A098449 -id:A098449 - cp's OEIS frontend

A340221 Constant whose decimal expansion is the concatenation of the smallest n-digit semiprime A098449(n), for n = 1, 2, 3, ...

4, 1, 0, 1, 0, 6, 1, 0, 0, 3, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 6, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

M. F. Hasler, Jan 01 2021

The terms of sequence A215647 converge to this sequence of digits, and to this constant, up to powers of 10.

			The smallest prime with 1, 2, 3, 4, ... digits is, respectively, 2, 11, 101, 1009, .... Here we list the sequence of digits of these numbers: 2: 1, 1; 1, 0, 1; 1, 0, 0, 9; ...
This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.2111011009...

Cf. A098449 (smallest n-digit semiprime), A215647 (has this as "limit"), A340206 (same for squares, limit of A215689), A340207 (similar, with largest n-digit squares, limit of A339978), A340208 (same for cubes, limit of A215692), A340209 (same with largest n-digit cube, limit of A340115), A340219 (same for primes, limit of A215641).

Cf. A033307 (Champernowne constant), A030190 (binary), A001191 (concatenation of all squares), A134724 (cubes), A033308 (primes: Copeland-Erdős constant).

concat([digits(A098449(k))|k<-[1..14]]) \\ as seq. of digits
c(N=20)=sum(k=1,N,.1^(k*(k+1)/2)*A098449(k)) \\ as constant

c = 0.410106100310001100001100000110000001100000001100000000610000000003...
  = Sum_{k >= 1} 10^(-k(k+1)/2)*A098449(k)
a(-n(n+1)/2) = 1 for all n >= 2, followed by increasingly more zeros.

A098450 Largest n-digit semiprime.

Original entry on oeis.org

9, 95, 998, 9998, 99998, 999997, 9999998, 99999997, 999999991, 9999999997, 99999999997, 999999999997, 9999999999989, 99999999999997, 999999999999998, 9999999999999994, 99999999999999989, 999999999999999993, 9999999999999999991, 99999999999999999983

Offset: 1

Rick L. Shepherd, Sep 07 2004

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A098449 (smallest n-digit semiprime), A003618 (largest n-digit prime), A001358 (semiprimes), A119320.

NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega[sp] != 2, If[sgn < 0, sp--, sp++]]; If[sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]]; f[n_] := NextSemiPrime[10^n, -1]; Array[f, 18] (* Robert G. Wilson v, Dec 18 2012 *)

apply( A098450(n)={n=10^n;until(bigomega(n-=1)==2,);n}, [1..20]) \\ M. F. Hasler, Jan 01 2021

from sympy import factorint
def semiprime(n): f = factorint(n); return sum(f[p] for p in f) == 2
def a(n):
  an = 10**n - 1
  while not semiprime(an): an -= 1
  return an
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Apr 10 2021

a(n) = 10^n - A119320(n). - Amiram Eldar, Sep 18 2021

A180922 Smallest n-digit number that is divisible by exactly 3 primes (counted with multiplicity).

Original entry on oeis.org

8, 12, 102, 1001, 10002, 100006, 1000002, 10000005, 100000006, 1000000003, 10000000001, 100000000006, 1000000000001, 10000000000001, 100000000000018, 1000000000000002, 10000000000000006, 100000000000000007, 1000000000000000001, 10000000000000000007

Offset: 1

Jonathan Vos Post, Jan 23 2011

This is to 3 as smallest n-digit semiprime A098449 is to 2, and as smallest n-digit prime A003617 is to 1. Smallest n-digit triprime. Smallest n-digit 3-almost prime.

			a(1) = 8 because 8=2^3 is the smallest (only) 1-digit number divisible by exactly 3 primes (counted with multiplicity).
a(2) = 12 because 12 = 2^2 * 3 is the smallest of the (21) 2-digit numbers divisible by exactly 3 primes (counted with multiplicity).
a(3) = 102 because 102 = 2 * 3 * 17 is the smallest 3-digit numbers divisible by exactly 3 primes (counted with multiplicity).

Cf. A003617, A014612, A098449.

A180922(n)=for(n=10^(n-1),10^n-1,bigomega(n)==3&return(n)) \\ M. F. Hasler, Jan 23 2011

from sympy import factorint
def triprimes(n): f = factorint(n); return sum(f[p] for p in f) == 3
def a(n):
  an = max(1, 10**(n-1))
  while not triprimes(an): an += 1
  return an
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Apr 10 2021

A347818 Smallest n-digit brilliant number.

Original entry on oeis.org

4, 10, 121, 1003, 10201, 100013, 1018081, 10000043, 100140049, 1000000081, 10000600009, 100000000147, 1000006000009, 10000000000073, 100000380000361, 1000000000000003, 10000001400000049, 100000000000000831, 1000000014000000049, 10000000000000000049, 100000000380000000361

Offset: 1

Eric Chen, Sep 15 2021

A brilliant number is a semiprime (products of two primes, A001358) whose two prime factors have the same number of decimal digits. For an n-digit brilliant number, the two prime factors must each have ceiling(n/2) decimal digits.

Since all brilliant numbers are semiprimes, a(n) >= A098449(n), also, a(n) = A098449(n) for n = 1, 2, 4, 16, 78, ..., are there infinitely many n such that a(n) = A098449(n)?

			a(6) =    100013 =   103 * 971.
a(7) =   1018081 =  1009 * 1009.
a(8) =  10000043 =  2089 * 4787.
a(9) = 100140049 = 10007 * 10007.

Dario Alejandro Alpern, Brilliant numbers
World of Numbers, Smallest n-digit prp

Cf. A078972, A003617, A083289, A084475, A084476, A083128, A098449.

Join[{4,10},Table[Module[{k=1},While[PrimeOmega[10^n+k]!=2||Length[ Union[ IntegerLength/@ FactorInteger[ 10^n+k][[;;,1]]]]!=1,k+=2];10^n+k],{n,2,20}]] (* Harvey P. Dale, Jan 09 2024 *)

isA078972(n)=my(f=factor(n)); (#f[, 1]==1 && f[1, 2]==2) || (#f[, 1]==2 && f[1, 2]==1 && f[2, 2]==1 && #Str(f[1, 1])==#Str(f[2, 1]))
A084476(n)=for(k=0,10^n,if(isA078972(10^(2*n-1)+k),return(k)))
a(n)=if(n%2,nextprime(10^((n-1)/2))^2,10^(n-1)+A084476(n/2)) \\ after Charles R Greathouse IV in A078972

a(n) = 10^(n-1) + A083289(n).
a(2*n) = 10^(2*n-1) + A084476(n).
a(2*n+1) = A003617(n+1)^2.
a(n) >= A098449(n).

cp's OEIS Frontend

A340221 Constant whose decimal expansion is the concatenation of the smallest n-digit semiprime A098449(n), for n = 1, 2, 3, ...

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A098450 Largest n-digit semiprime.

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A180922 Smallest n-digit number that is divisible by exactly 3 primes (counted with multiplicity).

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PARI

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A347818 Smallest n-digit brilliant number.

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Mathematica

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