A033992 -id:A033992 - cp's OEIS frontend

A364265 The first term in a chain of at least 3 consecutive numbers each with exactly 6 distinct prime factors (i.e., belonging to A074969).

323567034, 431684330, 468780388, 481098980, 577922904, 639336984, 715008644, 720990620, 726167154, 735965384, 769385252, 808810638, 822981560, 831034918, 839075510, 847765554, 879549670, 895723268, 902976710, 903293468, 904796814, 918520420, 940737005, 944087484, 982059364

Offset: 1

R. J. Mathar, Jul 16 2023

nonn

To distinguish this from A259349: "Numbers n with exactly k distinct prime factors" means numbers with A001221(n) = omega(n) = k, which specifies that in the prime factorization n = Product_{i>=1} p_i^(e_i), e_i >= 1, the exponents are ignored, and only the size of the set of the (distinct) p_i is considered. In A259349, the numbers n are products of k distinct primes, which means in the prime factorization of n, all exponents e_i are equal to 1. (If all exponents e_i = 1, the n are squarefree, i.e., in A005117.) Rephrased: the n which are products of k distinct primes have A001221(n) = omega(n) = A001222(n) = bigomega(n) = k, whereas the n which have exactly k distinct prime factors are the superset of (weaker) requirement A001221(n) = omega(n) = k. - R. J. Mathar, Jul 18 2023

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A259349 (requires squarefree). Subsequence of A273879.
Cf. A364266 (5 distinct factors).
See also A001221, A001222, A005117.
Numbers divisible by d distinct primes: A246655 (d=1), A007774 (d=2), A033992 (d=3), A033993 (d=4), A051270 (d=5), A074969 (d=6), A176655 (d=7), A348072 (d=8), A348073 (d=9).

omega := proc(n)
    nops(numtheory[factorset](n)) ;
end proc:
for k from 1 do
    if omega(k) = 6 then
        if omega(k+1) = 6 then
            if omega(k+2) = 6 then
                print(k) ;
            end if;
        end if;
    end if;
end do:

upto(n) = {my(res = List(), streak = 0); forfactored(i = 2, n, if(#i[2]~ == 6, streak++; if(streak >= 3, listput(res, i[1] - 2)), streak = 0)); res} \\ David A. Corneth, Jul 18 2023

a(1) = A138206(3).

{k: A001221(k) = A001221(k+1) = A001221(k+2) = 6}.

More terms from David A. Corneth, Jul 18 2023

A365795 Numbers k such that omega(k) = 3 and its prime factors satisfy the equation p_1 + p_2 = p_3.

Original entry on oeis.org

30, 60, 70, 90, 120, 140, 150, 180, 240, 270, 280, 286, 300, 350, 360, 450, 480, 490, 540, 560, 572, 600, 646, 700, 720, 750, 810, 900, 960, 980, 1080, 1120, 1144, 1200, 1292, 1350, 1400, 1440, 1500, 1620, 1750, 1798, 1800, 1920, 1960, 2160, 2240, 2250, 2288, 2400, 2430, 2450

Offset: 1

Stefano Spezia, Sep 19 2023

nonn

The lower prime factor p_1 is equal to 2 and the other two are twin primes: p_3 - p_2 = 2.

			60 is a term since 60 = 2^2*3*5 and 2 + 3 = 5.
286 is a term since 286 = 2*11*13 and 2 + 11 = 13.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A001221, A001359, A006512, A071142.

Subsequence of A033992 and of A071140.

Select[Range[2500],PrimeNu[#]==3&&Part[First/@FactorInteger[#],1]+Part[First/@FactorInteger[#],2]==Part[First/@FactorInteger[#],3]&]

isok(k) = if (omega(k)==3, my(f=factor(k)[,1]); f[1] + f[2] == f[3]); \\ Michel Marcus, Sep 19 2023

A225228 Numbers with prime signatures (1,1,1) or (2,2,1) or (3,2,2).

