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Search: a003325 -id:a003325
Displaying 1-10 of 135 results found. page 1 2 3 4 5 6 7 8 9 10 ... 14
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A085366 Semiprimes that are the sum of two positive cubes. Common terms of A003325 and A046315. +20
5
9, 35, 65, 91, 133, 217, 341, 407, 559, 737, 793, 1027, 1241, 1339, 1343, 1843, 1853, 2071, 2413, 2771, 2869, 3197, 3383, 3439, 3473, 4097, 4439, 5129, 5833, 6119, 6641, 7471, 7859, 8027, 8587, 9773, 10261, 10649, 10991, 11377, 12679, 12913, 14023 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Sum of two positive cubes x^3 + y^3 such that both x+y and x^2 - x*y + y^2 are primes.
The only square is 9. Also, all terms have a unique representation as a sum of two distinct positive cubes. - Zak Seidov, Jun 02 2011
LINKS
Zak Seidov, Table of n, a(n) for n = 1..1356
EXAMPLE
a(2) = 35 because 3^3 + 2^3 = 5*7.
a(5) = 133 = 5^3 + 2^3 = (5+2)*(5^2 - 5*2 + 2^2) = 7*19.
CROSSREFS
Cf. A003325, A046388, A001358, A085367.
KEYWORD
nonn
AUTHOR
Hugo Pfoertner, Jun 25 2003
STATUS
approved
A267702 Numbers that are the sum of 3 nonzero squares (A000408) and the sum of 2 positive cubes (A003325). +20
4
9, 35, 54, 65, 72, 91, 126, 133, 152, 189, 217, 224, 243, 250, 280, 341, 344, 370, 432, 468, 513, 539, 576, 637, 686, 728, 730, 737, 756, 793, 854, 945, 1001, 1027, 1064, 1072, 1125, 1216, 1241, 1332, 1339, 1358, 1395, 1456, 1458, 1512, 1547, 1674, 1729, 1736, 1755, 1843, 1853 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Intersection of A000408 and A003325.
Sequence focuses on the solutions of equation x^3 + y^3 = a^2 + b^2 + c^2 where x, y, a, b, c > 0.
LINKS
Robert Israel, Table of n, a(n) for n = 1..3649
EXAMPLE
9 is a term because 9 = 1^3 + 2^3 = 1^2 + 2^2 + 2^2.
35 is a term because 35 = 2^3 + 3^3 = 1^2 + 3^2 + 5^2.
54 is a term because 54 = 3^3 + 3^3 = 3^2 + 3^2 + 6^2.
MAPLE
N:= 1000: # to get all terms <= N
S3:= {seq(seq(seq(a^2+b^2+c^2, c = b .. floor(sqrt(N-a^2-b^2))),
b=a .. floor(sqrt((N-a^2)/2))), a = 1 .. floor(sqrt(N/3)))}:
C2:= {seq(seq(a^3+b^3, b = a .. floor((N-a^3)^(1/3))), a = 1 .. floor((N/2)^(1/3)))}:
sort(convert(S3 intersect C2, list)); # Robert Israel, Jan 25 2016
PROG
(PARI) isA000408(n) = {my(a, b); a=1; while(a^2+1<n, b=1; while(b<=a && a^2+b^2<n, if(issquare(n-a^2-b^2), return(1)); b++; ); a++; ); return(0); }
T=thueinit('z^3+1);
isA003325(n)=#select(v->min(v[1], v[2])>0, thue(T, n))>0;
for(n=3, 1e4, if(isA000408(n) && isA003325(n), print1(n, ", ")));
CROSSREFS
Cf. A000408, A003325.
