A007774 -id:A007774 - cp's OEIS frontend

A079275 Number of divisors of n that are semiprimes with distinct factors.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 3, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 3, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 3, 0, 1, 1, 0, 1, 3, 0, 1, 1, 3, 0, 1, 0, 1, 1, 1, 1, 3, 0, 1, 0, 1, 0, 3, 1, 1, 1, 1, 0, 3, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 3, 0, 1, 3

Offset: 1

Reinhard Zumkeller, Feb 07 2003

Number of pairs of prime factors of n, (p,q), such that p < q. For example, the prime factors of 30 are 2, 3 and 5, so we have the ordered pairs (2,3), (2,5) and (3,5). - Wesley Ivan Hurt, Sep 14 2020

Inverse Möbius transform of A280710(n). - Wesley Ivan Hurt, Jul 06 2025

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000005, A000961, A001221, A001358, A006881, A007774, A033992, A033993, A280710.

A079275 := proc(n)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if A001221(d) = 2 and A001222(d) = 2 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A079275(n),n=1..40) ; # R. J. Mathar, Jan 18 2021

f[n_]:=Module[{c=PrimeNu[n]},(c(c-1))/2]; Array[f,110] (* Harvey P. Dale, Oct 05 2011 *)

a(n) = sumdiv(n, d, (bigomega(d)==2) && (omega(d)==2)); \\ Michel Marcus, Sep 15 2020

a(n) = binomial(omega(n),2) \\ David A. Corneth, Sep 15 2020

a(A000961(n)) = 0; a(A007774(n)) = 1; a(A033992(n)) = 3; a(A033993(n)) = 6.
a(n) = omega(n)*(omega(n)-1)/2, where omega(n) is the number of distinct prime factors of n.
a(n) = Sum_{p|n, q|n, p,q prime, pWesley Ivan Hurt, Sep 14 2020
a(n) = Sum_{d|n} A280710(d). - Wesley Ivan Hurt, Jul 06 2025

A100367 Even numbers with two prime factors, not counting multiplicity.

Original entry on oeis.org

6, 10, 12, 14, 18, 20, 22, 24, 26, 28, 34, 36, 38, 40, 44, 46, 48, 50, 52, 54, 56, 58, 62, 68, 72, 74, 76, 80, 82, 86, 88, 92, 94, 96, 98, 100, 104, 106, 108, 112, 116, 118, 122, 124, 134, 136, 142, 144, 146, 148, 152, 158, 160, 162, 164, 166, 172, 176, 178, 184, 188

Offset: 1

Labos Elemer, Nov 22 2004

nonn

This is a proper subset of A098902. E.g., 210 is not here, but it is there.

Numbers of the form 2^k*p^h where k > 0, h > 0, and p is an odd prime. [corrected by Lei Zhou, Oct 29 2018]

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001221.
Intersection of A007774 and A098902.
The even terms of A336101.
A100368 is a subsequence.

Select[Range[2, 200, 2], PrimeNu@ # == 2 &] (* Michael De Vlieger, Oct 31 2018 *)

isok(n) = !(n%2) && (omega(n) == 2); \\ Michel Marcus, Feb 02 2018

A162142 Numbers that are the cube of a product of two distinct primes (p^3*q^3).

Original entry on oeis.org

216, 1000, 2744, 3375, 9261, 10648, 17576, 35937, 39304, 42875, 54872, 59319, 97336, 132651, 166375, 185193, 195112, 238328, 274625, 328509, 405224, 456533, 551368, 614125, 636056, 658503, 753571, 804357, 830584, 857375, 1191016, 1367631, 1520875, 1643032

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 25 2009

nonn

Subset of A046306, of A000578, and of A007774. - R. J. Mathar, Jun 27 2009

			216=2^3*3^3. 1000=2^3*5^3. 2744=2^3*7^3.

Cf. A000578, A006881, A007774, A046306.

