Displaying 71-80 of 113 results found.
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Numbers that are the sum of 2 cubes in more than 1 way (primitive solutions).
+10
4
1729, 4104, 20683, 39312, 40033, 64232, 65728, 134379, 149389, 171288, 195841, 216027, 327763, 402597, 439101, 443889, 515375, 684019, 704977, 805688, 842751, 920673, 955016, 984067, 994688, 1009736, 1016496, 1073375, 1092728, 1331064
COMMENTS
Nakao's table has more entries because he lists nonprimitive numbers if they are the sum of two cubes in three ways.
Rajesh Bhowmick, Dec 12 2011: The odd number 40533595075161 can be represented as sum of two cubes in just two different ways: (34314)^(3)+(5073)^(3) = (34321)^(3)+(4730)^(3). Here, the cubes are greater than 1, there is no common factor between the odd numbers, there is no common factor between the L.H.S & the R.H.S, the even number is greater than 2, the cubes are in their primitive form, and they are not of the form (27)^(3) or (121)^(3) (which are actually (3)^(9) & (11)^(6)).
Products of three consecutive primes of the form 6n+1 (see A002476).
+10
4
1729, 7657, 21793, 49321, 97051, 175741, 298351, 386389, 559399, 789289, 1089019, 1425829, 1924177, 2665603, 3295273, 3864241, 4631971, 5694079, 6951667, 8103877, 9363547, 10775137, 12307147, 14956219, 18091147, 21243961, 24066037
COMMENTS
Note that a(1)=1729 is the Hardy-Ramanujan number (see taxicab numbers in A001235, A011541).
EXAMPLE
13, 19, 31 are three consecutive primes of the form 6n+1 and 13*19*31 = 7657. - Emeric Deutsch, Jan 21 2009
MAPLE
a := proc (n) if `mod`(ithprime(n), 6) = 1 then ithprime(n) else end if end proc: A := [seq(a(n), n = 1 .. 100)]: seq(A[j]*A[j+1]*A[j+2], j = 1 .. 30); # Emeric Deutsch, Jan 21 2009
MATHEMATICA
Times@@@Partition[Select[Prime[Range[100]], IntegerQ[(#-1)/6]&], 3, 1] (* Harvey P. Dale, Jan 13 2019 *)
Numbers with at least two 3s in their prime signature.
+10
3
216, 1000, 1080, 1512, 2376, 2744, 2808, 3000, 3375, 3672, 4104, 4968, 5400, 6264, 6696, 6750, 7000, 7560, 7992, 8232, 8856, 9000, 9261, 9288, 10152, 10584, 10648, 11000, 11448, 11880, 12744, 13000, 13176, 13500, 13720, 14040, 14472, 15336, 15768, 16632, 17000, 17064, 17576, 17928, 18360, 18522, 19000, 19224, 19656, 20520, 20952, 21000, 21816, 22248, 23000, 23112, 23544, 23625, 24408, 24696, 24840, 25704, 26136, 27000
COMMENTS
In other words, if the canonical prime factorization of a term into prime powers is Product p(i)^e(i), then e(i) = 3 for at least two values of i.
Does not include all numbers with at least two unitary prime power divisors that are cubes (see example section).
The asymptotic density of this sequence is 1 - (1 + Sum_{p prime} ((p-1)/(p^4-p+1))) * Product_{p prime} (1-1/p^3+1/p^4) = 0.0024593812036570543518... . - Amiram Eldar, Jul 22 2024
EXAMPLE
216 = 2^3*3^3, 1000 = 2^3*5^3, 1080 = 2^3*3^3*5, ...
On the other hand, 1728 = 2^6*3^3, 8000 = 2^6*5^3 and 21952 = 2^6*7^3 are not in the sequence.
MATHEMATICA
f[n_]:=Count[Last/@FactorInteger[n], 3]>1; Select[Range[8!], f]
9, 28, 35, 65, 72, 91, 126, 133, 152, 189, 217, 224, 243, 280, 341, 344, 351, 370, 407, 468, 513, 520, 539, 559, 576, 637, 728, 855, 1001, 1008, 1027, 1064, 1125, 1216, 1332, 1339, 1343, 1358, 1395, 1456, 1512, 1547, 1674, 1729, 1736, 1755, 1792, 1843, 1853
COMMENTS
Not a supersequence of A001235; 7094269 is the smallest number that is in A001235 but not in this sequence (see third example below), the next number is 11261376.
