Search: a003325 -id:a003325
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A267702
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Numbers that are the sum of 3 nonzero squares (A000408) and the sum of 2 positive cubes (A003325).
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+20
4
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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
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OFFSET
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1,1
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COMMENTS
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Sequence focuses on the solutions of equation x^3 + y^3 = a^2 + b^2 + c^2 where x, y, a, b, c > 0.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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
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PROG
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(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, ", ")));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A273498
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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.
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+20
2
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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
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OFFSET
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1,1
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COMMENTS
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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, ...
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LINKS
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EXAMPLE
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793 is a term because 793 = 3^2 + 28^2 = 8^2 + 9^3 = 4^3 + 9^3.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A282872
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Numbers in A003325 whose 4th power is the sum of two positive cubes in a nontrivial way.
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+20
2
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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
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OFFSET
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1,1
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LINKS
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FORMULA
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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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
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OFFSET
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1,1
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LINKS
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EXAMPLE
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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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
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OFFSET
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1,1
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LINKS
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EXAMPLE
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A197719
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Position of n-th taxi-cab number A001235(n) in the sequence A003325 of sums of two positive cubes.
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+20
1
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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
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OFFSET
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1,1
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LINKS
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FORMULA
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EXAMPLE
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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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
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OFFSET
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1,2
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LINKS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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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;
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listen;
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OFFSET
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1,1
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LINKS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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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;
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listen;
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OFFSET
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1,2
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LINKS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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