A001235 -id:A001235 - cp's OEIS frontend

A352222 First members C of two non-consecutive numbers such that the sums of their cubes are equal to centered cube numbers and to at least one other sum of two cubes, i.e., A = B^3 + (B+1)^3 = C^3 + D^3 = E^3 + F^3.

153, 206, 292, 710, 1307, 1623, 4230, 7170, 19275, 20331, 20063, 30486, 55572, 101135, 199614, 238806, 317427, 3145700, 4450334, 10163157, 146173525, 808182534

Offset: 1

Vladimir Pletser, Mar 07 2022

Numbers C such that A = B^3 + (B+1)^3 = C^3 + D^3 = E^3 + F^3 with C <> (D +- 1), E <> (F +- 1), E > C > B, C > |D| and E > |F|, where A = A352220(n), B = A352221(n), C = a(n) (this sequence), D = A352223(n), E = A352224(n) and F = A352225(n).
Terms are ordered according to increasing order of A352220(n) or A352221(n).
Subsequence of A352135.

			153 belongs to the sequence as 153^3 + 18^3 = 121^3 + 122^3 = 369^3 + (-360)^3 = 3587409.

A. Grinstein, Ramanujan and 1729, University of Melbourne Dept. of Math and Statistics Newsletter: Issue 3, 1998.
Eric Weisstein's World of Mathematics, Centered Cube Number

Cf. A005898, A001235, A272885, A352133, A352134, A352135, A352136, A352220, A352221, A352223, A352224, A352225.

a(n)^3 + A352223(n)^3 = A352221(n)^3 + (A352221(n) + 1)^3 = A352224(n)^3 + A352225(n)^3 = A352220(n).

a(21) from Chai Wah Wu, Mar 17 2022

a(22) from Bert Dobbelaere, Apr 18 2022

A008917 Numbers that are the sum of 3 positive cubes in more than one way.

Original entry on oeis.org

251, 1009, 1366, 1457, 1459, 1520, 1730, 1737, 1756, 1763, 1793, 1854, 1945, 2008, 2072, 2241, 2414, 2456, 2458, 2729, 2736, 3060, 3391, 3457, 3592, 3599, 3655, 3745, 3926, 4105, 4112, 4131, 4168, 4229, 4320, 4376, 4402, 4437, 4447

Offset: 1

N. J. A. Sloane and J. H. Conway

Of course reordering the terms does not count.

A025456(a(n)) > 1. [Reinhard Zumkeller, Apr 23 2009]

			a(2) = 1009 = 1^3+2^3+10^3 = 4^3+6^3+9^3.

T. D. Noe and Christian N. K. Anderson, Table of n, a(n) for n = 1..10000 (first 1000 terms are from T. D. Noe)
Christian N. K. Anderson, Decomposition of first 10000 terms into multiple cube triples.

Cf. A001235, A003072, A024796, A025396, A025398, A025406, A309762.

Select[Range[4450], 1 < Length @ Cases[PowersRepresentations[#, 3, 3], {?Positive, ?Positive, ?Positive}] &]  (* _Jean-François Alcover, Apr 04 2011 *)

is(n)=k=ceil((n-2)^(1/3)); d=0; for(a=1, k, for(b=a, k, for(c=b, k, if(a^3+b^3+c^3==n, d++)))); d
n=3; while(n<5000, if(is(n)>1, print1(n, ", ")); n++) \\ Derek Orr, Aug 27 2015

A078128 Number of ways to write n as sum of cubes > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 1, 0, 0, 1, 0, 2, 1, 0, 2, 0, 0, 1, 0, 2, 1, 0, 2, 0, 0, 1, 0, 2, 1

Offset: 1

Reinhard Zumkeller, Nov 19 2002

nonn

a(A078129(n))=0; a(A078130(n))=1; a(A078131(n))>0;

Conjecture (lower bound): for all k exists b(k) such that a(n)>k for n>b(k); see b(0)=A078129(83)=154 and b(1)=A078130(63)=218.

			a(160)=4: 160 = 20*2^3 = 4^3+12*2^3 = 2*4^3+4*2^3 = 5^3+3^3+2^3.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of cubes
Eric Weisstein's World of Mathematics, Cubic Number.