Cf. A085541, A085966.

fQ[n_]:=Last/@FactorInteger[n]=={3,3}; lst={};Do[If[fQ[n],AppendTo[lst, n]],{n,6*9!}];lst
With[{nn=30},Select[Union[(Times@@@Subsets[Prime[Range[nn]],{2}])^3],#<= (2Prime[ nn])^3&]](* Harvey P. Dale, May 27 2024 *)

from math import isqrt
from sympy import primepi, primerange
def A162142(n):
    def f(x): return int(n+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m**3 # Chai Wah Wu, Dec 09 2024

a(n) = (A006881(n))^3 = A000578(A006881(n)). - R. J. Mathar, Jun 27 2009

Sum_{n>=1} 1/a(n) = (P(3)^2 - P(6))/2 = (A085541^2 - A085966)/2 = 0.006735..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

Definition rephrased by R. J. Mathar, Jun 27 2009

A067003 Number of numbers <= n with same number of distinct prime factors as n.

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 5, 6, 7, 2, 8, 3, 9, 4, 5, 10, 11, 6, 12, 7, 8, 9, 13, 10, 14, 11, 15, 12, 16, 1, 17, 18, 13, 14, 15, 16, 19, 17, 18, 19, 20, 2, 21, 20, 21, 22, 22, 23, 23, 24, 25, 26, 24, 27, 28, 29, 30, 31, 25, 3, 26, 32, 33, 27, 34, 4, 28, 35, 36, 5, 29, 37, 30, 38, 39, 40, 41

Offset: 1

Henry Bottomley, Dec 21 2001

			a(11)=8 since 2,3,4,5,7,8,9,11 each have one distinct prime factor. a(12)=3 since 6,10,12 each have two distinct prime factors.
From _Gus Wiseman_, Dec 28 2018: (Start)
Column n lists the a(n) positive integers less than or equal to n with the same number of distinct prime factors as n:
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20
  ---------------------------------------------------------------------
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20
        2  3  4     5  7  8  6   9   10  11  12  14  13  16  15  17  18
           2  3     4  5  7      8   6   9   10  12  11  13  14  16  15
              2     3  4  5      7       8   6   10  9   11  12  13  14
                    2  3  4      5       7       6   8   9   10  11  12
                       2  3      4       5           7   8   6   9   10
                          2      3       4           5   7       8   6
                                 2       3           4   5       7
                                         2           3   4       5
                                                     2   3       4
                                                         2       3
                                                                 2
(End)

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Positions of 1's are A002110.
Cf. A001221, A008479, A058933, A067004. Inverse of A000961, A007774, A033992, A033993, A051270 etc.
Cf. A000010, A006049, A061142, A294277, A294278, A302242, A322837, A322841.

Table[Length[Select[Range[n],PrimeNu[#]==PrimeNu[n]&]],{n,100}] (* Gus Wiseman, Dec 28 2018 *)

a(n) = my(nb = #factor(n)~); sum(k=1, n, #factor(k)~ == nb); \\ Michel Marcus, Jul 13 2019

a(A002110(n)) = 1.

A070915 Numbers having at most two distinct prime factors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71

Offset: 1

Reinhard Zumkeller, May 20 2002

nonn

			88 and 77 are terms, as 88=11*2^3 and 77=11*7, but 66 is not a term, as 66 = 11*3*2.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Disjoint union of A000961 and A007774.
A003586 is a subsequence.
Cf. A001221.
Complement of A000977.

Select[Range[80],PrimeNu[#]<3&] (* Harvey P. Dale, Sep 19 2012 *)

is(n)=omega(n)<3 \\ Charles R Greathouse IV, Oct 16 2015

A001221(a(n)) <= 2.

a(n) ~ n log n/log log n. - Charles R Greathouse IV, Oct 16 2015

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).

Original entry on oeis.org

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

A112801 Number of ways of representing 2n-1 as sum of three integers, each with two distinct prime factors.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 2, 2, 4, 4, 4, 8, 7, 8, 11, 11, 13, 15, 16, 18, 23, 23, 26, 30, 31, 33, 40, 40, 45, 51, 53, 56, 62, 66, 66, 76, 79, 82, 88, 94, 96, 105, 111, 111, 124, 127, 132, 141, 145, 148, 164, 166, 170, 180, 187, 187, 206, 204, 208

Offset: 1

Jonathan Vos Post and Ray Chandler, Sep 19 2005

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.