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 180, 182, 186, 190, 195, 222, 230, 231, 238, 246, 252, 255, 258, 266, 273, 282, 285, 286, 290, 300, 310, 318, 322, 345, 354, 357, 366, 370, 374, 385, 396, 399, 402, 406, 410, 418, 426, 429

Offset: 1

Reinhard Zumkeller, May 03 2013

nonn

Union of A007304, A179643 and A179695; subsequence of A033992;
A001221(a(n)) = 3 and A051903(a(n)) <= A051904(a(n)) + 1 and A001222(a(n)) = 3 or 5 or 7;
A050326(a(n)) = 5.

			A007304(1) = 2*3*5 = 30, A206778(30,1..8)=[1,2,3,5,6,10,15,30]:
A050326(30) = #{30, 15*2, 10*3, 6*5, 5*3*2} = 5;
A179643(1) = 2^2*3^2*5 = 180, A206778(180,1..8)=[1,2,3,5,6,10,15,30]:
A050326(180) = #{30*6, 30*3*2, 15*6*2, 10*6*3, 6*5*3*2} = 5;
A179695(1) = 2^3*3^2*5^2 = 1800, A206778(1800,1..8)=[1,2,3,5,6,10,15,30]:
A050326(1800) = #{30*10*6, 30*6*5*2, 30*10*3*2, 15*10*6*2, 10*6*5*3*2} = 5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A124010.

a225228 n = a225228_list !! (n-1)
a225228_list = filter f [1..] where
   f x = length es == 3 && sum es `elem` [3,5,7] &&
                           maximum es - minimum es <= 1
         where es = a124010_row x

is(n)=my(f=vecsort(factor(n)[,2]~)); f==[1,1,1] || f==[1,2,2] || f==[2,2,3] \\ Charles R Greathouse IV, Jul 28 2016

a(n) ~ 2n log n / (log log n)^2. - Charles R Greathouse IV, Jul 28 2016

A304636 Numbers n with prime omicron 3, meaning A304465(n) = 3.

Original entry on oeis.org

8, 27, 30, 42, 66, 70, 78, 102, 105, 110, 114, 125, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 343, 345, 354, 357, 360, 366, 370, 374, 385, 399, 402, 406, 410, 418, 426

Offset: 1

Gus Wiseman, May 15 2018

nonn

If n > 1 is not a prime number, we have A056239(n) >= Omega(n) >= omega(n) >= A071625(n) >= ... >= omicron(n) > 1 where Omega = A001222, omega = A001221, and omicron = A304465.

			This is a list of normalized factorizations (see A112798) of selected entries:
     8: {1,1,1}
    30: {1,2,3}
   360: {1,1,1,2,2,3}
   720: {1,1,1,1,2,2,3}
   900: {1,1,2,2,3,3}
  1440: {1,1,1,1,1,2,2,3}
  2160: {1,1,1,1,2,2,2,3}
  2880: {1,1,1,1,1,1,2,2,3}
  4320: {1,1,1,1,1,2,2,2,3}
  5760: {1,1,1,1,1,1,1,2,2,3}
  8640: {1,1,1,1,1,1,2,2,2,3}
Starting with A112798(1801800) and repeatedly taking the multiset of multiplicities we have {1,1,1,2,2,3,3,4,5,6} -> {1,1,1,2,2,3} -> {1,2,3} -> {1,1,1} -> {3}, so 1801800 belongs to the sequence.

Cf. A001221, A001222, A005117, A007916, A014612, A033992, A071625, A112798, A181819, A182850, A182857, A304464, A304465, A304634, A304647.

Join@@Position[Table[Switch[n,1,0,?PrimeQ,1,,NestWhile[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Length[#]>1&]//First],{n,200}],3]

A112802 Number of ways of representing 2n-1 as sum of three integers with 3 distinct prime factors.

