cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A003386 Numbers that are the sum of 8 nonzero 8th powers.

Original entry on oeis.org

8, 263, 518, 773, 1028, 1283, 1538, 1793, 2048, 6568, 6823, 7078, 7333, 7588, 7843, 8098, 8353, 13128, 13383, 13638, 13893, 14148, 14403, 14658, 19688, 19943, 20198, 20453, 20708, 20963, 26248, 26503, 26758, 27013, 27268, 32808, 33063, 33318, 33573
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

Examples

			From _David A. Corneth_, Aug 01 2020: (Start)
9534597 is in the sequence as 9534597 = 2^8 + 3^8 + 3^8 + 3^8 + 5^8 + 6^8 + 6^8 + 7^8.
13209988 is in the sequence as 13209988 = 1^8 + 1^8 + 2^8 + 2^8 + 2^8 + 6^8 + 7^8 + 7^8.
19046628 is in the sequence as 19046628 = 2^8 + 2^8 + 3^8 + 4^8 + 6^8 + 7^8 + 7^8 + 7^8. (End)

Links

Crossrefs

A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).

Programs

  • Mathematica
    M = 92646056; m = M^(1/8) // Ceiling; Reap[
For[a = 1, a <= m, a++, For[b = a, b <= m, b++, For[c = b, c <= m, c++,
For[d = c, d <= m, d++, For[e = d, e <= m, e++, For[f = e, f <= m, f++,
For[g = f, g <= m, g++, For[h = g, h <= m, h++,
s = a^8 + b^8 + c^8 + d^8 + e^8 + f^8 + g^8 + h^8;
If[s <= M, Sow[s]]]]]]]]]]][[2, 1]] // Union (* Jean-François Alcover, Dec 01 2020 *)

Extensions

b-file checked by R. J. Mathar, Aug 01 2020
Incorrect program removed by David A. Corneth, Aug 01 2020

A046040 Numbers that are the sum of 6 but no fewer positive cubes.

Original entry on oeis.org

6, 13, 20, 34, 39, 41, 46, 48, 53, 58, 60, 69, 76, 79, 84, 86, 95, 98, 102, 104, 105, 110, 117, 121, 123, 124, 132, 139, 147, 151, 158, 165, 170, 173, 177, 184, 196, 202, 203, 210, 215, 221, 222, 228, 235, 236, 242, 247, 249, 263, 265, 268, 273, 275, 284, 287
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

According to the McCurley article, it is conjectured that there are exactly 3922 terms of which the largest is a(3922) = 1290740.

Links

Crossrefs

Cf. A000578, A003325, A003072, A003327, A003328, A018890, A018889.

Programs

  • Mathematica
    Select[Range[300], (pr = PowersRepresentations[#, 6, 3]; pr != {} && Count[pr, r_/; (Times @@ r) == 0] == 0)&] (* Jean-François Alcover, Jul 26 2011 *)

Extensions

Corrected by Arlin Anderson (starship1(AT)gmail.com).

A047703 Numbers that are the sum of 4 but no fewer positive cubes.

Original entry on oeis.org

4, 11, 18, 25, 30, 32, 37, 44, 51, 56, 63, 67, 70, 74, 82, 88, 89, 93, 100, 107, 108, 119, 130, 135, 137, 142, 144, 145, 149, 154, 156, 161, 163, 168, 180, 182, 187, 191, 193, 198, 200, 205, 206, 219, 226, 233, 240, 245, 252, 254, 256, 259, 261, 266, 271, 278
Offset: 1

Views

Author

Arlin Anderson (starship1(AT)gmail.com)

Keywords

Links

Crossrefs

Cf. A004215, A002376, A003327.

Programs

  • Mathematica
    Select[Range[300], (pr = PowersRepresentations[#, 4, 3]; pr != {} && Count[pr, r_/; (Times @@ r) == 0] == 0)&] (* Jean-François Alcover, Jul 26 2011 *)

A069137 Numbers which are sums of neither 1, 2, 3, 4, 5 or 6 nonnegative cubes.

Original entry on oeis.org

7, 14, 15, 21, 22, 23, 42, 47, 49, 50, 61, 77, 85, 87, 103, 106, 111, 112, 113, 114, 122, 140, 148, 159, 166, 167, 174, 175, 178, 185, 186, 204, 211, 212, 223, 229, 230, 231, 237, 238, 239, 276, 292, 295, 300, 302, 303, 311, 327, 329, 337, 340, 356, 363, 364
Offset: 1

Views

Author

N. J. A. Sloane, Apr 08 2002; edited Sep 15 2006

Keywords

Comments

Sequence is conjectured to be finite.

Examples

			Numbers which need at least seven terms to represent them as a sum of positive cubes: 14=8+1+1+1+1+1+1.

References

  • Bohman, Jan and Froberg, Carl-Erik; Numerical investigation of Waring's problem for cubes, Nordisk Tidskr. Informationsbehandling (BIT) 21 (1981), 118-122.
  • F. Romani, Computations concerning Waring's problem, Calcolo, 19 (1982), 415-431.