KEYWORD
nonn
AUTHOR
Altug Alkan, Jan 23 2016
STATUS
approved
A273498 Numbers that are, at the same time, the sum of: two positive squares, a positive square and a positive cube, and two positive cubes. In other words, intersection of A000404, A003325 and A055394. +20
2
2, 65, 72, 128, 468, 730, 793, 1241, 1332, 1458, 2000, 2745, 3528, 4097, 4160, 4608, 4825, 5096, 5840, 5913, 6344, 8125, 8192, 9000, 9325, 9928, 12168, 13357, 13498, 14824, 15626, 15633, 15689, 16354, 17640, 18369, 18737, 19721, 19773, 21953, 22681, 27792, 29449 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Numbers n such that n = x^a + y^b where x,y > 0, is soluble for all 1 < a <= b < 4.
Perfect power terms are 128, 8192, 97344, 140625, 524288, 1500625, ...
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
EXAMPLE
793 is a term because 793 = 3^2 + 28^2 = 8^2 + 9^3 = 4^3 + 9^3.
PROG
(PARI) isA003325(n)=for(k=1, sqrtnint(n\2, 3), ispower(n-k^3, 3) && return(1))
isA000404(n) = for( i=1, #n=factor(n)~%4, n[1, i]==3 && n[2, i]%2 && return); n && ( vecmin(n[1, ])==1 || (n[1, 1]==2 && n[2, 1]%2))
isA055394(n) = for(k=1, sqrtnint(n-1, 3), if(issquare(n-k^3), return(1))); 0
lista(nn) = for(n=1, nn, if(isA003325(n) && isA000404(n) && isA055394(n), print1(n, ", ")));
(PARI) isA000404(n)=my(f=factor(n)); for(i=1, #f~, if(f[i, 1]%4==3 && f[i, 2]%2, return(0))); n>1 && (vecmin(f[, 1]%4)==1 || (f[1, 1]==2 && f[1, 2]%2))
isA055394(n) = for(k=1, sqrtnint(n-1, 3), if(issquare(n-k^3), return(1))); 0
list(lim)=my(v=List(), n3, t); lim\=1; for(n=1, sqrtnint(lim-1, 3), n3=n^3; for(m=1, sqrtnint(lim-n3, 3), t=n3+m^3; if(isA000404(t) && isA055394(t), listput(v, t)))); Set(v) \\ Charles R Greathouse IV, May 31 2016
CROSSREFS
Cf. A000404, A003325, A055394.
KEYWORD
nonn,easy
AUTHOR
Altug Alkan, May 23 2016
STATUS
approved
A282872 Numbers in A003325 whose 4th power is the sum of two positive cubes in a nontrivial way. +20
2
2457, 4914, 4977, 8001, 8216, 10773, 15561, 16263, 19656, 39816, 64008, 66339, 80236, 86184, 124336, 124488, 127062, 130104, 132678, 132867, 157248, 166887, 201717, 221832, 238329, 252035, 290871, 307125, 318528, 338821, 358036, 406952, 411021, 420147, 421876 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..10016
FORMULA
A003325 INTERSECT A051387.
CROSSREFS
Cf. A001235, A003325, A051387, A282873.
KEYWORD
nonn
AUTHOR
Chai Wah Wu, Feb 24 2017
STATUS
approved
A145732 Terms in A003325 which are sum of two subsequents terms in A003325. +20
1
217, 341, 1458, 2457, 3059, 12005, 27216, 27683, 39520, 41965, 53128, 72296, 115505, 250559, 251378, 251775, 344728, 425024, 476747, 520000, 578368, 584136, 827099, 843661, 1033676, 1061333, 1185499, 1222039, 1228123, 1299942, 1395000 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..31.
EXAMPLE
a(1)=217=91+126=A003325(15)=A003325(9)+A003325(10), a(2)=341=152 + 189=A003325(20)=A003325(13)+A003325(14), a(3)=1458=728 + 730=A003325(15)=A003325(34)+A003325(35), a(69)=9776312=4887584 + 4888728=A003325(20016)=A003325(12602)+A003325(12603).
CROSSREFS
A003325
KEYWORD
nonn
AUTHOR
Zak Seidov, Oct 17 2008
STATUS
approved
A145755 Terms in A003325 which are sum of three subsequent terms in A003325. +20
1
1674, 3059, 5488, 24696, 29744, 50661, 67375, 69095, 109655, 148608, 164502, 247589, 248976, 407511, 421876, 421911, 684216, 762048, 877058, 884763, 942920, 1265544, 1725542, 1817179, 1975545, 2240000, 3133809, 3819905, 4120389 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..29.