Original entry on oeis.org

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, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 2

Offset: 1

Jonathan Vos Post and Ray Chandler, Sep 19 2005

nonn

Meng proves a remarkable generalization of the Goldbach-Vinogradov classical result that every sufficiently large odd integer N can be partitioned as the sum of three primes N = p1 + p2 + p3. The new proof is that every sufficiently large odd integer N can be partitioned as the sum of three integers N = a + b + c where each of a, b, c has k distinct prime factors for the same k.

			a(83) = 1 because the only partition into three integers each with 3 distinct prime factors of (2*83)-1 = 165 is 165 = 30 + 30 + 105 = (2*3*5) + (2*3*5) + (3*5*7). Coincidentally, 165 itself has three distinct prime factors 165 = 3 * 5 * 11.
a(89) = 1 because the only partition into three integers each with 3 distinct prime factors of (2*89)-1 = 177 = 30 + 42 + 105 = (2*3*5) + (2*3*7) + (3*5*7).
a(107) = 2 because the two partitions into three integers each with 3 distinct prime factors of (2*107)-1 = 213 are 213 = 30 + 78 + 105 = 42 + 66 + 105.

R. J. Mathar, Table of n, a(n) for n = 1..1290
Xianmeng Meng, On sums of three integers with a fixed number of prime factors, Journal of Number Theory, Vol. 114 (2005), pp. 37-65.

Cf. A000961, A007304, A112799, A112800, A112801.

isA033992 := proc(n)
    numtheory[factorset](n) ;
    if nops(%) = 3 then
        true;
    else
        false;
    end if;
end proc:
A033992 := proc(n)
    option remember;
    local a;
    if n = 1 then
        30;
    else
        for a from procname(n-1)+1 do
            if isA033992(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
A112802 := proc(n)
    local a,i,j,p,q,r,n2;
    n2 := 2*n-1 ;
    a := 0 ;
    for i from 1 do
        p := A033992(i) ;
        if 3*p > n2 then
            return a;
        else
            for j from i do
                q := A033992(j) ;
                r := n2-p-q ;
                if r < q then
                    break;
                end if;
                if isA033992(r) then
                    a := a+1 ;
                end if;
            end do:
        end if ;
    end do:
end proc:
for n from 1 do
    printf("%d %d\n",n,A112802(n));
end do: # R. J. Mathar, Jun 09 2014

Number of ways of representing 2n-1 as sum of three members of A033992. Number of ways of representing 2n-1 as a + b + c where omega(a) = omega(b) = omega(c) = 3, where omega=A001221.

A136152 Composites one larger than a prime and with exactly three distinct prime factors.

Original entry on oeis.org

30, 42, 60, 84, 90, 102, 110, 114, 132, 138, 140, 150, 168, 174, 180, 182, 198, 228, 230, 234, 240, 252, 258, 264, 270, 282, 294, 308, 312, 318, 348, 350, 354, 360, 374, 380, 402, 410, 434, 440, 444, 450, 468, 480, 492, 504, 522, 558, 564, 572, 588, 594, 600

Offset: 1

Enoch Haga, Dec 16 2007

			a(0)=30 because 30 follows the prime 29 and has three factors 2, 3 and 5.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A136151, A136153, A136154, A136155.

isA008864 := proc(n) if n -prevprime(n) = 1 then true ; else false ; fi ; end: isA033992 := proc(n) if nops(numtheory[factorset](n)) = 3 then true ; else false ; fi ; end: isA136152 := proc(n) isA008864(n) and isA033992(n) ; end: for n from 1 do p := ithprime(n) ; if isA136152(p+1) then print(p+1) ; fi ; od: # R. J. Mathar, Feb 20 2008

Select[Prime[Range[110]]+1,PrimeNu[#]==3&] (* Harvey P. Dale, Apr 08 2012 *)

Find primes followed by N with exactly three prime factors, without repetition.

Equals A008864 INTERSECT A033992. - R. J. Mathar, Feb 20 2008

Edited by R. J. Mathar, Feb 20 2008

A287483 Irregular triangle T(n,k) read by rows: row n lists numbers m with A002110(n) <= m < A002110(n+1) such that omega(m) = n.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 14, 15, 21, 22, 26, 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 210, 330, 390, 462, 510, 546, 570, 690, 714, 770, 798, 858, 870, 910, 930, 966, 1110, 1122, 1155, 1190, 1218, 1230, 1254, 1290

Offset: 0

Michael De Vlieger, May 25 2017

The primorial A002110(n) is the smallest squarefree number with n prime factors. Here the n-th row of the triangle is a list of squarefree numbers with n prime factors greater than and including A002110(n) but less than A002110(n+1).

