A224483 -id:A224483 - cp's OEIS frontend

A102618 Numbers which are the sum of two positive cubes and divisible by 37.

370, 407, 1332, 2331, 2960, 3256, 4921, 5957, 8029, 8288, 9990, 10656, 10989, 12691, 12913, 13357, 13949, 14023, 14911, 16021, 16354, 17353, 18648, 18907, 19684, 19721, 20683, 22681, 23680, 24605, 24901, 26048, 27343, 30007, 30303, 32893, 34965, 35964, 36001, 36556, 37259, 39331, 39368, 39627

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Jan 31 2005

nonn

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

Cf. A003325. Other sequences of the form "sum of two positive cubes and divisible by ...": A224484, A224485, A101421, A101852, A094447, A099178, A102619, A101806, A224483, A102658.

N:= 200000: # for terms <= N
G:= expand(add(x^(i^3),i=1..floor(N^(1/3)))^2):
select(t -> coeff(G,x,t) > 0, [seq(i,i=37..N,37)]); # Robert Israel, Jun 12 2020

upto[n_] := Block[{t}, Union@ Reap[ Do[If[ Mod[t = x^3 + y^3, 37] == 0, Sow@ t], {x, n^(1/3)}, {y, Min[x, (n - x^3)^(1/3) ]}]][[2, 1]]]; upto[40000] (* Giovanni Resta, Jun 12 2020 *)
stpcQ[n_]:=Count[IntegerPartitions[n,{2}],?(AllTrue[CubeRoot[#],IntegerQ]&)]>0; Select[37* Range[1100],stpcQ] (* _Harvey P. Dale, Jul 10 2024 *)

Corrected by Robert Israel, Jun 12 2020

A102619 Numbers which are the sum of two positive cubes and divisible by 19.

Original entry on oeis.org

133, 152, 513, 855, 1064, 1216, 1729, 1843, 2071, 2261, 2413, 2869, 2926, 3059, 3439, 3591, 4104, 4123, 4921, 4940, 5833, 6175, 6840, 7163, 7657, 8512, 9386, 9728, 10773, 13167, 13357, 13718, 13832, 13851, 14174, 14364, 14744, 15542, 15561, 16568

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Jan 31 2005

If 12*h-1083 is a square then some values of 19*h are in this sequence. It is easy to verify that h is of the form 3*m^2-3*m+91, and therefore 19*(3*m^2-3*m+91) = (10-m)^3+(m+9)^3. - Vincenzo Librandi, May 10 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A003325, A101421 (divisible by k=7), A101852 (k=11), A094447 (k=13), A099178 (k=17), A101806 (k=23), A224483 (k=29), A102658 (k=31), A102618 (k=37).

[n: n in [2..2*10^4] | exists{i: i in [1..Iroot(n-1,3)] | IsPower(n-i^3,3) and IsZero(n mod 19)}]; // Bruno Berselli, May 10 2013

upto[n_] := Block[{t}, Union@ Reap[ Do[If[ Mod[t = x^3 + y^3, 19] == 0, Sow@t], {x, n^(1/3)}, {y, Min[x, (n - x^3)^(1/3)]}]][[2, 1]]]; upto[17000] (* Giovanni Resta, Jun 12 2020 *)

A102658 Numbers which are the sum of two positive cubes and divisible by 31.

Original entry on oeis.org

217, 341, 1395, 1674, 1736, 2728, 4123, 5425, 5642, 5859, 6293, 7471, 7657, 8029, 8587, 9207, 9269, 9331, 10261, 11160, 11377, 12679, 12896, 13392, 13888, 14167, 14911, 15841, 16120, 17577, 17701, 19747, 20026, 21824, 21979, 24149, 24397

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Feb 02 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A003325, A101421 (divisible by k=7), A101852 (k=11), A094447 (k=13), A099178 (k=17), A102619 (k=19), A101806 (k=23), A224483 (k=29), A102618 (k=37).

upto[n_] := Block[{t}, Union@ Reap[ Do[If[Mod[t = x^3 + y^3, 31] == 0, Sow@t], {x, n^(1/3)}, {y, Min[x, (n - x^3)^(1/3)]}]][[2, 1]]]; upto[24500] (* Giovanni Resta, Jun 12 2020 *)

A224484 Numbers which are the sum of two positive cubes and divisible by 3.