See A243751 for the range of this sequence, and A243750 for the indices of record values. - M. F. Hasler, Jun 09 2014

			a(14) = 1 because the only partition into three integers each with 2 distinct prime factors of (2*14)-1 = 27 is 27 = 6 + 6 + 15 = (2*3) + (2*3) + (3*5).
a(16) = 1 because the only partition into three integers each with 2 distinct prime factors of (2*16)-1 = 31 is 31 = 6 + 10 + 15 = (2*3) + (2*5) + (3*5).
a(17) = 2 because the two partitions into three integers each with 2 distinct prime factors of (2*17)-1 = 33 are 33 = 6 + 6 + 21 = 6 + 12 + 15.

R. J. Mathar, Table of n, a(n) for n = 1..1020
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, A112799, A112800, A112802.

A112801(n)={n=n*2-1;sum(a=6,n\3,if(omega(a)==2,sum(b=a,(n-a)\2, omega(b)==2 && omega(n-a-b)==2)))} \\ M. F. Hasler, Jun 09 2014

Number of ways of representing 2n-1 as a + b + c where a<=b<=c are elements of A007774.

A323055 Numbers with exactly two distinct exponents in their prime factorization, or two distinct parts in their prime signature.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200

Offset: 1

Gus Wiseman, Jan 03 2019

nonn

The first term is A006939(2) = 12.
First differs from A059404 in lacking 360, whose prime signature has three distinct parts.
Positions of 2's in A071625.
Numbers k such that A001221(A181819(k)) = 2.
The asymptotic density of this sequence is (6/Pi^2) * Sum_{n>=2, n squarefree} 1/((n-1)*psi(n)) = 0.3611398..., where psi is the Dedekind psi function (A001615) (Sanna, 2020). - Amiram Eldar, Oct 18 2020

			3000 = 2^3 * 3^1 * 5^3 has two distinct exponents {1, 3}, so belongs to the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carlo Sanna, On the number of distinct exponents in the prime factorization of an integer, Proceedings - Mathematical Sciences, Indian Academy of Sciences, Vol. 130, No. 1 (2020), Article 27, alternative link.

One distinct exponent: A062770 or A072774.
Two distinct exponents: this sequence.
Three distinct exponents: A323024.
Four distinct exponents: A323025.
Five distinct exponents: A323056.
Cf. A001221, A001222, A001615, A006939, A007774, A059404, A071625, A118914, A181819, A323014, A323022.

isA323055 := proc(n)
    local eset;
    eset := {};
    for pf in ifactors(n)[2] do
        eset := eset union {pf[2]} ;
    end do:
    simplify(nops(eset) = 2 ) ;
end proc:
for n from 12 to 1000 do
    if isA323055(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 09 2019

Select[Range[100],Length[Union[Last/@FactorInteger[#]]]==2&]

A082997 a(n) = card{ x <= n : omega(x) = 2 }.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 5, 6, 6, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 12, 12, 12, 13, 14, 15, 16, 16, 17, 18, 19, 19, 19, 19, 20, 21, 22, 22, 23, 23, 24, 25, 26, 26, 27, 28, 29, 30, 31, 31, 31, 31, 32, 33, 33, 34, 34, 34, 35, 36, 36, 36, 37, 37, 38, 39, 40, 41

Offset: 1

Benoit Cloitre, May 30 2003

nonn

G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, p. 203, Publications de l'Institut Cartan, 1990.

Daniel Suteu, Table of n, a(n) for n = 1..10000

Cf. A001221, A007774, A025528, A072000, A082998, A346621.

