A033992 -id:A033992 - cp's OEIS frontend

A143041 10^n-th number divisible by exactly 3 distinct primes.

Original entry on oeis.org

105, 455, 2988, 26128, 258764, 2677258, 28013887, 293553638, 3072062072, 32084334808

Offset: 1

Lekraj Beedassy, Jul 18 2008

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 455, pp 94, Ellipses, Paris 2008.

Cf. A033992.

a(n)=A033992(10^n).

a(8)-a(10) from Donovan Johnson, Mar 30 2010

A179938 Third largest prime factor of numbers that are divisible by at least three different primes (A000977).

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 3, 3, 3, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 3, 2, 5, 2, 3, 3, 2, 2, 2, 2, 2, 3, 2, 5, 3, 3, 3, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2, 2, 3, 5, 2, 2, 3, 3, 3, 2, 2, 2, 2, 3, 5, 2, 2, 3, 2, 3

Offset: 1

Jonathan Vos Post, Jan 12 2011

Third largest prime factor of numbers k such that omega(k) = A001221(k) > 2. The 3rd largest prime factor may equal the second largest. This is not identical to third largest distinct prime factor of numbers that are divisible by at least three different primes. Indices n where a(n) equals 2, 3, 5, 7, 11, 13, 17, 19, 23, ... for the first time are 1, 8, 72, 299, 905, 1718, 3302, 6020, 10330, ... the corresponding numbers from A000977 are 30, 90, 350, 1001, 2431, 4199, 7429, 12673, 20677, ...

			a(1) = 2 because 30 = 2 * 3 * 5 has third largest prime factor 2.
a(2) = 2 because 42 = 2 * 3 * 7 has third largest prime factor 2.
a(3) = 2 because 60 = 2 * 2 * 3 * 5 has both third and fourth largest prime factor 2.
a(8) = 3 because 90 = 2 * 3 * 3 * 5 has both second and third largest prime factor 3.

Cf. A000040, A000977, A001221, A002808, A033992, A007774, A000961, A033993, A051270, A087040, A088739, A179312.

b:= proc(n) option remember; local k;
      if n=1 then 30
      else for k from b(n-1)+1 while
              nops(ifactors(k)[2])<3 do od;
           k
      fi
    end:
a:= n-> sort(map(x-> x[1]$x[2], ifactors(b(n))[2]))[-3]:
seq(a(n), n=1..120);

b[n_] := b[n] = Module[{k}, If[n==1, 30, For[k = b[n-1]+1, PrimeNu[k] < 3, k++]; k]];
a[n_] := (Table[#[[1]], {#[[2]]}]& /@ FactorInteger[b[n]] // Flatten // Sort)[[-3]];
Array[a, 120] (* Jean-François Alcover, Nov 28 2020, after Alois P. Heinz *)

Edited by Alois P. Heinz, Jan 14 2011

A226168 Numbers n such that 1/a + 1/b + 1/c + 1/abc = m /(a+b+c) where a, b and c are the 3 distinct prime divisors of n, and m is a positive integer such that the equation has infinitely many solutions.

Original entry on oeis.org

42, 70, 84, 126, 140, 168, 231, 252, 280, 294, 336, 350, 378, 490, 504, 560, 588, 672, 693, 700, 756, 882, 980, 1008, 1120, 1134, 1176, 1344, 1400, 1512, 1617, 1750, 1764, 1960, 2016, 2058, 2079, 2240, 2268, 2352, 2450, 2541, 2646, 2688, 2800, 3024, 3402, 3430

Offset: 1

Michel Lagneau, May 29 2013

nonn

Subset of A033992.

The value m = 12 is probably unique. We find only 3 primitive values of n: 42 = 2*3*7, 70 = 2*5*7 and 231 = 3*7*11.

			42 is in the sequence because the prime divisors of 42 are 2, 3 and 7 => 1/2 + 1/3 + 1/7 + 1/(2*3*7) = 12/(2+3+7) = 1.