Original entry on oeis.org

9, 54, 72, 126, 189, 243, 351, 432, 468, 513, 576, 756, 855, 945, 1008, 1125, 1332, 1395, 1458, 1512, 1674, 1755, 1944, 2205, 2322, 2331, 2457, 2709, 2745, 2808, 3087, 3402, 3456, 3528, 3591, 3744, 4104, 4221, 4608, 4914, 4941

Offset: 1

Vincenzo Librandi, May 10 2013

If 12*h-27 is a square then some values of 3*h are in this sequence. It is easy to verify that h is of the form 3*m^2-3*m+3, and therefore 9*(m^2-m+1) = (2-m)^3+(m+1)^3.

All entries are multiples of 9. [Proof: the cubes mod 3 are A010872. So the two cubes are either of the form (3i)^3 and (3j)^3 or (3i+1)^3 and (3j+2)^3. The same 3-periodic pattern is seen in the cubes modulo 9, A167176.] - R. J. Mathar, Aug 24 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A224485 (divisible by k=5), A101421 (k=7), A101852 (k=11), A094447 (k=13), A099178 (k=17), A102619 (k=19), A101806 (k=23), A224483 (k=29), A102658 (k=31), A102618 (k=37).

upto[n_] := Block[{t}, Union@ Reap[ Do[If[Mod[t = x^3 + y^3, 3] == 0, Sow@ t], {x, n^(1/3)}, {y, Min[x, (n - x^3)^(1/3)]}] ][[2, 1]]]; upto[5000] (* Giovanni Resta, Jun 12 2020 *)
Module[{nn=20},Select[Union[Total/@Tuples[Range[nn]^3,2]],Mod[#,3]==0 && #Harvey P. Dale, Mar 06 2022 *)

A224485 Numbers which are the sum of two positive cubes and divisible by 5.

Original entry on oeis.org

35, 65, 250, 280, 370, 520, 730, 855, 945, 1125, 1395, 1755, 2000, 2060, 2205, 2240, 2540, 2745, 2960, 3500, 3925, 4075, 4160, 4375, 4825, 4940, 5425, 5840, 6175, 6750, 6840, 6860, 7075, 7110, 7560, 8125, 8190, 9000, 9325, 9990

Offset: 1

Vincenzo Librandi, May 10 2013

If 12*h-75 is a square then some values of 5*h are in this sequence. It is easy to verify that h is of the form 3*m^2-3*m+7, and therefore 5*(3*m^2-3*m+7) = (3-m)^3+(m+2)^3.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A224484 (divisible by k=3), A101421 (k=7), A101852 (k=11), A094447 (k=13), A099178 (k=17), A102619 (k=19), A101806 (k=23), A224483 (k=29), A102658 (k=31), A102618 (k=37).

upto[n_] := Block[{t}, Union@ Reap[ Do[If[Mod[t = x^3 + y^3, 5] == 0, Sow@t], {x, n^(1/3)}, {y, Min[x, (n - x^3)^(1/3)]}]][[2, 1]]]; upto[10000] (* Giovanni Resta, Jun 12 2020 *)

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A102618 Numbers which are the sum of two positive cubes and divisible by 37.

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A102619 Numbers which are the sum of two positive cubes and divisible by 19.

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A102658 Numbers which are the sum of two positive cubes and divisible by 31.

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A224484 Numbers which are the sum of two positive cubes and divisible by 3.

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A224485 Numbers which are the sum of two positive cubes and divisible by 5.

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