A287484(n) gives row lengths.

			The sequence begins with 1 as it is equal to A002110(0) and has 0 prime factors. The first primes less than 6 come next, followed by the first squarefree semiprimes (A006881) less than 30 and the smallest terms of A033992 less than 210, etc.
Triangle begins:
n   Row n
0:   1;
1:   2,  3,  5;
2:   6, 10, 14, 15, 21,  22,  26;
3:  30, 42, 66, 70, 78, 102, 105, 110, 114, 130, ..., 195;
    ...
In each row n, the squarefree terms m must have omega(m) = n.

Michael De Vlieger, Table of n, a(n) for n = 0..10309 (rows 0 <= n <= 9).
Eric Weisstein's World of Mathematics, Primorial
Eric Weisstein's World of Mathematics, Squarefree

Cf. A001221, A002110, A005117, A006881, A033992, A287484, A287691.

Table[Select[Range[#, Prime[n + 1] # - 1] &@ Product[Prime@ i, {i, n}], And[SquareFreeQ@ #, PrimeOmega@ # == n] &], {n, 0, 4}] // Flatten

Edited by N. J. A. Sloane, Jun 05 2017

A064040 Integers whose number of distinct prime divisors is prime.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105

Offset: 1

Lior Manor, Aug 23 2001

For all terms below 210 this sequence and A024619 are identical.

			210 = 2*3*5*7 has 4 prime factors, hence it is not here, but it is part of A024619.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A001221, A024619, A000977, A033992, A007774, A033993, A051270, A063989.

q:= n-> isprime(nops(ifactors(n)[2])):
select(q, [$1..210])[];  # Alois P. Heinz, Apr 18 2024

Select[Range[200], PrimeQ[PrimeNu[#]] &] (* Paolo Xausa, Mar 28 2024 *)

n=0; for (m=1, 10^9, if (isprime(omega(m)), write("b064040.txt", n++, " ", m); if (n==1000, break))) \\ Harry J. Smith, Sep 06 2009

is(n)=isprime(omega(n)) \\ Charles R Greathouse IV, Sep 18 2015

Edited by Charles R Greathouse IV, Mar 18 2010

Name edited by Michel Marcus, Oct 16 2023

A137563 Fibonacci numbers with three distinct prime divisors.

Original entry on oeis.org

610, 987, 2584, 10946, 3524578, 9227465, 24157817, 39088169, 63245986, 1836311903, 7778742049, 20365011074, 591286729879, 4052739537881, 17167680177565, 44945570212853, 61305790721611591, 420196140727489673, 1500520536206896083277, 6356306993006846248183

Offset: 1

Parthasarathy Nambi, Apr 25 2008

nonn

			The distinct prime divisors of the Fibonacci number 610 are 2, 5 and 61.
The distinct prime divisors of the Fibonacci number 44945570212853 are 269, 116849 and 1429913.

Michel Marcus and Amiram Eldar, Table of n, a(n) for n = 1..83 (terms 1..80 from Michel Marcus)
Ron Knott, Fibonacci Numbers.
Ron Knott, Prime divisors of Fibonacci numbers.

Cf. A053409, A114841.
Intersection of A033992 and A000045. - Michel Marcus, Mar 24 2018
Column k=3 of A303218.

P1:=List([1..110],n->Fibonacci(n));;
P2:=List([1..Length(P1)],i->Filtered(DivisorsInt(P1[i]),IsPrime));;
a:=List(Filtered([1..Length(P2)],i->Length(P2[i])=3),j->P1[j]); # Muniru A Asiru, Mar 25 2018

with(numtheory): with(combinat): a:=proc(n) if nops(factorset(fibonacci(n)))= 3 then fibonacci(n) else end if end proc: seq(a(n),n=1..110); # Emeric Deutsch, May 18 2008

Select[Array[Fibonacci, 120], PrimeNu@ # == 3 &] (* Michael De Vlieger, Apr 10 2018 *)

lista(nn) = for (n=1, nn, if (omega(f=fibonacci(n))==3, print1(f, ", "))); \\ Michel Marcus, Mar 24 2018

a(n) = A000045(A114841(n)). - Michel Marcus, Mar 24 2018

More terms from Emeric Deutsch, May 18 2008

A306908 Numbers k with exactly three distinct prime factors and such that phi(k) is a square.

