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Multiples of 1729, the Hardy-Ramanujan number.
+10
1
0, 1729, 3458, 5187, 6916, 8645, 10374, 12103, 13832, 15561, 17290, 19019, 20748, 22477, 24206, 25935, 27664, 29393, 31122, 32851, 34580, 36309, 38038, 39767, 41496, 43225, 44954, 46683, 48412, 50141, 51870, 53599, 55328, 57057
COMMENTS
About 1729: "No," said Ramanujan, "It is a very interesting number..."
REFERENCES
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1997, 153.
Powers of 1729, the Hardy-Ramanujan number.
+10
1
1, 1729, 2989441, 5168743489, 8936757492481, 15451653704499649, 26715909255079893121, 46191807102033135206209, 79865634479415290771535361, 138087682014909037743984639169, 238753602203777726259349441123201, 412804978210331688702415183702014529
COMMENTS
About 1729: "No," said Ramanujan, "It is a very interesting number..."
REFERENCES
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1997, 153.
FORMULA
a(n) = 1729^n.
a(n) = 1729*a(n-1) for n > 0.
G.f.: 1/(1 - 1729*x). (End)
Numbers that are the sum of 2 (not-distinct) numbers; nonzero power3 and power5, including repetitions.
+10
1
2, 9, 28, 33, 40, 59, 65, 96, 126, 157, 217, 244, 248, 251, 270, 307, 344, 368, 375, 459, 513, 544, 586, 730, 755, 761, 972, 1001, 1025, 1032, 1032, 1051, 1088, 1149, 1240, 1243, 1332, 1363, 1367, 1536, 1574, 1729, 1753, 1760, 1971, 2024, 2198, 2229, 2355
COMMENTS
40=2^3+2^5, 1032=2^3+4^5 = 1032=10^3+2^5, 1971=12^3+3^5, ...
MATHEMATICA
lst={}; Do[Do[Do[a=x^3+y^5; If[a>n, Break[]]; If[a==n, AppendTo[lst, n]], {y, 5!}], {x, 5!}], {n, 7!}]; lst
CROSSREFS
Cf. A088719, A088677, A088703, A088687, A001235, A024670, A025320, A025319, A025318, A025317, A025316, A025315, A025314, A025313, A024508, A004431, A024507, A155468, A155469, A155470
Numbers that are both Poulet and Proth.
+10
1
1729, 4033, 8321, 12801, 65281, 130561, 348161, 3225601, 8355841, 8384513, 16773121, 40280065, 104988673, 2147418113, 4294901761, 4294967297, 53282340865, 68719214593, 137439477761, 1099510579201, 1911029760001, 2199021158401, 8796097216513, 281474959933441, 9007199388958721, 576460753377165313, 2305843011361177601, 18446744073709551617
COMMENTS
a(1) = 1729 is known as the Hardy-Ramanujan number (see A001235). - Omar E. Pol, Jun 14 2014
Triangle read by rows: T(n,k) = 6*k + 1, n>=0, 0<=k<=(2^n-1).
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1, 1, 7, 1, 7, 13, 19, 1, 7, 13, 19, 25, 31, 37, 43, 1, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 1, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103, 109, 115, 121, 127, 133, 139, 145, 151, 157, 163, 169, 175, 181, 187, 1, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103
COMMENTS
Row n lists the first 2^n terms of A016921, n >= 0. The sum of all terms of the first k rows gives A060867(k). The product of the terms of the third row is equal to the Hardy-Ramanujan number: 1 * 7 * 13 * 19 = 1729.
EXAMPLE
Triangle begins:
1;
1,7;
1,7,13,19;
1,7,13,19,25,31,37,43;
1,7,13,19,25,31,37,43,49,55,61,67,73,79,85,91;
...
Illustration of initial terms in the fourth quadrant of the square grid:
------------------------------------------------------------------------
n a(n) Compact diagram
------------------------------------------------------------------------
. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
0 1 |_|_ |_ _ _ |_ _ _ _ _ _ _ |
1 1 | |_| |_ _ | |_ _ _ _ _ _ | |
2 7 |_ _ _|_ | | |_ _ _ _ _ | | |
3 1 | | | |_| | | |_ _ _ _ | | | |
4 7 | | |_ _ _| | |_ _ _ | | | | |
5 13 | |_ _ _ _ _| |_ _ | | | | | |
6 19 |_ _ _ _ _ _ _|_ | | | | | | |
7 1 | | | | | | | |_| | | | | | | |
8 7 | | | | | | |_ _ _| | | | | | |
9 13 | | | | | |_ _ _ _ _| | | | | |
10 19 | | | | |_ _ _ _ _ _ _| | | | |
11 25 | | | |_ _ _ _ _ _ _ _ _| | | |
12 31 | | |_ _ _ _ _ _ _ _ _ _ _| | |
13 37 | |_ _ _ _ _ _ _ _ _ _ _ _ _| |
14 43 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
a(n) is also the number of cells in the n-th region of the diagram.
MATHEMATICA
With[{rows=7}, Array[Range[1, 6*2^#, 6]&, rows, 0]] (* Paolo Xausa, Sep 26 2023 *)
1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 96, 108, 144, 192, 216, 288, 432, 576, 864, 1728
COMMENTS
A000578(12) = 1728 is the cube of 12.
The number of divisors of 1728 is A000005(1728) = 28. The sum of the divisors of 1728 is A000203(1728) = 5080. The prime factorization of 1728 is 2^6 * 3^3.
1728 + 1 = A001235(1) = A011541(2) = 1729 is the Hardy-Ramanujan number. Three examples related to cellular automata:
1728 is also the number of ON cells after 32 generations of the cellular automata A160239 and A253088. 1728 is also the total number of ON cells around the central ON cell after 24 generations of the cellular automata A160414 and A256530. 1728 is also the total number of ON cells around the central ON cell after 43 generations of the cellular automata A160172 and A255366.