Cf. A000578, A003108, A001235, A078132, A078133, A078134.

nmax = 105; CoefficientList[Series[Product[1/(1 - x^(k^3)), {k, 1, nmax}], {x, 0, nmax}], x] // Differences (* Jean-François Alcover, Mar 01 2019, after Vaclav Kotesovec *)

a(n) = 1/n*Sum_{k=1..n} (b(k)-1)*a(n-k), a(0) = 1, where b(k) is sum of cube divisors of k. - Vladeta Jovovic, Nov 20 2002
From Vaclav Kotesovec, Jan 05 2017: (Start)
a(n) = A003108(n) - A003108(n-1).
a(n) ~ exp(4*(Gamma(1/3) * Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) * (Gamma(1/3) * Zeta(4/3))^(3/2) / (8 * 3^(5/2) * Pi^2 * n^2).
(End)

A086119 Numbers of the form p^3 + q^3, p, q primes.

Original entry on oeis.org

16, 35, 54, 133, 152, 250, 351, 370, 468, 686, 1339, 1358, 1456, 1674, 2205, 2224, 2322, 2540, 2662, 3528, 4394, 4921, 4940, 5038, 5256, 6244, 6867, 6886, 6984, 7110, 7202, 8190, 9056, 9826, 11772, 12175, 12194, 12292, 12510, 13498, 13718, 14364

Offset: 1

Hollie L. Buchanan II, Jul 11 2003

nonn

			133 belongs to the sequence because it can be written as 2^3 + 5^3.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A001235, A003325, A045636, A045699, A086120, A086121.

sumList[x_List, y_List] := Module[{t = {}}, Do[t = Union[t, x[[i]] + y], {i, Length[x]}];  t]; nn = 10; Select[sumList[Prime[Range[nn]]^3, Prime[Range[nn]]^3], # < Prime[nn]^3 &]

More terms from Alexander Adamchuk, Nov 10 2006

A176297 Numbers with at least one 3 in their prime signature.

Original entry on oeis.org

8, 24, 27, 40, 54, 56, 72, 88, 104, 108, 120, 125, 135, 136, 152, 168, 184, 189, 200, 216, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 360, 375, 376, 378, 392, 408, 424, 432, 440, 456, 459, 472, 488, 500, 504, 513, 520, 536, 540, 552, 568, 584, 594, 600, 616, 621, 632, 648, 664, 675, 680, 686, 696, 702, 712

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 07 2010

nonn

That is, if n = p1^e1 p2^e2 ... pr^er for distinct primes p1, p2,..., pr, then one of the exponents must be 3 for n to be in this sequence.

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/p^3 + 1/p^4) = 0.0952910730... - Amiram Eldar, Nov 14 2020

			8=2^3, 24=2^3*3, 27=3^3, 40=2^3*5, ...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000578, A001235, A176313, A176350, A109399, A176359.

filter:= proc(x) local F; F:= map(t->t[2],ifactors(x)[2]);has(F,3) end proc:
select(filter, [$1..1000]); # Robert Israel, Jan 11 2015
# alternative:
isA176297 := proc(n)
    local p;
    for p in ifactors(n)[2] do
        if op(2,p) = 3 then
            return true;
        end if;
    end do:
    false ;
end proc: # R. J. Mathar, Dec 08 2015

f[n_]:=MemberQ[Last/@FactorInteger[n],3]; Select[Range[6!],f]

isok(n) = vecsearch(vecsort(factor(n)[,2]), 3); \\ Michel Marcus, Jan 11 2015

from sympy import factorint
def ok(n): return 3 in [e for e in factorint(n).values()]
print(list(filter(ok, range(713)))) # Michael S. Branicky, Aug 24 2021

A280618 Expansion of (Sum_{k>=1} x^(k^3))^2.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Ilya Gutkovskiy, Jan 06 2017

nonn

Number of ways to write n as an ordered sum of two positive cubes.

			a(9) = 2 because we have [8, 1] and [1, 8].

Antti Karttunen, Table of n, a(n) for n = 0..65537
Eric Weisstein's World of Mathematics, Cubic Number
Index entries for sequences related to sums of cubes

Cf. A000578, A001235 (positions of terms > 3), A003325 (of nonzero terms), A010057, A063725, A173677.

nmax = 150; CoefficientList[Series[(Sum[x^(k^3), {k, 1, nmax}])^2, {x, 0, nmax}], x]

A010057(n) = ispower(n, 3);
A280618(n) = if(n<2, 0, sum(r=1,sqrtnint(n-1,3),A010057(n-(r^3)))); \\ Antti Karttunen, Nov 30 2021

G.f.: (Sum_{k>=1} x^(k^3))^2.