Zak Seidov, Table with full details for 36 first terms in A145755
EXAMPLE
a(1)=1674=539+ 559+ 576, or A003325(60)=A003325(29)+A003325(30)+A003325(31); a(2)=3059=1008+ 1024+ 1024, or A003325(91)=A003325(43)+A003325(44)+A003325(45); a(36)=7968512=2656064+ 2656151+ 2656297, or A003325(17465)=A003325(8385)+A003325(8386)+A003325(8387).
CROSSREFS
Cf. A145732 Terms in A003325 which are sum of two subsequent terms in A003325, A003325 Numbers that are the sum of 2 positive cubes.
KEYWORD
nonn
AUTHOR
Zak Seidov, Oct 17 2008
STATUS
approved
A197719 Position of n-th taxi-cab number A001235(n) in the sequence A003325 of sums of two positive cubes. +20
1
61, 110, 248, 328, 445, 499, 510, 561, 697, 708, 1001, 1004, 1145, 1226, 1309, 1342, 1470, 1563, 1565, 1785, 2012, 2042, 2065, 2259, 2372, 2515, 2540, 2795, 2800, 2806, 2840, 2958, 3076, 3390, 3448, 3779, 3896, 4022, 4031, 4135, 4235, 4320, 4345, 4396, 4412 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..1000
FORMULA
A001235(n) = A003325(a(n)).
EXAMPLE
First taxi-cab number A001235(1)=1729 is A003325(61) hence a(1)=61; 2nd taxi-cab number A001235(2)=4104 is A003325(110) hence a(2)=110.
CROSSREFS
Cf. A001235, A003325.
KEYWORD
nonn
AUTHOR
Zak Seidov, Oct 17 2011
STATUS
approved
A024665 Positions of even numbers in A003325. +20
0
1, 3, 4, 6, 8, 10, 11, 13, 16, 18, 19, 21, 23, 25, 26, 28, 31, 33, 34, 35, 37, 39, 43, 44, 46, 47, 49, 51, 54, 56, 57, 58, 60, 62, 64, 67, 68, 69, 71, 73, 74, 76, 80, 81, 83, 85, 87, 89, 90, 94, 95, 97, 99, 101, 102, 104, 105, 110, 112, 114, 116, 118, 119, 120, 122, 124, 127, 128, 131, 134 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..70.
KEYWORD
nonn
AUTHOR
Clark Kimberling
STATUS
approved
A024666 Positions of odd numbers in A003325. +20
0
2, 5, 7, 9, 12, 14, 15, 17, 20, 22, 24, 27, 29, 30, 32, 36, 38, 40, 41, 42, 45, 48, 50, 52, 53, 55, 59, 61, 63, 65, 66, 70, 72, 75, 77, 78, 79, 82, 84, 86, 88, 91, 92, 93, 96, 98, 100, 103, 106, 107, 108, 109, 111, 113, 115, 117, 121, 123, 125, 126, 129, 130, 132, 133, 138, 140, 142 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..67.
KEYWORD
nonn
AUTHOR
Clark Kimberling
STATUS
approved
A024667 a(n) = position of 2*n^3 in A003325. +20
0
1, 3, 6, 11, 18, 25, 33, 44, 57, 68, 81, 99, 116, 134, 152, 177, 200, 223, 246, 276, 304, 331, 360, 397, 433, 465, 501, 541, 579, 617, 662, 707, 749, 793, 845, 895, 944, 995, 1051, 1105, 1161, 1214, 1279, 1337, 1397, 1456, 1528, 1591, 1657, 1722, 1799, 1870 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..52.
KEYWORD
nonn
AUTHOR
Clark Kimberling
STATUS
approved
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Last modified August 21 21:14 EDT 2024. Contains 375353 sequences. (Running on oeis4.)