A112801 -id:A112801 - cp's OEIS frontend

A243750 Indices of records in A112801 = number of partitions of 2n-1 into three numbers with two distinct prime factors each.

14, 17, 20, 23, 26, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 82, 83, 86, 88, 89, 92, 94, 95, 97, 98, 101, 103, 104, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 137, 140, 143, 146, 149, 152, 155

Offset: 1

M. F. Hasler, Jun 09 2014

nonn

It appears that starting from a(64)=104 on, the sequence consists exactly in the numbers (larger than 103) congruent to 2 mod 3.

m=0;for(i=1,1000,m<(m=max(A112801(i),m))&&print1(i","))

For n >= 64, a(n) = 3n - 88. (Conjectured.)

A243751 Range of A112801 (number of partitions of 2n-1 into three summands with two distinct prime factors each).

Original entry on oeis.org

0, 1, 2, 4, 7, 8, 11, 13, 15, 16, 18, 23, 26, 30, 31, 33, 40, 45, 51, 53, 56, 62, 66, 76, 79, 82, 88, 94, 96, 105, 111, 124, 127, 132, 141, 145, 148, 164, 166, 170, 180, 187, 204, 206, 208, 228, 229, 237, 253, 260, 275, 278, 285, 300, 303, 314, 328, 338

Offset: 1

M. F. Hasler, Jun 09 2014

nonn

A007774 Numbers that are divisible by exactly 2 different primes; numbers n with omega(n) = A001221(n) = 2.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 104, 106, 108, 111, 112, 115, 116, 117, 118

Offset: 1

Luke Pebody (ltp1000(AT)hermes.cam.ac.uk)

nonn

Every group of order p^a * q^b is solvable (Burnside, 1904). - Franz Vrabec, Sep 14 2008

Characteristic function for a(n): floor(omega(n)/2) * floor(2/omega(n)) where omega(n) is the number of distinct prime factors of n. - Wesley Ivan Hurt, Jan 10 2013

			20 is a term because 20 = 2^2*5 with two distinct prime divisors 2, 5.

T. D. Noe, Table of n, a(n) for n = 1..1000
W. Burnside, On groups of order p^alpha q^beta, Proc. London Math. Soc. (2) 1 (1904), 388-392.
Hans Montanus and Ron Westdijk, Cellular Automation and Binomials, Green Blue Mathematics (2022), p. 90.

Subsequence of A085736; A256617 is a subsequence.
Row 2 of A125666.
Cf. A001358 (products of two primes), A014612 (products of three primes), A014613 (products of four primes), A014614 (products of five primes), where the primes are not necessarily distinct.
Cf. A006881, A046386, A046387, A067885 (product of exactly 2, 4, 5, 6 distinct primes respectively).
Cf. A000040, A112801.

a007774 n = a007774_list !! (n-1)
a007774_list = filter ((== 2) . a001221) [1..]
-- Reinhard Zumkeller, Aug 02 2012

with(numtheory,factorset):f := proc(n) if nops(factorset(n))=2 then RETURN(n) fi; end;

Select[Range[0,6! ],Length[FactorInteger[ # ]]==2&] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2010 *)
Select[Range[120],PrimeNu[#]==2&] (* Harvey P. Dale, Jun 03 2020 *)

is(n)=omega(n)==2 \\ Charles R Greathouse IV, Apr 01 2013

from sympy import primefactors
A007774_list = [n for n in range(1,10**5) if len(primefactors(n)) == 2] # Chai Wah Wu, Aug 23 2021

Expanded definition. - N. J. A. Sloane, Aug 22 2021

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.

A112800 Number of ways of representing 2n-1 as sum of three integers with 1 distinct prime factor.

Original entry on oeis.org

0, 0, 0, 1, 3, 4, 6, 8, 9, 10, 12, 14, 14, 16, 18, 18, 20, 23, 25, 26, 28, 30, 30, 32, 32, 34, 37, 36, 40, 43, 42, 44, 46, 46, 46, 50, 51, 53, 59, 57, 57, 61, 62, 62, 66, 68, 69, 71, 72, 71, 73, 76, 74, 81, 81, 78, 87, 90, 87, 91, 93, 90, 94, 97, 94, 100, 107, 103, 114, 115

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(4) = 1 because the only partition into nontrivial prime powers of (2*4)-1 = 7 is 7 = 2 + 2 + 3.
a(5) = 3 because the 3 partitions into nontrivial prime powers of (2*5)-1 = 9 are 9 = 2 + 2 + 5 = 2 + 3 + 4 = 3 + 3 + 3. The middle one of those partitions has "4" which is not a prime, but is a power of a prime.
a(6) = 4 because the 4 partitions into nontrivial prime powers of (2*6)-1 = 11 are 11 = 2 + 2 + 7 = 2 + 4 + 5 = 3 + 3 + 5.
a(7) = 6 because the 6 partitions into nontrivial prime powers of (2*7)-1 = 13 are 13 = 2 + 2 + 9 = 2 + 3 + 8 = 2 + 4 + 7 = 3 + 3 + 7 = 3 + 5 + 5 = 4 + 4 + 5.

R. J. Mathar, Table of n, a(n) for n = 1..1655
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. A000040, A112799, A112801, A112802.

isA000961 := proc(n)
    if n = 1 then
        return true;
    end if;
    numtheory[factorset](n) ;
    if nops(%) = 1 then
        true;
    else
        false;
    end if;
end proc:
A000961 := proc(n)
    option remember;
    local a;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA000961(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
A112800 := proc(n)
    local a,i,j,p,q,r,n2;
    n2 := 2*n-1 ;
    a := 0 ;
    for i from 2 do
        p := A000961(i) ;
        if 3*p > n2 then
            return a;
        else
            for j from i do
                q := A000961(j) ;
                r := n2-p-q ;
                if r < q then
                    break;
                end if;
                if isA000961(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,A112800(n));
end do: # R. J. Mathar, Jun 09 2014

Number of ways of representing 2n-1 as sum of three primes (A000040) or powers of primes (A000961 except 1). Number of ways of representing 2n-1 as a + b + c where omega(a) = omega(b) = omega(c) = 1.

A112799 Least odd number such that all greater odd numbers can be represented as sum of three integers with n distinct prime factors (conjectured).

Original entry on oeis.org

5, 29, 283, 4409, 95539, 2579897, 88149143

Offset: 1

Jonathan Vos Post and Ray Chandler, Sep 19 2005

Strangely, the first 5 values of this sequence are all primes. 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(5) = 95539; all odd numbers up to 200000 checked, no larger term found that could not be represented as sum of three integers each with 5 distinct prime factors.
a(1)-a(3): checked odd numbers < 10^5. a(4): checked odd numbers < 10^6. a(5): checked odd numbers < 3*10^6. a(6): checked odd numbers < 3*10^7. a(7): checked odd numbers between 8*10^7 and 2*10^8. [From Donovan Johnson, Feb 04 2009]

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. A112800, A112801, A112802.

a(6)-a(7) from Donovan Johnson, Feb 04 2009

cp's OEIS Frontend

A243750 Indices of records in A112801 = number of partitions of 2n-1 into three numbers with two distinct prime factors each.

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A243751 Range of A112801 (number of partitions of 2n-1 into three summands with two distinct prime factors each).

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A007774 Numbers that are divisible by exactly 2 different primes; numbers n with omega(n) = A001221(n) = 2.

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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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A112800 Number of ways of representing 2n-1 as sum of three integers with 1 distinct prime factor.

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A112799 Least odd number such that all greater odd numbers can be represented as sum of three integers with n distinct prime factors (conjectured).

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