A005088 Number of primes = 1 mod 3 dividing n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2

Offset: 1

N. J. A. Sloane

nonn

The first instance of a(n)=2 is for n=91; the first instance of a(n)=3 is for n=1729. 1729 is famously Ramanujan's taxi cab number -- see A001235. - Harvey P. Dale, Jun 25 2013

Antti Karttunen, Table of n, a(n) for n = 1..10000
S. R. Finch and Pascal Sebah, Squares and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2006-2016.

Cf. A001221, A005070, A005090, A020639, A028234, A079978.

Cf. A121940 (first number having n such factors).

A005088 := proc(n)
    local a,pe;
    a := 0 ;
    for pe in ifactors(n)[2] do
        if modp(op(1,pe),3)= 1 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, May 19 2020

Join[{0},Table[Count[Transpose[FactorInteger[n]][[1]],?(Mod[#-1,3] == 0&)],{n,2,100}]] (* _Harvey P. Dale, Sep 22 2021 *)
Array[DivisorSum[#, 1 &, And[PrimeQ@ #, Mod[#, 3] == 1] &] &, 91] (* Michael De Vlieger, Jul 11 2017 *)

a(n)=my(f=factor(n)[,1]); sum(i=1,#f,f[i]%3==1) \\ Charles R Greathouse IV, Jan 16 2017

(define (A005088 n) (if (= 1 n) 0 (+ (modulo (modulo (A020639 n) 3) 2) (A005088 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

Additive with a(p^e) = 1 if p = 1 (mod 3), 0 otherwise.
From Antti Karttunen, Jul 10 2017: (Start)
a(1) = 0; for n > 1, ((A020639(n) mod 3) mod 2) + a(A028234(n)).
a(n) = A001221(n) - A005090(n) - A079978(n).
(End)

A083737 Pseudoprimes to bases 2, 3 and 5.

Original entry on oeis.org

1729, 2821, 6601, 8911, 15841, 29341, 41041, 46657, 52633, 63973, 75361, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 294409, 314821, 334153, 340561, 399001, 410041, 488881, 512461, 530881, 552721, 658801, 670033, 721801, 748657

Offset: 1

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), May 05 2003

a(n) = n-th positive integer k(>1) such that 2^(k-1) == 1 (mod k), 3^(k-1) == 1 (mod k) and 5^(k-1) == 1 (mod k)
See A153580 for numbers k > 1 such that 2^k-2, 3^k-3 and 5^k-5 are all divisible by k but k is not a Carmichael number (A002997).
Note that a(1)=1729 is the Hardy-Ramanujan number. - Omar E. Pol, Jan 18 2009

			a(1)=1729 since it is the first number such that 2^(k-1) == 1 (mod k), 3^(k-1) == 1 (mod k) and 5^(k-1) == 1 (mod k).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 102 from R. J. Mathar)
J. Bernheiden, Pseudoprimes (Text in German)
F. Richman, Primality testing with Fermat's little theorem
Index entries for sequences related to pseudoprimes

Proper subset of A052155. Superset of A230722. Cf. A153580, A002997, A001235, A011541.

Select[ Range[838200], !PrimeQ[ # ] && PowerMod[2, # - 1, # ] == 1 && PowerMod[3, 1 - 1, # ] == 1 && PowerMod[5, # - 1, # ] == 1 & ]

is(n)=!isprime(n)&&Mod(2,n)^(n-1)==1&&Mod(3,n)^(n-1)==1&&Mod(5,n)^(n-1)==1 \\ Charles R Greathouse IV, Apr 12 2012

Edited by Robert G. Wilson v, May 06 2003

Edited by N. J. A. Sloane, Jan 14 2009

A018786 Numbers that are the sum of two 4th powers in more than one way.