Partial sums of A215480.

a:= proc(n) option remember; `if`(n=0, 0,
      a(n-1)+`if`(nops(ifactors(n)[2])=2, 1, 0))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 23 2021

a[n_] := Count[PrimeNu[Range[n]], 2];
Array[a, 100] (* Jean-François Alcover, Mar 02 2022 *)

PARI
```
a(n)=sum(i=1,n,if(omega(i)-2,0,1))
```

a(n) = my(s = sqrtint(n), p = 2, j = 1, count = 0); while(p <= s, my(r = nextprime(p+1)); my(t = p); while (t <= n, my(w = n\t); if(r > w, break); count += primepi(w) - j; my(r2 = r); while(r2 <= w, my(u = t*r2*r2); if(u > n, break); while (u <= n, count += 1; u *= r2); r2 = nextprime(r2+1)); t *= p); p = r; j += 1); count; \\ Daniel Suteu, Jul 21 2021

from sympy import factorint
from itertools import accumulate
def cond(n): return int(len(factorint(n))==2)
def aupto(nn): return list(accumulate(map(cond, range(1, nn+1))))
print(aupto(77)) # Michael S. Branicky, Jul 21 2021

a(n) ~ (n/log(n))*log(log(n)).

a(A007774(n)) = n. - Daniel Suteu, Jul 21 2021

A284318 Triangle read by rows in which row n lists divisors d of n such that n divides d^n.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 7, 2, 4, 8, 3, 9, 10, 11, 6, 12, 13, 14, 15, 2, 4, 8, 16, 17, 6, 18, 19, 10, 20, 21, 22, 23, 6, 12, 24, 5, 25, 26, 3, 9, 27, 14, 28, 29, 30, 31, 2, 4, 8, 16, 32, 33, 34, 35, 6, 12, 18, 36, 37, 38, 39, 10, 20, 40, 41, 42, 43, 22, 44, 15, 45, 46, 47, 6, 12, 24, 48, 7, 49, 10, 50

Offset: 1

Juri-Stepan Gerasimov, Mar 25 2017

Row n lists divisors of n that are divisible by A007947(n). - Robert Israel, Apr 27 2017

			Triangle begins:
    1;
    2;
    3;
    2, 4;
    5;
    6;
    7;
    2, 4, 8;
    3, 9;
    10;
    11;
    6, 12;
    13;
    14;
    15;
    2, 4, 8, 16.

Robert Israel, Table of n, a(n) for n = 1..10002 (rows 1 to 5250, flattened)

Cf. A000961 (1 together with k such that k divides p^k for some prime divisor p of k), A005361 (row length), A007774 (m such that m divides s^m for some semiprime divisor s of m), A007947 (smallest u such that u^n|n and n|u, or divisor k such that A000005(k) = 2^A001221(n)), A057723 (row sums), A066503 (difference between largest x and smallest y such that x^n|n, n|x, y^n|n and n|y).

Cf. A003557, A027750.

[[u: u in [1..n] | Denominator(n/u) eq 1 and Denominator(u^n/n) eq 1]: n in [1..50]];

f:= proc(n) local r;
    r:= convert(numtheory:-factorset(n),`*`);
    op(sort(convert(map(`*`, numtheory:-divisors(n/r),r),list)))
end proc:
map(f, [$1..100]); # Robert Israel, Apr 27 2017

Flatten[Table[Select[Range[n], Divisible[n, #] && Divisible[#^n, n] &], {n, 50}]] (* Indranil Ghosh, Mar 25 2017 *)

for(n=1, 50, for(i=1, n, if(n%i==0 & (i^n)%n==0, print1(i,", "););); print();); \\ Indranil Ghosh, Mar 25 2017

for n in range(1, 51):
    print([i for i in range(1, n + 1) if n%i==0 and (i**n)%n==0]) # Indranil Ghosh, Mar 25 2017

T(n,k) = A007947(n) * A027750(A003557(n), k). - Robert Israel, Apr 27 2017

cp's OEIS Frontend

A079275 Number of divisors of n that are semiprimes with distinct factors.

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A100367 Even numbers with two prime factors, not counting multiplicity.

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A162142 Numbers that are the cube of a product of two distinct primes (p^3*q^3).

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A067003 Number of numbers <= n with same number of distinct prime factors as n.

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A070915 Numbers having at most two distinct prime factors.

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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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A112801 Number of ways of representing 2n-1 as sum of three integers, each with two distinct prime factors.

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