EXAMPLE
9 is the sum of two distinct nonzero cubes in exactly one way: 9 = 1^3 + 2^3. 9 is not in A031980 because 1 and 2 are earlier terms of A031980. Therefore 9 is a term of this sequence. 1729 is the sum of two distinct nonzero cubes in exactly two ways: 1729 = 9^3 + 10^3 = 1^3 + 12^3. 1729 is not in A031980 because 1 and 12 are earlier terms of A031980. Therefore 1729 is a term of this sequence. 7094269 is the sum of two distinct nonzero cubes in exactly two ways: 7094269 = 70^3 + 189^3 = 133^3 + 168^3. 7094269 is in A031980 because it not the sum of cubes of two earlier terms of A031980; in the first case 189 and in the second case 133 is not a term of A031980. Therefore 7094269 is not a term of this sequence.
MATHEMATICA
max = 2000; A031980 = {1}; Do[ m = Ceiling[(n - 1)^(1/3)]; s = Select[ A031980, # <= m &]; ls = Length[s]; sumOfCubes = Union[Flatten[ Table[s[[i]]^3 + s[[j]]^3, {i, 1, ls}, {j, i + 1, ls}]]]; If[FreeQ[sumOfCubes, n], AppendTo[ A031980, n] ], {n, 2, max}]; Complement[Range[max], A031980] (* Jean-François Alcover, Sep 03 2013 *)
PROG
(Magma) m:=1853; a:=[]; a2:={}; for n in [1..m] do p:=1; u:= a2 join { x: x in a }; while p in u do p:=p+1; end while; if p gt m then break; end if; a2:=a2 join { x^3 + p^3: x in a | x^3 + p^3 le m }; Append(~a, p); end for; print a2;
CROSSREFS
Cf. A141806, A031980 (smallest number not occurring earlier and not the sum of cubes of two distinct earlier terms), A024670 (sums of cubes of two distinct positive integers), A001235 (sums of two cubes in more than one way).
Products of three consecutive happy primes A035497.
+10
3
1729, 5681, 13547, 56327, 237553, 789289, 1089019, 1560553, 2530217, 4480109, 7703209, 12131401, 18417101, 24119467, 30355679, 38022301, 46039783, 53272619, 57627329, 62188859, 79075651, 112140029, 169169677, 226833263, 271152373, 300157327, 325898231
COMMENTS
Note that a(1)=1729 is the Hardy-Ramanujan number (see taxicab numbers in A001235, A011541).
Products of three distinct happy primes A035497.
+10
3
1729, 2093, 2821, 3059, 4123, 4991, 5681, 7189, 7657, 8827, 9269, 9373, 9919, 10507, 12649, 12719, 12901, 13547, 13699, 14497, 15197, 15617, 16583, 17143, 17549, 17563, 18487, 19513, 21049, 21749, 22211, 22351, 22379, 23621, 23653, 23933, 23959, 25441
COMMENTS
Note that a(1)=1729 is the Hardy-Ramanujan number (see taxicab numbers in A001235, A011541).
Numbers that are the sum of 2 (not-distinct) numbers; nonzero square and cube, including repetitions.
+10
3
2, 5, 9, 10, 12, 17, 17, 24, 26, 28, 31, 33, 36, 37, 43, 44, 50, 52, 57, 63, 65, 65, 68, 72, 73, 76, 80, 82, 89, 89, 91, 100, 101, 108, 108, 113, 122, 126, 127, 128, 129, 129, 134, 141, 145, 145, 148, 150, 152, 161, 164, 170, 171, 174, 177, 185, 189, 196, 197, 204
COMMENTS
5=2^2+1^3, 12=2^2+2^3, 17=3^2+2^3, 31=2^2+3^3, 43=4^2+3^3, 65=1^2+4^3, 65=8^2+1^3, 100=6^2+4^3, ...
MATHEMATICA
lst={}; Do[Do[Do[a=x^2+y^3; If[a>n, Break[]]; If[a==n, AppendTo[lst, n]], {y, 5!}], {x, 5!}], {n, 4*5!}]; lst
CROSSREFS
Cf. A088719, A088677, A088703, A088687, A001235, A024670, A025320, A025319, A025318, A025317, A025316, A025315, A025314, A025313, A024508, A004431, A024507, A155468
Numbers with at least three 3s in their prime signature.