EXAMPLE
a(3) * a(26) = 3 * 576 = 1728.
a(4) * a(25) = 4 * 432 = 1728.
a(5) * a(24) = 6 * 288 = 1728.
PROG
(Sage) divisors(1728);
(PARI) divisors(1728)
CROSSREFS
Cf. A000005, A000203, A000578, A001235, A011541, A133029, A160172, A160239, A160414, A246030, A253088, A255366, A256530.
1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
COMMENTS
Related to the distribution of noncube terms in A273555. What is the distribution of 0's in this sequence as n goes to infinity?
PROG
(PARI) T = thueinit(x^3+1, 1);
isA001235(n) = my(v=thue(T, n)); sum(i=1, #v, v[i][1]>=0 && v[i][2]>=v[i][1])>1;
lista(nn) = for(n=1, nn, if(isA001235(n), print1(ispower(n-sqrtnint(n, 3)^3, 3), ", ")));
1729, 13832, 46683, 110656, 216125, 373464, 593047, 885248, 1260441, 1729000, 2301299, 2987712, 3798613, 4744376, 5835375, 7081984, 8494577, 10083528, 11859211, 13832000, 16012269, 18410392, 21036743, 23901696, 27015625, 30388904, 34031907, 37955008, 42168581, 46683000
COMMENTS
Previous name was: Taxi-cab numbers of form n^3*1729; in other words, taxi-cab numbers of form n^3* A001235(1).
FORMULA
a(n) = n^3 * 1729.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4.
G.f.: x*(1729*x^2 + 6916*x + 1729)/(x - 1)^4. (End)
EXAMPLE
For n = 11, a(11) = 11^3 * 1729 = 2301299 and a(11) = A001235(67).
PROG
(PARI) vector(n, 30, 1729*n^3) \\ Derek Orr, Mar 05 2017
a(n) is the smallest Carmichael number k such that gpf(p-1) = prime(n) for all prime factors p of k.
+10
1
1729, 252601, 1152271, 1615681, 4335241, 172947529, 214852609, 79624621, 178837201, 775368901, 686059921, 985052881, 5781222721, 10277275681, 84350561, 5255104513, 492559141, 74340674101, 9293756581, 1200778753, 129971289169, 2230305949, 851703301, 8714965001, 6693621481
COMMENTS
The first term is the Hardy-Ramanujan number. - Omar E. Pol, Nov 25 2019
EXAMPLE
a(2) = 1729 = (2*3 + 1)(2*2*3 + 1)(2*3*3 + 1).
a(3) = 252601 = (2*2*2*5 + 1)(2*2*3*5 + 1)(2*2*5*5 + 1).
a(4) = 1152271 = (2*3*7 + 1)(2*3*3*7 + 1)(2*3*5*7 + 1).
a(5) = 1615681 = (2*11 + 1)(2*3*3*11 + 1)(2*2*2*2*2*11 + 1).
MATHEMATICA
carmQ[n_] := CompositeQ[n] && Divisible[n - 1, CarmichaelLambda[n]]; gpf[n_] := FactorInteger[n][[-1, 1]]; g[n_] := If[Length[(u = Union[gpf /@ (FactorInteger[n][[;; , 1]] - 1)])] == 1, u[[1]], 1]; m = 5; c = 0; k = 0; v = Table[0, {m}]; While[c < m, k++ If[! carmQ[k], Continue[]]; If[(p = g[k]) > 1, i = PrimePi[p] - 1; If[i <= m && v[[i]] == 0, c++; v[[i]] = k]]]; v (* Amiram Eldar, Oct 08 2019 *)
PROG
(Perl) use ntheory ":all"; sub a { my $p = nth_prime(shift); for(my $k = 1; ; ++$k) { return $k if (is_carmichael($k) and vecall { (factor($_-1))[-1] == $p } factor($k)) } }
for my $n (2..10) { print "a($n) = ", a($n), "\n" }
Numbers k such that k^3 = x^3 + y^3 + z^3, x > y > z >= 0, has at least 2 distinct solutions.
+10
1
18, 36, 41, 46, 54, 58, 60, 72, 75, 76, 81, 82, 84, 87, 88, 90, 92, 96, 100, 108, 114, 116, 120, 123, 126, 132, 134, 138, 140, 142, 144, 145, 150, 152, 156, 159, 160, 162, 164, 168, 170, 171, 174, 176, 178, 180, 184, 185, 186, 189, 190, 192, 198, 200, 201, 202, 203
COMMENTS
This sequence is based on a generalization of Fermat's last theorem with n=3, in which three terms are added. Fermat's Theorem states that there are no solution with only two terms, this sequence shows there are many integers for which there are multiple solutions if three terms are allowed. The sequence is also related to the Taxicab numbers.
EXAMPLE
41 is in the sequence because 41^3 = 33^3 + 32^3 + 6^3 = 40^3 + 17^3 + 2^3.
MATHEMATICA
q[k_] := Count[IntegerPartitions[k^3, {3}, Range[0, k-1]^3], _?(UnsameQ @@ # &)] > 1; Select[Range[200], q] (* Amiram Eldar, Sep 03 2021 *)
PROG
(Python)
from itertools import combinations
from collections import Counter
from sympy import integer_nthroot
def icuberoot(n): return integer_nthroot(n, 3)[0]
def aupto(kmax):
cubes = [i**3 for i in range(kmax+1)]
cands, cubesset = (sum(c) for c in combinations(cubes, 3)), set(cubes)
c = Counter(s for s in cands if s in cubesset)
return sorted(icuberoot(s) for s in c if c[s] >= 2)
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