A098450 -id:A098450 - cp's OEIS frontend

A340222 Constant whose decimal expansion is the concatenation of the largest n-digit semiprime A098450(n), for n = 1, 2, 3, ...

9, 9, 5, 9, 9, 8, 9, 9, 9, 8, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 7, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 7, 9, 9, 9, 9, 9, 9, 9, 9, 1, 9, 9, 9, 9, 9, 9, 9, 9, 9, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 0

M. F. Hasler, Jan 01 2021

			The smallest prime with 1, 2, 3, 4, ... digits is, respectively, 9 = 3^2, 95 = 5*19, 998 = 2*499, 9998 = 2*4999, .... Here we list the sequence of digits of these numbers: 9: 9, 5; 9, 9, 8; 9, 9, 9, 8; ...
This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.9959989998...

Cf. A098450 (largest n-digit semiprime), A340221 (similar, with smallest semiprime, limit of A215647), A340207 (same for squares, limit of A339978), A340206 (similar, with smallest n-digit squares, limit of A215689), A340209 (same with cubes, limit of A340115), A340208 (similar, with smallest n-digit cubes, limit of A215692), A340220 (same for primes, limit of A338968), A340219 (similar for smallest n-digit primes, limit of A215641).

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

concat([digits(A098450(k))|k<-[1..14]]) \\ as seq. of digits
c(N=20)=sum(k=1,N,.1^(k*(k+1)/2)*A098450(k)) \\ as constant

c = 0.995998999899998999997999999899999997999999991999999999799999999997...
  = Sum_{k >= 1} 10^(-k(k+1)/2)*A098450(k)
a(-n(n+1)/2) = 9 for all n >= 0, followed by increasingly more 9s.

A098449 Smallest n-digit semiprime.

Original entry on oeis.org

4, 10, 106, 1003, 10001, 100001, 1000001, 10000001, 100000001, 1000000006, 10000000003, 100000000007, 1000000000007, 10000000000015, 100000000000013, 1000000000000003, 10000000000000003, 100000000000000015, 1000000000000000007, 10000000000000000001

Offset: 1

Rick L. Shepherd, Sep 07 2004

Derek Orr, Table of n, a(n) for n = 1..60

Cf. A098450 (largest n-digit semiprime), A003617 (smallest n-digit prime), A001358 (semiprimes).

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, 19, 0] (* Robert G. Wilson v, Dec 18 2012 *)
Table[Module[{k=0},While[PrimeOmega[10^n+k]!=2,k++];10^n+k],{n,0,20}] (* Harvey P. Dale, Jun 15 2025 *)

a(n)=for(k=10^(n-1),10^n-1,if(bigomega(k)==2,return(k)))
vector(50, n, a(n)) \\ Derek Orr, Aug 15 2014

from sympy import factorint
def semiprime(n): f = factorint(n); return sum(f[p] for p in f) == 2
def a(n):
  an = max(1, 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

A119320 Difference between 10^n and the largest n-digit semiprime.

Original entry on oeis.org

1, 5, 2, 2, 2, 3, 2, 3, 9, 3, 3, 3, 11, 3, 2, 6, 11, 7, 9, 17, 3, 3, 11, 3, 3, 17, 5, 5, 9, 33, 11, 17, 17, 3, 11, 41, 39, 11, 29, 7, 11, 6, 21, 11, 17, 17, 9, 17, 11, 7, 11, 83, 49, 3, 2, 11, 3, 38, 23, 7, 11, 14, 9, 33, 3, 11, 11, 11, 31, 66, 9, 47, 21, 17, 29

Offset: 1

Zak Seidov, May 14 2006

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

Cf. A098450.

a(n) = my(k=10^n-1); while (bigomega(k) != 2, k--); 10^n - k; \\ Michel Marcus, Sep 18 2021

a(n) = 10^n - A098450(n).

More terms from Amiram Eldar, Sep 18 2021

A182648 a(n) is the largest n-digit number with exactly 4 divisors.

Original entry on oeis.org

8, 95, 998, 9998, 99998, 999997, 9999998, 99999997, 999999991, 9999999997, 99999999997, 999999999997, 9999999999989, 99999999999997, 999999999999998, 9999999999999994, 99999999999999989, 999999999999999993, 9999999999999999991, 99999999999999999983

Offset: 1

Jaroslav Krizek, Nov 27 2010

a(n) is the largest n-digit number of the form p^3 or p^1*q^1, (p, q = distinct primes).

Large overlap with A098450 which considers p^2 and p*q with n digits. - R. J. Mathar, Apr 23 2024

Michael S. Branicky, Table of n, a(n) for n = 1..62

Subsequence of A030513.

Cf. A174322, A098450.

Table[k=10^n-1; While[DivisorSigma[0,k] != 4, k--]; k, {n,10}]
lnd4[n_]:=Module[{k=10^n-1},While[DivisorSigma[0,k]!=4,k--];k]; Array[lnd4,20] (* Harvey P. Dale, Aug 20 2024 *)

from sympy import divisors
def a(n):
    k = 10**n - 1
    divs = -1
    while divs != 4:
      k -= 1
      divs = 0
      for d in divisors(k, generator=True):
        divs += 1
        if divs > 4: break
    return k
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Jun 10 2021

A000005(a(n)) = 4.

a(19) and beyond from Michael S. Branicky, Jun 10 2021

A180927 Largest n-digit number that is divisible by exactly 3 primes (counted with multiplicity).

Original entry on oeis.org

8, 99, 994, 9994, 99997, 999994, 9999994, 99999994, 999999998, 9999999995, 99999999998, 999999999998, 9999999999998, 99999999999998, 999999999999995, 9999999999999998, 99999999999999998, 999999999999999987, 9999999999999999999

Offset: 1

Jonathan Vos Post, Jan 23 2011

This is to 3 and A014612, as 2 and A098450 (largest n-digit semiprime), and as 1 and A003618 (largest n-digit prime). Largest n-digit triprime. Largest n-digit 3-almost prime.

			a(1) = 8 because 8 = 2^3 is the largest (only) 1-digit number that is divisible by exactly 3 primes (counted with multiplicity).
a(2) = 99 because 99 = 3^2 * 11 is the largest 2-digit number (of 21) that is divisible by exactly 3 primes (counted with multiplicity).
a(3) = 994 because 994 = 2 * 7 * 71 is the largest 3-digit number that is divisible by exactly 3 primes (counted with multiplicity).

Cf. A003618, A014612, A098450, A180922.

lndn3[n_]:=Module[{k=10^n-1},While[PrimeOmega[k]!=3,k--];k]; Array[ lndn3,20] (* Harvey P. Dale, Jul 25 2019 *)

A180927(n)=forstep(n=10^n-1,10^(n-1),-1,bigomega(n)==3&return(n)) \\ M. F. Hasler, Jan 23 2011

cp's OEIS Frontend

A340222 Constant whose decimal expansion is the concatenation of the largest n-digit semiprime A098450(n), for n = 1, 2, 3, ...

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A098449 Smallest n-digit semiprime.

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A119320 Difference between 10^n and the largest n-digit semiprime.

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A182648 a(n) is the largest n-digit number with exactly 4 divisors.

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A180927 Largest n-digit number that is divisible by exactly 3 primes (counted with multiplicity).

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