+10
3
27000, 74088, 189000, 287496, 297000, 343000, 351000, 370440, 459000, 474552, 513000, 621000, 783000, 814968, 837000, 963144, 999000, 1029000, 1061208, 1107000, 1157625, 1161000, 1259496, 1269000, 1323000, 1331000, 1407672, 1431000, 1437480, 1481544, 1593000, 1647000, 1704024, 1809000, 1852200, 1917000, 1971000, 2012472, 2079000, 2133000, 2148552
COMMENTS
In other words, if the canonical prime factorization of a term into prime powers is Product p(i)^e(i), then e(i) = 3 for at least three values of i.
The asymptotic density of this sequence is 1 - (1 + s(1) + s(1)^2/2 - s(2)/2) * Product_{p prime} (1-1/p^3+1/p^4) = 0.000018992895371889141564..., where s(k) = Sum_{p prime} ((p-1)/(p^4-p+1))^k. - Amiram Eldar, Jul 22 2024
EXAMPLE
27000 is a term since 27000 = 2^3 * 3^3 * 5^3.
74088 is a term since 74088 = 2^3 * 5^3 * 7^3.
MATHEMATICA
f[n_]:=Count[Last/@FactorInteger[n], 3]>2; Select[Range[10!], f]
PROG
(PARI) is(n) = #select(x -> x == 3, factor(n)[, 2]) > 2; \\ Amiram Eldar, Jul 22 2024
Smallest product of three distinct primes of the form n*k+1.
+10
3
30, 105, 1729, 1105, 13981, 1729, 88537, 50881, 51319, 13981, 137149, 29341, 548497, 88537, 285541, 186337, 3372529, 51319, 18326641, 252601, 1152271, 137149, 1809641, 1366633, 3828001, 548497, 4814857, 645569, 4797703, 285541, 79230049, 4811297
COMMENTS
Note that the Hardy-Ramanujan is the first and the smallest repeated number: a(3)=a(6)=1729.
EXAMPLE
a(1) = 2*3*5 = 30
a(2) = 3*5*7 = 105
a(3) = 7*13*19 = 1729
a(4) = 5*13*17 = 1105
a(5) = 11*31*41 = 13981
PROG
(PARI) a(n)=my(p, q, k=1); while(!isprime(k+=n), ); p=k; while(!isprime(k+=n), ); q=k; while(!isprime(k+=n), ); p*q*k \\ Charles R Greathouse IV, Sep 03 2011
a(n) > n is the smallest integer such that there exist integers n < c < d satisfying n^3 + a(n)^3 = c^3 + d^3.
+10
3
12, 16, 36, 32, 60, 48, 84, 53, 34, 27, 93, 40, 156, 112, 80, 106, 39, 68, 228, 54, 238, 176, 94, 80, 167, 156, 102, 224, 99, 67, 246, 166, 279, 78, 98, 120, 174, 304, 468, 108, 319, 69, 516, 352, 170, 188, 97, 160, 282, 96, 82, 312, 550, 204, 113, 371, 180, 198, 708, 134, 600
COMMENTS
Since the identity n^3 + (12n)^3 = (9n)^3 + (10n)^3 holds, n < a(n) <= 12n.
EXAMPLE
For n = 11, a(11) = 93, since, first, 11^3 + 93^3 = 30^3 + 92^3. Second, for any integral y in the range [12, 92] there does not exist c, d, 11 < c < d < y, satisfying 11^3 + y^3 = c^3 + d^3.
MAPLE
a :=proc(n::integer) local found::boolean; local N, SQ, i;
found:=false; N:=n+1; SQ:={};
while not found do SQ:=SQ union {N^3}; N:=N+1;
for i from n+1 to N-1 do if evalb(N^3+n^3-i^3 in SQ) then
found:=true; end if; end do; end do; N end proc;
MATHEMATICA
a[n_] := a[n] = Module[{found, m, SQ, i}, found = False; m = n+1; SQ = {}; While[!found, SQ = SQ ~Union~ {m^3}; m = m+1; For[i = n+1, i <= m-1, i++, If[MemberQ[SQ, m^3+n^3-i^3], found = True]]]; m];
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