Links

Crossrefs

Cf. A057907, A003329, A003325, A003072, A003327, A003328, A000578, A085334.

Formula

Natural numbers remaining if union of A003325, A003072, A003327, A003328, A003329 and A000578 sets were deleted. Remark: this sequence itself does not include cubes, in contrast to A085334.

A057905 Positive integers that are not the sum of exactly four positive cubes.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 31, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 64, 65, 66, 68, 69, 71, 72, 73, 75, 76, 77, 78, 79, 80, 83, 84, 85, 86, 87
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

It is conjectured that this sequence is finite, with 7373170279850 as its last member. - Charles R Greathouse IV, Jan 14 2017

Links

Crossrefs

Complement is A003327.

Programs

  • Mathematica
    pr[n_] := Select[ PowersRepresentations[n, 4, 3], FreeQ[#, 0] &]; Select[ Range[90], pr[#] == {} &] (* Jean-François Alcover, Nov 08 2012 *)
  • PARI
    list(lim)=my(v=List(),e=1+lim\1,x='x,t); t=sum(i=1,sqrtnint(e-4,3), x^i^3, O(x^e))^4; for(n=1,lim, if(polcoeff(t,n)==0, listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Jan 14 2017

A084355 Least number of positive cubes needed to represent n!.

Original entry on oeis.org

1, 1, 2, 6, 3, 5, 5, 4, 4, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3
Offset: 0

Views

Author

Hugo Pfoertner, Jun 22 2003

Keywords

Examples

			a(4)=3 because 4!=24=2^3+2^3+2^3.
a(0)=1 because 0!=1=1^3.
a(1)=1 because 1!=1=1^3.
a(2)=2 because 2!=2=1^3+1^3.
a(3)=6 because 3!=6=1^3+1^3+1^3+1^3+1^3+1^3.
a(4)=3 because 4!=24=2^3+2^3+2^3.
a(5)=5 because 5!=120=1^3+3^3+3^3+4^3+1^3.
a(6)=5 because 6!=720=4^3+6^3+6^3+6^3+2^3.
a(7)=4 because 7!=5040=1^3+5^3+17^3+1^3.
a(8)=4 because 8!=40320=2^3+10^3+34^3+2^3.
a(9)=3 because 9!=362880=52^3+56^3+36^3.
a(10)=3 because 10!=3628800=96^3+140^3+4^3.
a(11)=3 because 11!=39916800=222^3+303^3+105^3.
a(12)=3 because 12!=479001600=214^3+777^3+47^3.
a(13)=4 because 13!=6227020800=106^3+255^3+1838^3+33^3.
a(14)=3 because 14!=87178291200=1344^3+4392^3+312^3.
a(15)=3 because 15!=1307674368000=2040^3+10908^3+1092^3.
a(16)=3 because 16!=20922789888000=8400^3+27040^3+8240^3.
a(17)=3 because 17!=355687428096000=22848^3+69984^3+9984^3.
a(18)=3 because 18!=6402373705728000=54060^3+184080^3+18900^3.
From _Donovan Johnson_, May 17 2010: (Start)
a(19)=3 because 19!=121645100408832000=131040^3+331200^3+436320^3.
a(20)=3 because 20!=2432902008176640000=87490^3+1034430^3+1098440^3.
(End)

Crossrefs

Cf. A000142, A002376, A002325, A003072, A003327, A003328, A003329.

Programs

  • PARI
    a(n,up,dw,k)=local(i,m);if(k==1,if(n==round(sqrtn(n,3))^3,return(1),return(-1)),forstep(i=up,dw,-1,m=n-i^3;if(a(m,min(i,floor(sqrtn(m,3))),ceil(sqrtn(m/(k-1),3)),k-1)==1,return(1)))) for(n=0,18,for(k=1,9,if(a(n!,floor(sqrtn(n!,3)),ceil(sqrtn(n!/k,3)),k)==1,print1(k", ");break))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 01 2007

Formula

a(n)=A002376(n!).

Extensions

More terms from David W. Wilson, Jun 23 2003
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 01 2007
a(19)-a(20) from Donovan Johnson, May 17 2010

A085321 First difference sequence of A003337, i.e., consecutive differences between those consecutive numbers which are sums of three 4th powers.

Original entry on oeis.org

15, 15, 15, 35, 15, 15, 50, 15, 65, 15, 15, 15, 50, 15, 65, 95, 15, 65, 34, 15, 15, 50, 15, 46, 19, 95, 15, 65, 175, 114, 15, 32, 15, 15, 3, 47, 15, 65, 48, 47, 15, 65, 175, 67, 47, 15, 65, 175, 226, 15, 15, 50, 15, 48, 17, 30, 15, 50, 15, 65, 110, 65, 114, 15, 65, 110, 65
Offset: 1

Views

Author

Labos Elemer, Jul 01 2003

Keywords

Examples

			48 = 16 + 16 + 16, 83 = 81 + 1 + 1, 81 - 48 = 35 = a(4);
Certain differences occur consequently like 15, 30, 31, 32, 49, 50, 64, 65, 175, etc.;
Distance of closest neighbors equals 1,
like those of 7202 = 6561 + 625 + 16 and 7203 = 2401 + 2401 + 2401.

