A011541 -id:A011541 - cp's OEIS frontend

A123075 Smallest number expressible as the sum of three 4th powers in exactly n ways.

1, 2673, 811538, 5978882, 137149922, 292965218, 779888018, 5745705602, 105760443698, 49511121842, 1872511131218, 281539574498, 17673688436978, 17873514984962, 253930825318898, 7865870969138, 768054952462322

Offset: 1

Tom Womack (tom(AT)womack.net), Sep 19 2006

nonn

Cf. A085559, A011541, A122540.

A133029 Divisors of 1729, the 2nd taxicab number (also called the Hardy-Ramanujan number).

Original entry on oeis.org

1, 7, 13, 19, 91, 133, 247, 1729

Offset: 1

Omar E. Pol, Oct 23 2007, Nov 07 2007

Note that 19 * 91 = 1729. For products of n-th prime and n-th prime written backwards, see A133019.

			7 * 247 = 1729 and 13 * 133 = 1729.

Index entries for sequences related to divisors of numbers

Cf. A000005, A018487. Taxicab numbers: A011541.

Divisors[1729] (* Alonso del Arte, Oct 11 2019 *)

divisors(1729) \\ Charles R Greathouse IV, Jun 21 2017

A230561 Smallest number that is the sum of two positive n-th powers in >= n ways.

Original entry on oeis.org

2, 50, 87539319

Offset: 1

Jonathan Sondow, Oct 23 2013

Guy, 2004: "Euler knew that 635318657 = 133^4 + 134^4 = 59^4 + 158^4, and Leech showed this to be the smallest example. No one knows of three such equal sums." Thus no one knows whether a(4) exists, which requires four such equal sums.

a(4) > 10^21 (if it exists). There is no number <= 10^21 that is the sum of two positive 4th powers in >= three ways. - Donovan Johnson, Jan 07 2014

			2 = 1^1 + 1^1.
50 = 1^2 + 7^2 = 5^2 + 5^2.
87539319 = 167^3 + 436^3 = 228^3 + 423^3 = 255^3 + 414^3.

R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, D1.

Cf. A048610, A011541 for a(2), a(3).

Cf. also A016078, A230477.

a(n) >= A016078(n) for n > 1, with equality at least for n = 2, and inequality at least for n = 3.

A343708 Numbers that are the sum of two positive cubes in exactly two ways.

Original entry on oeis.org

1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, 65728, 110656, 110808, 134379, 149389, 165464, 171288, 195841, 216027, 216125, 262656, 314496, 320264, 327763, 373464, 402597, 439101, 443889, 513000, 513856, 515375, 525824, 558441, 593047, 684019, 704977, 805688, 842751, 885248, 886464

Offset: 1

David Consiglio, Jr., Apr 26 2021

This sequence differs from A001235 at term 455 because 87539319 = 167^3 + 436^3 = 228^3 + 423^3 = 255^3 + 414^3 = A011541(3). Thus, this term is not in this sequence but is in A001235.

			13832 is in this sequence because 13832 = 2^3 + 24^3 = 18^3 + 20^3.

David Consiglio, Jr., Table of n, a(n) for n = 1..1000

Cf. A001235, A025285, A025396, A338667, A344804.

Select[Range@70000,Length@Select[PowersRepresentations[#,2,3],FreeQ[#,0]&]==2&] (* Giorgos Kalogeropoulos, Apr 26 2021 *)

from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**3 for x in range(1,1000)]#n
for pos in cwr(power_terms,2):#m
    tot = sum(pos)
    keep[tot] += 1
rets = sorted([k for k,v in keep.items() if v == 2])#s
for x in range(len(rets)):
    print(rets[x])

A154728 Products of three consecutive primes of the form 6n+1 (see A002476).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jan 18 2009, Jan 21 2009

Note that a(1)=1729 is the Hardy-Ramanujan number (see taxicab numbers in A001235, A011541).

			13, 19, 31 are three consecutive primes of the form 6n+1 and 13*19*31 = 7657. - _Emeric Deutsch_, Jan 21 2009

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001235, A002476, A011541, A154716, A154717, A154729.

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

Times@@@Partition[Select[Prime[Range[100]],IntegerQ[(#-1)/6]&],3,1] (* Harvey P. Dale, Jan 13 2019 *)

Extended by Emeric Deutsch, Jan 21 2009

A180088 Primes which are the sum of three distinct positive cubes in two or more distinct ways.

