 5, 6, 8} and {0, 3, 6, 9} digit collections.

			n=1: {0,2,4,6,8} represent the relevant 1-subsets so a[1]=5.
Total number of prime irrelevant subset-classes from the 1023 nonempty k-digit-subsets equals 5 + 21 + 24 + 16 + 6 + 1 = 73 = 1023 - 950. See also A099653.
The "antiprime n-digit-collections" are taken from {0,2,4,5,6,8} or {0,3,6,9}, of which only composites can be constructed.

Cf. A099651, A099654, A099756.

Table[Binomial[6, n] + Binomial[4, n] - 5 Boole[n == 1], {n, 100}] (* Michael De Vlieger, Mar 26 2017 *)

a(n) = binomial(6, n) + binomial(4, n) - 5*(n==1); \\ Indranil Ghosh, Mar 27 2017

from sympy import binomial
def a(n): return binomial(6, n) + binomial(4, n) - 5*(n==1) # Indranil Ghosh, Mar 27 2017

a(n) = binomial(6,n) + binomial(4,n) for n > 1.

A099653 a(n) is the number of n-subsets (n=1,2,...,10) of the 10 decimal digits from which prime numbers can be constructed including all n distinct digits either with or without repetitions; a(n) <= binomial(10,n).

Original entry on oeis.org

5, 24, 96, 194, 246, 209, 120, 45, 10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Labos Elemer, Nov 15 2004

			n=1: {11,2,3,5,7} represent the 1-subsets; a(1) = 5;
n=2: A099651 includes least terms of each a(2) = 24 subsets;
n=5: a(5) = binomial(10,5) - binomial(6,5) - binomial(4,5) = 210 - 6 - 0 = 246;
n=6: each of the 6-subsets is good for primes except {0,2,4,5,6,8} so a(6) = 210 - 1.
n=7,8,9,10: a(n) = binomial(10,n).
Total number of relevant subset classes from the 1023 nonempty k-digit subsets equals 950. See also A099654.

Cf. A099651, A099654, A099756.

Table[5 Boole[n == 1] + Binomial[10, n] - Binomial[6, n] - Binomial[4, n], {n, 83}] (* Michael De Vlieger, Jul 24 2017 *)

a(n) = binomial(10,n) - binomial(6,n) - binomial(4,n); number of n-digit subsets minus "antiprime-digit subclasses" selected from {0, 2, 4, 5, 6, 8} and {0, 3, 6, 9} digit collections.

A099268 For each single digit {0,1,...,9} record the smallest prime made up of copies of that digit or 0 if no such prime exists; repeat for all of the C(10,2) = 45 pairs of distinct decimal digits; then for all triples; etc.

Original entry on oeis.org

0, 11, 2, 3, 0, 5, 0, 7, 0, 0, 101, 0, 0, 0, 0, 0, 0, 0, 0, 211, 13, 41, 151, 61, 17, 181, 19, 23, 0, 0, 0, 227, 0, 29, 43, 53, 0, 37, 83, 0, 0, 0, 47, 0, 499, 0, 557, 0, 59, 67, 0, 0, 787, 79, 89, 1021, 103, 401, 5101, 601, 701, 8101, 109, 2003, 0, 0, 0, 2027, 0, 2029, 4003, 503

Offset: 1

Robert G. Wilson v, Nov 16 2004

			There are no primes that consist of copies of the digit 4, or 6, or 8, or 9, or {0,2}, or {0,3}, etc.

Cf. A099651, A099756, A016112.

A100372 Build up the least positive nonprime number from all subsets of decimal digits {0,1,2,3,4,5,6,7,8,9}. The terms are ordered as follows: 1. for fixed k, the k-digit-subsets for design are ordered lexicographically; 2. choose k=1,2,...,9,10-subsets.

Original entry on oeis.org

1, 22, 33, 4, 55, 6, 77, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 12, 133, 14, 15, 16, 117, 18, 91, 32, 24, 25, 26, 27, 28, 92, 34, 35, 36, 377, 38, 39, 45, 46, 74, 48, 49, 56, 57, 58, 95, 76, 68, 69, 78, 779, 98, 102, 130, 104, 105, 106, 170, 108, 190, 203, 204, 205, 206

Offset: 1

Labos Elemer, Dec 01 2004

Composite analog of A099756. From the 1023 nonempty digit subsets 1022 terms can be designed because 0 is not permitted.

			For 2-subsets of {1,3},{1,7},{3,7},{7,9} the least composites should have at least two copies of a digit; that is why the solutions {133,117,377,779} have 3 digits.

Cf. A099756.

Mathematica
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A100369 Largest primes arising in A099756 which were built up from n distinct digits. This sequence differs from A007810 because more than one copy of each digit is permitted.

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A100370 Primes in A099756, sorted.

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A099651 For each of the C(10,2) = 45 pairs of decimal digits, record the smallest prime containing only these digits (if one exists); sort.

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A099654 a(n) is the number of n-subsets [n=1,2,...,10] of the 10 decimal digits from which no prime numbers can be constructed. See also A099653.

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A099653 a(n) is the number of n-subsets (n=1,2,...,10) of the 10 decimal digits from which prime numbers can be constructed including all n distinct digits either with or without repetitions; a(n) <= binomial(10,n).

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A099268 For each single digit {0,1,...,9} record the smallest prime made up of copies of that digit or 0 if no such prime exists; repeat for all of the C(10,2) = 45 pairs of distinct decimal digits; then for all triples; etc.

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A100372 Build up the least positive nonprime number from all subsets of decimal digits {0,1,2,3,4,5,6,7,8,9}. The terms are ordered as follows: 1. for fixed k, the k-digit-subsets for design are ordered lexicographically; 2. choose k=1,2,...,9,10-subsets.

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