Peter Vandendriessche, Hojoo Lee, Problems in Elementary Number Theory (see problem H67, p. 40). [Via Wayback Machine]

Cf. A033992.

with(numtheory): for n from 2 to 3500 do:x:=factorset(n): n1:=nops(x): if n1=3 then x1:=x[1]:x2:=x[2]:x3:=x[3]:s:=1/x1+ 1/x2+ 1/x3+1/(x1*x2*x3): for m from 1 to 500 do:if s=m/(x1+x2+x3) then printf ( "%d %d \n",n,m):else fi:od:fi:od:

A348882 Numbers that are expressible as the product of the number of distinct prime factors of preceding integers.

Original entry on oeis.org

16, 48, 72, 96, 144, 432, 576, 1296, 2592, 5184, 20736, 32805, 221184, 1555200, 11197440, 55987200, 95551488, 268738560, 302330880, 382205952, 524880000, 671846400, 6718464000, 34012224000, 155520000000, 403107840000, 6856864358400, 107495424000000, 110075314176000

Offset: 1

Metin Sariyar, Nov 02 2021

nonn

			The number of distinct prime factors of the numbers 15, 14, 13, 12, 11, 10 are respectively 2, 2, 1, 2, 1, 2 and 2*2*1*2*1*2 = 16, hence 16 is a term.

Cf. A001221, A000961, A007774, A033992, A033993, A051270, A074969, A176655, A348266.

om[n_] := om[n] = PrimeNu[n]; q[n_] := Module[{m = n, k = n - 1}, While[k > 1 && Divisible[m, om[k]], m /= om[k]; k--]; m == 1]; Select[Range[2, 10^6], q] (* Amiram Eldar, Nov 02 2021 *)

a(13)-a(17) from Amiram Eldar, Nov 02 2021

More terms from David A. Corneth, Nov 02 2021

A357075 Numbers sandwiched between numbers with exactly three distinct prime factors.

Original entry on oeis.org

131, 139, 155, 169, 181, 221, 229, 239, 259, 265, 281, 307, 309, 311, 341, 349, 365, 371, 373, 379, 407, 409, 439, 441, 443, 469, 475, 491, 493, 505, 517, 519, 521, 529, 531, 533, 551, 559, 573, 581, 589, 599, 601, 611, 617, 619, 637, 643, 645, 664, 671, 679, 681, 683

Offset: 1

Tanya Khovanova, Sep 10 2022

Number k such that both k-1 and k+1 are in A033992.

			131 is sandwiched between 130 = 2*5*13 and 132 = 2^2*3*11. Both 130 and 132 have exactly three prime factors. Thus, 131 is in this sequence.

Cf. A033992, A357074, A080569.

Select[Range[1000],Length[FactorInteger[# + 1]] == 3 && Length[FactorInteger[# - 1]] == 3 &]
Mean/@SequencePosition[Table[If[PrimeNu[n]==3,1,0],{n,700}],{1,,1}] (* _Harvey P. Dale, Jul 06 2025 *)

is(n)=omega(n-1)==3 && omega(n+1)==3 \\ Charles R Greathouse IV, Sep 11 2022

list(lim)=my(v=List(),a=3,b,c); forfactored(n=132,lim\1+1, c=#n[2]~; if(c==3 && a==3, listput(v,n[1]-1)); a=b; b=c); Vec(v) \\ Charles R Greathouse IV, Sep 28 2022

from sympy import factorint
def isA033992(n): return len(factorint(n)) == 3
def ok(n): return isA033992(n-1) and isA033992(n+1)
print([k for k in range(700) if ok(k)]) # Michael S. Branicky, Sep 10 2022

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A143041 10^n-th number divisible by exactly 3 distinct primes.

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A179938 Third largest prime factor of numbers that are divisible by at least three different primes (A000977).

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A226168 Numbers n such that 1/a + 1/b + 1/c + 1/a*b*c = m /(a+b+c) where a, b and c are the 3 distinct prime divisors of n, and m is a positive integer such that the equation has infinitely many solutions.

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A348882 Numbers that are expressible as the product of the number of distinct prime factors of preceding integers.

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A357075 Numbers sandwiched between numbers with exactly three distinct prime factors.

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A226168 Numbers n such that 1/a + 1/b + 1/c + 1/abc = m /(a+b+c) where a, b and c are the 3 distinct prime divisors of n, and m is a positive integer such that the equation has infinitely many solutions.