Original entry on oeis.org

635318657, 3262811042, 8657437697, 10165098512, 51460811217, 52204976672, 68899596497, 86409838577, 138519003152, 160961094577, 162641576192, 264287694402, 397074160625, 701252453457, 823372979472, 835279626752

Offset: 1

David W. Wilson

nonn

Since 4th powers are squares, this is a subsequence of A024508, the analog for squares. Sequence A001235 is the analog for third powers (taxicab numbers). Sequence A255351 lists max {a,b,c,d} where a^4 + b^4 = c^4 + d^4 = a(n), while A255352 lists the whole quadruples (a,b,c,d). - M. F. Hasler, Feb 21 2015

			a(1) = 59^4 + 158^4 = 133^4 + 134^4.
a(2) = 7^4 + 239^4 = 157^4 + 227^4. Note the remarkable coincidence that here all of {7, 239, 157, 227} are primes. The next larger solution with this property is 17472238301875630082 = 62047^4 + 40351^4 = 59693^4 + 46747^4. - _M. F. Hasler_, Feb 21 2015

R. K. Guy, Unsolved Problems in Number Theory, D1.

Mia Muessig, Table of n, a(n) for n = 1..30000 (terms 1..111 from Vincenzo Librandi, terms 112..4359 from Sean A. Irvine)
J. Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780.
Mia Muessig, Julia code for finding general taxicab numbers
Eric Weisstein's World of Mathematics, Biquadratic Number.
Eric Weisstein's World of Mathematics, Diophantine Equation.

Subsequence of A003336 (and hence A004831) and A024508 (and hence A001481).

Cf. A001235, A003336, A003824, A309762.

Select[ Split[ Sort[ Flatten[ Table[x^4 + y^4, {x, 1, 1000}, {y, 1, x}]]]], Length[#] > 1 & ][[All, 1]] (* Jean-François Alcover, Jul 26 2011 *)

n=4;L=[];for(b=1,999,for(a=1,b,t=a^n+b^n;for(c=a+1,sqrtn(t\2,n),ispower(t-c^n,n)||next;print1(t",")))) \\ M. F. Hasler, Feb 21 2015

list(lim)=my(v=List()); for(a=134,sqrtnint(lim,4)-1, my(a4=a^4); for(b=sqrtnint((4*a^2 + 6*a + 4)*a,4)+1,min(sqrtnint(lim-a4,4),a), my(t=a4+b^4); for(c=a+1,sqrtnint(lim,4), if(ispower(t-c^4,4), listput(v,t); break)))); Set(v) \\ Charles R Greathouse IV, Jul 12 2024

A weak lower bound: a(n) >> n^2. - Charles R Greathouse IV, Jul 12 2024

A023051 Numbers that are the sum of two positive cubes in at least four ways (all solutions).

Original entry on oeis.org

6963472309248, 12625136269928, 21131226514944, 26059452841000, 55707778473984, 74213505639000, 95773976104625, 101001090159424, 159380205560856, 169049812119552, 174396242861568, 188013752349696

Offset: 1

David W. Wilson (revised Oct 15 1997)

nonn

R. K. Guy, Unsolved Problems in Number Theory, D1.

Uwe Hollerbach, Table of n, a(n) for n=1..16939
Uwe Hollerbach, Taxi, Taxi! [Original link, broken]
Uwe Hollerbach, Taxi, Taxi! [Replacement link to Wayback Machine]
Uwe Hollerbach, Taxi! Taxi! [Cached copy from Wayback Machine, html version of top page only]
D. W. Wilson, The Fifth Taxicab Number is 48988659276962496, J. Integer Sequences, Vol. 2, 1999, #99.1.9.

Cf. A001235, A003826, A011541, A018787, A023050, A025295, A051167, A343968, A345865.

b-file extended by Ray Chandler, Jan 19 2009

cp's OEIS Frontend

A352222 First members C of two non-consecutive numbers such that the sums of their cubes are equal to centered cube numbers and to at least one other sum of two cubes, i.e., A = B^3 + (B+1)^3 = C^3 + D^3 = E^3 + F^3.

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A008917 Numbers that are the sum of 3 positive cubes in more than one way.

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A078128 Number of ways to write n as sum of cubes > 1.

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A086119 Numbers of the form p^3 + q^3, p, q primes.

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A176297 Numbers with at least one 3 in their prime signature.

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A280618 Expansion of (Sum_{k>=1} x^(k^3))^2.

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A005088 Number of primes = 1 mod 3 dividing n.

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