Original entry on oeis.org

60, 114, 126, 170, 204, 240, 273, 285, 315, 364, 370, 380, 438, 444, 456, 468, 504, 540, 680, 816, 825, 902, 960, 969, 978, 1010, 1026, 1071, 1095, 1100, 1134, 1212, 1258, 1292, 1358, 1456, 1460, 1480, 1500, 1520, 1729, 1746, 1752, 1776, 1824, 1836, 1872

Offset: 1

Bernard Schott, Mar 16 2019

nonn

This sequence is the intersection of A033992 and A039770.

The integers with only one prime factor and whose totient is a square are in A002496 and A054755, the integers with two prime factors and whose totient is a square are in A324745, A324746 and A324747.

			1st family: 273 = 3 * 7 * 13 and phi(273) = 12^2.
2nd family: 816 = 2^4 * 3 * 17 and phi(816) = 16^2.
3rd family: 6975 = 3^2 * 5^2 * 31 and phi(6975) = 60^2.

Robert Israel, Table of n, a(n) for n = 1..10000
Bernard Schott, Subfamilies and subsequences

Intersection of A033992 and A039770.

Cf. A002496, A054755 (only one prime factor), A324745, A324746, A324747 (two prime factors).

filter:= n -> issqr(numtheory:-phi(n)) and nops
(numtheory:-factorset(n))=3:
select(filter, [$1..2000]); # after Robert Israel in A324745

Select[Range[2000], And[PrimeNu@ # == 3, IntegerQ@ Sqrt@ EulerPhi@ #] &] (* Michael De Vlieger, Mar 31 2019 *)

isok(n) = (omega(n)==3) && issquare(eulerphi(n)); \\ Michel Marcus, Mar 19 2019

1st family: The primitive terms are p*q*r with p,q,r primes and phi(p*q*r) = (p-1)*(q-1)*(r-1) = m^2. These primitives generate the entire family formed by the numbers k = p^(2s+1) * q^(2t+1) * r^(2u+1) with s,t,u >= 0, and phi(k) = (p^s * q^t * r^u * m)^2.
2nd family: The primitive terms are p^2 * q * r with p,q,r primes and phi(p^2 * q * r) = p*(p-1)*(q-1)*(r-1) = m^2. These primitives generate the entire family formed by the numbers k = p^(2s) * q^(2t+1) * r^(2u+1) with s >= 1, t,u >= 0, and phi(k) = (p^(s-1) * q^t * r^u * m)^2.
3rd family: The primitive terms are p^2 * q^2 * r with p,q,r primes and phi(p^2 * q^2 * r) = p*q*(p-1)*(q-1)*(r-1) = m^2. These primitives generate the entire family formed by the numbers k = p^(2s) * q^(2t) * r^(2u+1) with s,t> = 1, u >= 0, and phi(k) = (p^(s-1) * q^(t-1) * r^u * m)^2.

cp's OEIS Frontend

A364265 The first term in a chain of at least 3 consecutive numbers each with exactly 6 distinct prime factors (i.e., belonging to A074969).

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A365795 Numbers k such that omega(k) = 3 and its prime factors satisfy the equation p_1 + p_2 = p_3.

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A225228 Numbers with prime signatures (1,1,1) or (2,2,1) or (3,2,2).

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A304636 Numbers n with prime omicron 3, meaning A304465(n) = 3.

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A112802 Number of ways of representing 2n-1 as sum of three integers with 3 distinct prime factors.

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Maple

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A136152 Composites one larger than a prime and with exactly three distinct prime factors.

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Maple

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A287483 Irregular triangle T(n,k) read by rows: row n lists numbers m with A002110(n) <= m < A002110(n+1) such that omega(m) = n.

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A064040 Integers whose number of distinct prime divisors is prime.