Original entry on oeis.org

1009, 4229, 4447, 4733, 6301, 7561, 10657, 12377, 12979, 13103, 13859, 14561, 15569, 15667, 17207, 20663, 20747, 20899, 21673, 22511, 24137, 24499, 25999, 27793, 27917, 28001, 28493, 28729, 29917, 31123, 32579, 32833, 32957, 33119

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 09 2010

nonn

1^3+2^3+10^3=1009=4^3+6^3+9^3

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A011541, A024974, A025400, A122723

lst1=Sort@Select[Flatten[Table[a^3+b^3+c^3,{a,1,66},{b,a-1,1,-1},{c,b-1,1,-1}]],PrimeQ[ # ]&]; lst={};Do[If[lst1[[n]]==lst1[[n+1]],AppendTo[lst,lst1[[n]]]],{n,Length[lst1]-1}];lst

A080642 Cubefree taxicab numbers: the smallest cubefree number that is the sum of 2 positive cubes in n ways.

Original entry on oeis.org

2, 1729, 15170835645, 1801049058342701083

Offset: 1

Stuart Gascoigne (Stuart.G(AT)scoigne.com), Feb 28 2003

A necessary condition for the sum to be cubefree is that each pair of cubes be relatively prime.

If the sequence is infinite, then the Mordell-Weil rank of the elliptic curve of rational solutions to x^3 + y^3 = a(n) tends to infinity with n. In fact, the rank exceeds C*log(n) for some constant C>0 (see Silverman p. 339). - Jonathan Sondow, Oct 22 2013

			2 = 1^3 + 1^3,
1729 = 12^3 + 1^3 = 10^3 + 9^3,
15170835645 = 2468^3 + 517^3 = 2456^3 + 709^3 = 2152^3 + 1733^3,
1801049058342701083 = 1216500^3 + 92227^3 = 1216102^3 + 136635^3 = 1207602^3 + 341995^3 = 1165884^3 + 600259^3.

Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), Article #A38, 39 pp.; arXiv:1902.11283 [math.NT], 2019.
J. H. Silverman, Taxicabs and Sums of Two Cubes, Amer. Math. Monthly, 100 (1993), 331-340.

Cf. A011541.

a(n) >= A011541(n) for n > 0, with equality for n = 1, 2 (only?). - Jonathan Sondow, Oct 25 2013

Name clarified by Jeppe Stig Nielsen, Aug 21 2020

A154716 Products of three consecutive happy primes A035497.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jan 18 2009

Note that a(1) = 1729 is the Hardy-Ramanujan number (see taxicab numbers in A001235, A011541).

Cf. A001235, A011541, A035497, A154717, A154728, A154729.

happyQ[n_, b_] := NestWhile[Total[IntegerDigits[#, b]^2] &, n, UnsameQ, All] == 1; Times @@@ Partition[Select[Prime[Range[150]], happyQ[#, 10] &], 3, 1] (* Amiram Eldar, Jan 17 2025 *)

a(5)-a(27) from Nathaniel Johnston, Apr 30 2011

A154717 Products of three distinct happy primes A035497.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jan 18 2009

Note that a(1)=1729 is the Hardy-Ramanujan number (see taxicab numbers in A001235, A011541).

Cf. A001235, A011541, A035497, A154716, A154728, A154729.

a(5) - a(38) from Nathaniel Johnston, Apr 30 2011

A180089 Semiprimes which are the sum of three distinct positive cubes in two or more distinct ways.

Original entry on oeis.org

1366, 1457, 1793, 1945, 3599, 4105, 5435, 7379, 8315, 9017, 10261, 10963, 11773, 12706, 13957, 15163, 15371, 15553, 15751, 15758, 16271, 17263, 17354, 17947, 18649, 19027, 19369, 19657, 19729, 19774, 19781, 19907, 21026, 21167, 22411

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 09 2010

nonn

2^3+3^3+11^3=5^3+8^3+9^3=1366=2*683, 1^3+5^3+11^3=6^3+8^3+9^3=1457=31*47,..

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A011541, A024974, A025400, A122723, A180088

f[n_]:=Last/@FactorInteger[n]=={1,1}; lst={};Do[Do[Do[If[f[p=a^3+b^3+c^3],AppendTo[lst,p]],{c,b-1,1,-1}],{b,a-1,1,-1}],{a,55}];lst1=Sort@lst; lst={};Do[If[lst1[[n]]==lst1[[n+1]],AppendTo[lst,lst1[[n]]]],{n,Length[lst1]-1}];lst

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A123075 Smallest number expressible as the sum of three 4th powers in exactly n ways.

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A133029 Divisors of 1729, the 2nd taxicab number (also called the Hardy-Ramanujan number).

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A230561 Smallest number that is the sum of two positive n-th powers in >= n ways.

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A343708 Numbers that are the sum of two positive cubes in exactly two ways.

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A154728 Products of three consecutive primes of the form 6n+1 (see A002476).

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A180088 Primes which are the sum of three distinct positive cubes in two or more distinct ways.

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A080642 Cubefree taxicab numbers: the smallest cubefree number that is the sum of 2 positive cubes in n ways.

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A154716 Products of three consecutive happy primes A035497.

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A154717 Products of three distinct happy primes A035497.

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