Crossrefs

Cf. A003337, A003327-A003399.

Programs

  • Mathematica
    {m=12, k=5, m^k}; t=Union[Flatten[Table[Table[Table[Table[w^k+q^k+t^k+u^k, {w, 1, m}], {q, 1, m}], {t, 1, m}], {u, 1, m}]]]; Length[t]; dt=Delete[ -RotateRight[t]+t, 1]; Sort[dt]

A085322 Terms m of A003337 such that m+1 is also in A003337. I.e., smaller one of two consecutive numbers, both equal to a sum of three 4th powers.

Original entry on oeis.org

4802, 7202, 10257, 14802, 15522, 38577, 45602, 57122, 57202, 76832, 86002, 90337, 94817, 109777, 112162, 116177, 131697, 135712, 136897, 155617, 163697, 171137, 188577, 243777, 260642, 284562, 296882, 332417, 388962, 390962, 391922, 459457, 474402, 617057, 637712
Offset: 1

Views

Author

Labos Elemer, Jul 01 2003

Keywords

Examples

			Distance of closest neighbors in A003337 equals 1: 7202 = 6561 + 625 + 16 and 7203 = 2401 + 2401 + 2401 are corresponding neighbors, so 7202 is a term.

Links

Crossrefs

Cf. A003337, A003327-A003399, A085321.

Programs

  • Mathematica
    {m=25, k=4, m^k} t=Union[Flatten[Table[Table[Table[w^k+q^k+t^k, {w, 1, m}], {q, 1, m}], {t, 1, m}]]] dt=Delete[ -RotateRight[t]+t, 1]; Part[t, Flatten[Position[dt, 1]]]
Select[Partition[Union[Total/@Tuples[Range[25]^4,3]],2,1],#[[2]]-#[[1]] == 1&][[All,1]] (* Harvey P. Dale, Jul 31 2020 *)

Extensions

More terms from Amiram Eldar, Mar 08 2025

A085334 Numbers which are neither sums of 2,3,4,5 or that of 6 nonnegative cubes.

Original entry on oeis.org

1, 7, 8, 14, 15, 21, 22, 23, 42, 47, 49, 50, 61, 77, 85, 87, 103, 106, 111, 112, 113, 114, 122, 125, 140, 148, 159, 166, 167, 174, 175, 178, 185, 186, 204, 211, 212, 223, 229, 230, 231, 237, 238, 239, 276, 292, 295, 300, 302, 303, 311, 327, 329, 337, 340, 356
Offset: 1

Views

Author

Labos Elemer, Jul 07 2003

Keywords

Crossrefs

Cf. A057907, A003329, A003325, A003072, A003327, A003328, A069137.

Formula

Remaining set of all natural numbers if union of A003325, A003072, A003327, A003328 and A003329 sets were deleted.Remark: this sequence includes those cubes too, which are sums of 7 or more cubes.

A122728 Primes that are the sum of 4 positive cubes.

Original entry on oeis.org

11, 37, 67, 89, 107, 137, 149, 163, 191, 193, 233, 271, 317, 353, 367, 379, 383, 409, 439, 461, 467, 479, 503, 523, 541, 587, 593, 601, 613, 631, 641, 653, 691, 709, 739, 751, 773, 809, 821, 839, 857, 863, 883, 887, 919, 929, 947, 971, 983, 991, 1033, 1069
Offset: 1

Views

Author

Jonathan Vos Post, Sep 23 2006

Keywords

Comments

By parity, there must be an odd number of odds in the sum. Hence this sequence is the union of primes which are the sum of the cube of an even number and the cubes of three odd numbers (such as 11 = 1^3 + 1^3 + 1^3 + 2^3) and the primes which are the sum of the cube of an odd number and the cubes of three even numbers (such as 149 = 2^3 + 2^3 + 2^3 + 5^3). A subset of this sequence is the primes which are the sum of the cubes of four distinct primes (i.e. of the form 2^3 + p^3 + q^3 + r^3 for p, q, r, distinct odd primes) such as 503 = 2^3 + 3^3 + 5^3 + 7^3; or 2357 = 2^3 + 3^3 + 5^3 + 13^3. No prime can be the sum of two cubes (by factorization of the sum of two cubes).

Examples

			a(1) = 11 = 1^3 + 1^3 + 1^3 + 2^3.
a(2) = 37 = 1^3 + 1^3 + 2^3 + 3^3.
a(3) = 67 = 1^3 + 1^3 + 1^3 + 4^3.

Links

Crossrefs

Cf. A000040, A003327.

Programs

  • Mathematica
    mx = 1000; lim = Floor[(mx - 3)^(1/3)]; Select[Union[Total /@ Tuples[Range[lim]^3, {4}]], # <= mx && PrimeQ[#] &] (* Harvey P. Dale, May 25 2011 *)

Formula

A000040 INTERSECTION A003327.

Extensions

More terms from Harvey P. Dale, May 25 2011
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