A099653 -id:A099653 - cp's OEIS frontend

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.

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.

A099756 For each single digit {0,1,...,9} record the smallest prime made up of copies of that digit (if there is one); repeat for all of the C(10,2) = 45 pairs of distinct decimal digits; then for all triples; etc.

Original entry on oeis.org

11, 2, 3, 5, 7, 101, 211, 13, 41, 151, 61, 17, 181, 19, 23, 227, 29, 43, 53, 37, 83, 47, 449, 557, 59, 67, 787, 79, 89, 1021, 103, 401, 1051, 601, 107, 1801, 109, 2003, 2027, 2029, 4003, 503, 307, 3083, 4007, 409, 5077, 509, 607, 8087, 709, 809, 1123, 241, 251, 1621

Offset: 1

N. J. A. Sloane, Nov 11 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}, ..., {0,2,4,5,6,8}.

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

Inspired by A099651. Cf. A016112.

Cf. A099653, A099654.

ss = Subsets[Range[0, 9], 10]; dlt =  {1, 2, 6, 8, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 30, 31, 32, 34, 38, 41, 42, 43, 45, 47, 49, 52, 53, 66, 67, 68, 70, 74, 77, 78, 79, 81, 83, 85, 88, 89, 127, 128, 130, 132, 134, 137, 153, 157, 159, 162, 168, 211, 212, 214, 216, 218, 221, 237, 241, 243, 246, 252, 332, 334, 337, 343, 373, 458, 460, 463, 469, 499, 604, 730}; ss = Delete[ss, {#} & /@ dlt]; k = 1; lst = {}; f[n_] := Block[{id = ss[[n]], p = NextPrime[ NestWhileList[ Quotient[#, 10] &, FromDigits[ss[[n]]], # > 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; Array[ f, 950] (* Robert G. Wilson v, Apr 06 2024 *)

More terms from Labos Elemer, Nov 15 2004

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.

Original entry on oeis.org

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]

cp's OEIS Frontend

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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A099756 For each single digit {0,1,...,9} record the smallest prime made up of copies of that digit (if there is one); repeat for all of the C(10,2) = 45 pairs of distinct decimal digits; then for all triples; etc.

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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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