A085366 -id:A085366 - cp's OEIS frontend

A272935 Taxi-cab numbers (A001235) that are the product of exactly three (not necessarily distinct) primes.

1729, 20683, 149389, 195841, 327763, 2418271, 6058747, 7620661, 9443761, 10765603, 13623913, 18406603, 32114143, 68007673, 105997327, 106243219, 166560193, 216226981, 446686147, 584504191, 813357253, 959281759, 1098597061, 1736913439, 2072769211, 2460483307

Offset: 1

Altug Alkan, May 11 2016

nonn

Note that the sum of two positive cubes cannot be prime except 2, obviously. Additionally, if the sum of two positive cubes is a semiprime, then, all corresponding semiprimes have a unique representation as a sum of two distinct positive cubes (see comment section of A085366). Since we know that 1729 is the first member of A001235 and it has three prime divisors, the minimum value of the number of prime divisors of a taxi-cab number must be three. This was the motivation of the definition of this sequence.

			Taxi-cab number 1729 is a term because 1729 = 7*13*19.
Taxi-cab number 20683 is a term because 20683 = 13*37*43.
Taxi-cab number 149389 is a term because 149389 = 31*61*79.

Chai Wah Wu, Table of n, a(n) for n = 1..131

Cf. A001235, A003325, A014612, A085366.

a(7)-a(26) from Chai Wah Wu, May 22 2016

A085367 Semiprimes that can be expressed as the sum or difference of two cubes: intersection of A001358 and A045980.

Original entry on oeis.org

9, 26, 35, 65, 91, 133, 169, 215, 217, 218, 335, 341, 386, 407, 469, 485, 511, 559, 721, 737, 793, 817, 866, 973, 1027, 1115, 1141, 1241, 1261, 1267, 1339, 1343, 1385, 1387, 1538, 1603, 1685, 1727, 1843, 1853, 1981, 2071, 2189, 2402, 2413, 2611, 2743, 2771

Offset: 1

Hugo Pfoertner, Jun 25 2003

nonn

			a(1)=9 because 2^3+1^3=3*3, a(2)=26=3^3-1^3=2*13.
a(5)=91 is the smallest semiprime expressible in two different ways: 91=4^3+3^3=6^3-5^3=7*13.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001358, A045980, A085366.

T=thueinit('z^3+1);
is(n)=bigomega(n)==2 && #thue(T, n)
list(lim)=my(v=List()); forprime(p=2,lim\2, forprime(q=2,min(lim\p,p), if(#thue(T, p*q), listput(v,p*q)))); Set(v) \\ Charles R Greathouse IV, Nov 29 2014

A122732 3-almost primes that are the sum of 2 positive cubes. Sums of 2 positive cubes, with the sums having exactly 3 prime divisors counted with multiplicity.

Original entry on oeis.org

28, 370, 539, 637, 730, 854, 1001, 1358, 1547, 1729, 2198, 2261, 3059, 3887, 3925, 4075, 4123, 4706, 4825, 4921, 5038, 5957, 6293, 6886, 6923, 7075, 7163, 7202, 7657, 8029, 8729, 9262, 9269, 9325, 9331, 10745, 10955, 11458, 12175, 12383, 12845

Offset: 1

Jonathan Vos Post, Sep 23 2006

3-almost prime analog of A085366 Semiprimes that are the sum of two positive cubes. The sum of two positive cubes cannot be prime.

			a(1) = 28 = 2^2 * 7 = 1^3 + 3^3.
a(2) = 370 = 2 * 5 * 37 = 3^3 + 7^3.
a(3) = 539 = 7^2 * 11 = 2^3 + 8^3.
a(4) = 637 = 7^2 * 13 = 5^3 + 8^3.
a(5) = 730 = 2 * 5 * 73 = 1^3 + 9^3.
a(6) = 854 = 2 * 7 * 61 = 5^3 + 9^3.
a(7) = 1001 = 7 * 11 * 13 = 1^3 + 10^3.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000578, A001222, A003325, A014612.

is(n)=bigomega(n)==3 && #select(v->min(v[1], v[2])>0, thue('x^3+1, n))>0 \\ Charles R Greathouse IV, Feb 05 2017

A003325 INTERSECTION A014612. {x = a^3 + b^3 for positive integers a, b, such that A001222(x) = 3}.

More terms from R. J. Mathar, Jan 27 2009

A122733 Least sum of n positive cubes to have exactly n prime factors, with multiplicity.

Original entry on oeis.org

9, 66, 56, 108, 144, 192, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152

Offset: 2

Jonathan Vos Post, Sep 23 2006

nonn

Sequence begins with n = 2 because a(1) is undefined (sum of one positive cube cannot have exactly one prime factor, i.e., be prime).

			a(2) = least semiprime in A003325 = 9 = 3 * 3 = 1^3 + 2^3 = A085366(1).
a(3) = least 3-almost prime in A003072 = 66 = 2 * 3 * 11 = 1^3 + 1^3 + 4^3 = A003072(10).
a(4) = least 4-almost prime in A003327 = 56 = 2^3 * 7 = 1^3 + 1^3 + 3^3 + 3^3 = A003327(10).
a(5) = least 5-almost prime in A003328 = 108 = 2^2 * 3^3 = 4^3 + 3^3 + 2^3 + 2^3 + 1^3 = A003328(25).
a(6) = least 6-almost prime in A003329 = 144 = 2^4 * 3^2 = 5^3 + 2^3 + 2^3 + 1^3 + 1^3 + 1^3 = A003329(46).

Cf. A000578, A001222, A003072, A003325, A003327, A003328, A003329, A085366.

isSumcPosC := proc(n,c,minb)
        local nrt ;
        if c = 1 then nrt := iroot(n,3) ; if nrt^3 = n  and n>= minb then true; else false; end if;
        else for b from minb do if b^3 > n then return false; end if; if isSumcPosC(n-b^3,c-1,b) then return true; end if; end do: end if;
end proc:
A122733 := proc(n)
        for a from 1 do if numtheory[bigomega](a) = n then if isSumcPosC(a,n,1) then return a; end if; end if;
        end do:
end proc:
for n from 2 do print(A122733(n)) ; end do: # R. J. Mathar, Dec 22 2010

a(n) = Min{x = (c_1)^3 + (c_2)^3 + ... + (c_n)^3 such that omega(x) = A001222(x) = n}.

a(17) from Giovanni Resta, Jun 13 2016

a(18)-a(21) more terms from R. J. Mathar, Jan 31 2017

cp's OEIS Frontend

A272935 Taxi-cab numbers (A001235) that are the product of exactly three (not necessarily distinct) primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A085367 Semiprimes that can be expressed as the sum or difference of two cubes: intersection of A001358 and A045980.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A122732 3-almost primes that are the sum of 2 positive cubes. Sums of 2 positive cubes, with the sums having exactly 3 prime divisors counted with multiplicity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A122733 Least sum of n positive cubes to have exactly n prime factors, with multiplicity.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

Extensions