A036337 -id:A036337 - cp's OEIS frontend

A036336 Smallest positive integer with n digits and exactly n prime factors (counted with multiplicity).

2, 10, 102, 1012, 10010, 100040, 1000125, 10000096, 100000032, 1000000080, 10000000080, 100000000512, 1000000001280, 10000000014336, 100000000004096, 1000000000010880, 10000000000008192, 100000000000008192, 1000000000000010240, 10000000000000045056

Offset: 1

Patrick De Geest, Dec 15 1998

David A. Corneth, Table of n, a(n) for n = 1..29
Carlos Rivera, Puzzle 25. Composed primes, The Prime Puzzles and Problems Connection.

Cf. A036335, A036337, A036338.

f:= proc(n) local k;
  for k from 10^(n-1) do
    if numtheory:-bigomega(k) = n then return k fi
  od
end proc:
map(f, [$1..20]); # Robert Israel, May 31 2018

npf[n_]:=Module[{k=1,st=10^(n-1)-1},While[PrimeOmega[st+k]!=n,k++];st+k]; Array[npf,20] (* Harvey P. Dale, Mar 25 2012 *)

from sympy import factorint
def a(n):
  for m in range(10**(n-1), 10**n):
    if sum(factorint(m).values()) == n: return m
print([a(n) for n in range(1, 13)]) # Michael S. Branicky, Feb 10 2021

More terms from Matthew Conroy, May 27 2001

Offset corrected, and a(19)-a(20) from Robert Israel, May 31 2018

A036335 Total number of composite numbers with n digits and n prime factors (counted with multiplicity).

Original entry on oeis.org

0, 31, 225, 1563, 10222, 63030, 374264, 2160300, 12196405, 67724342, 371233523, 2014305995

Offset: 1

Patrick De Geest, Dec 15 1998

Essentially the same as A124033.

			a(1) = 0, since any single-digit number with 1 prime factor is a prime!

Carlos Rivera, Puzzle 25. Composed primes (by G.L. Honaker, Jr.), The Prime Puzzles and Problems Connection. (A related puzzle.)

Cf. A036336, A036337, A036338.

Table[Total[Table[If[CompositeQ[n]&&PrimeOmega[n]==x,1,0],{n,10^(x-1),10^x-1}]],{x,8}] (* The program generates the first 8 terms of the sequence. *) (* Harvey P. Dale, Jun 19 2022 *)

One more term from Naohiro Nomoto, Jul 31 2001

a(9)-a(12) from Ray Chandler, Apr 12 2011

A036338 Composites whose digit length is equal to their number of prime factors (counted with multiplicity).

Original entry on oeis.org

10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 102, 105, 110, 114, 116, 117, 124, 125, 130, 138, 147, 148, 153, 154, 164, 165, 170, 171, 172, 174, 175, 182, 186, 188, 190, 195, 207

Offset: 1

Patrick De Geest, Dec 15 1998

Carlos Rivera, Composed primes, The Prime Puzzles and Problems Connection.

Cf. A036335, A036336, A036337.

1 removed by Sean A. Irvine, Oct 26 2020

A342109 Largest positive integer m with n digits and such that omega(m) = bigomega(m) = n.

Original entry on oeis.org

7, 95, 994, 9982, 99858, 999570, 9998142, 99953490, 999068070, 9592993410

Offset: 1

Bernard Schott, Feb 28 2021

Equivalently: largest n-digit squarefree number with n distinct prime factors (A167050).
Differs from A036337 where length(m) = bigomega(m) = n, when length(m) is the number of digits of m (A055642) and the n prime factors of m are counted with multiplicity (A001222).
Differs from A070843 where length(m) = omega(m) = n, when length(m) is the number of digits of m (A055642) and omega(m) is the number of distinct prime factors dividing m (A001221).
The first index for which these three sequences give three distinct terms is 4:
-> a(4) = 9982 = 2 * 7 * 23 * 31 with omega(9982) = bigomega(9982) = 4.
-> A036337(4) = 9999 = 3 * 3 * 11* 101 with bigomega(9999) = 4 > omega(9999) = 3.
-> A070843(4) = 9996 = 2^2 * 3 * 7^2 *17 with omega(9996) = 4 < bigomega(9996) = 6.
As these terms are the largest n-digit numbers in A167050 that is finite, this sequence is also finite with 10 terms, as for A070843.

			9592993410 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 43 and length(9592993410) = omega(9592993410) = bigomega(9592993410) = 10, so, a(10) = 9592993410 is a term; it is also the largest squarefree number with as many decimal digits as distinct prime factors (A167050).

Subsequence of A167050.
Cf. A001221, A001222, A055642.
Cf. A036337, A070843, A342108.

a={}; For[n=1,n<=10,n++,For[m=10^n-1,m>=10^(n-1),m--,If[PrimeOmega[m]==PrimeNu[m]==n,AppendTo[a, m];Break[]]]]; a (* Stefano Spezia, Mar 06 2021 *)

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A036336 Smallest positive integer with n digits and exactly n prime factors (counted with multiplicity).

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A036335 Total number of composite numbers with n digits and n prime factors (counted with multiplicity).

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A036338 Composites whose digit length is equal to their number of prime factors (counted with multiplicity).

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A342109 Largest positive integer m with n digits and such that omega(m) = bigomega(m) = n.

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