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.

Showing 1-6 of 6 results.

A048930 Numbers that are the sum of 6 positive cubes in exactly 2 ways.

Original entry on oeis.org

158, 165, 184, 228, 235, 247, 256, 261, 268, 273, 275, 280, 282, 284, 287, 291, 294, 306, 310, 313, 317, 324, 331, 332, 343, 345, 347, 350, 352, 362, 371, 373, 376, 378, 380, 385, 387, 388, 392, 395, 399, 404, 406, 408, 418, 425, 430, 432, 436, 437, 441
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

It appears that this sequence has 1094 terms, the last of which is 21722. - Donovan Johnson, Jan 09 2013

Examples

			158 is in the sequence since 158 = 64+64+27+1+1+1 = 125+8+8+8+8+1.

Links

Crossrefs

Cf. A003329, A048927, A048929, A048931, A345511, A345774, A345814.

Programs

  • Mathematica
    Reap[For[n = 1, n <= 500, n++, pr = Select[ PowersRepresentations[n, 6, 3], Times @@ # != 0 &]; If[pr != {} && Length[pr] == 2, Print[n, pr]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jul 31 2013 *)
  • PARI
    mx=10^6; ct=vector(mx); cb=vector(99); for(i=1, 99, cb[i]=i^3); for(i1=1, 99, s1=cb[i1]; for(i2=i1, 99, s2=s1+cb[i2]; if(s2+4*cb[i2]>mx, next(2)); for(i3=i2, 99, s3=s2+cb[i3]; if(s3+3*cb[i3]>mx, next(2)); for(i4=i3, 99, s4=s3+cb[i4]; if(s4+2*cb[i4]>mx, next(2)); for(i5=i4, 99, s5=s4+cb[i5]; if(s5+cb[i5]>mx, next(2)); for(i6=i5, 99, s6=s5+cb[i6]; if(s6>mx, next(2)); ct[s6]++)))))); n=0; for(i=6, mx, if(ct[i]==2, n++; write("b048930.txt", n " " i))) /* Donovan Johnson, Jan 09 2013 */

Extensions

Terms corrected by Donovan Johnson, Jan 09 2013

A345512 Numbers that are the sum of six cubes in three or more ways.

Original entry on oeis.org

221, 254, 369, 411, 443, 469, 495, 502, 576, 595, 600, 626, 648, 658, 684, 704, 711, 720, 739, 746, 753, 760, 765, 767, 772, 774, 779, 786, 793, 811, 818, 828, 830, 835, 837, 844, 854, 856, 863, 866, 873, 874, 880, 884, 886, 891, 892, 893, 899, 905, 910, 919
Offset: 1

Views

Author

David Consiglio, Jr., Jun 20 2021

Keywords

Examples

			254 is a term because 254 = 1^3 + 1^3 + 1^3 + 1^3 + 4^3 + 4^3 = 1^3 + 1^3 + 1^3 + 2^3 + 3^3 + 5^3 = 2^3 + 3^3 + 3^3 + 3^3 + 3^3 + 3^3.

Links

Crossrefs

Cf. A048931, A343704, A344807, A345511, A345513, A345521, A345560.

Programs

  • Python
    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)]
for pos in cwr(power_terms, 6):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v >= 3])
    for x in range(len(rets)):
        print(rets[x])

A345559 Numbers that are the sum of six fourth powers in two or more ways.

Original entry on oeis.org

261, 276, 291, 341, 356, 421, 516, 531, 596, 771, 885, 900, 965, 1140, 1361, 1509, 1556, 1571, 1636, 1811, 2180, 2596, 2611, 2661, 2676, 2691, 2706, 2721, 2741, 2756, 2771, 2786, 2836, 2851, 2916, 2931, 2946, 2961, 3011, 3026, 3091, 3186, 3201, 3220, 3266
Offset: 1

Views

Author

David Consiglio, Jr., Jun 20 2021

Keywords

Examples

			276 is a term because 276 = 1^4 + 1^4 + 1^4 + 1^4 + 2^4 + 4^4 = 1^4 + 2^4 + 2^4 + 3^4 + 3^4 + 3^4.

Links

Crossrefs

Cf. A003340, A344238, A345507, A345511, A345560, A345568, A345814.

Programs

  • Python
    from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**4 for x in range(1, 1000)]
for pos in cwr(power_terms, 6):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v >= 2])
    for x in range(len(rets)):
        print(rets[x])

A343702 Numbers that are the sum of five positive cubes in two or more ways.

Original entry on oeis.org

157, 220, 227, 246, 253, 260, 267, 279, 283, 286, 305, 316, 323, 342, 344, 361, 368, 377, 379, 384, 403, 410, 435, 440, 442, 468, 475, 487, 494, 501, 523, 530, 531, 549, 562, 568, 586, 592, 594, 595, 599, 602, 621, 625, 640, 647, 657, 658, 683, 703, 710, 712, 719, 729, 731, 738, 745, 752, 759, 764, 766, 771, 773, 778, 785
Offset: 1

Views

Author

David Consiglio, Jr., Apr 26 2021

Keywords

Comments

This sequence differs from A048927:
766 = 1^3 + 1^3 + 2^3 + 3^3 + 9^3
= 1^3 + 4^3 + 4^3 + 5^3 + 8^3
= 2^3 + 2^3 + 4^3 + 7^3 + 7^3.
So 766 is a term, but not a term of A048927.

Examples

			227 = 1^3 + 1^3 + 1^3 + 2^3 + 6^3
    = 2^3 + 3^3 + 4^3 + 4^3 + 4^3
so 227 is a term of this sequence.

Links

Crossrefs

Cf. A003328, A025406, A048927, A343704, A344238, A344795, A345511.

Programs

  • Mathematica
    Select[Range@1000,Length@Select[PowersRepresentations[#,5,3],FreeQ[#,0]&]>1&] (* Giorgos Kalogeropoulos, Apr 26 2021 *)
  • Python
    from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**3 for x in range(1,50)]#n
for pos in cwr(power_terms,5):#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])

A345520 Numbers that are the sum of seven cubes in two or more ways.

Original entry on oeis.org

131, 159, 166, 173, 185, 192, 211, 222, 229, 236, 243, 248, 255, 257, 262, 264, 269, 274, 276, 281, 283, 285, 288, 290, 292, 295, 299, 300, 302, 307, 309, 311, 314, 318, 320, 321, 325, 332, 333, 337, 339, 340, 344, 346, 348, 351, 353, 355, 358, 359, 360, 363
Offset: 1

Views

Author

David Consiglio, Jr., Jun 20 2021

Keywords

Examples

			159 is a term because 159 = 1^3 + 1^3 + 1^3 + 1^3 + 3^3 + 3^3 + 3^3 = 1^3 + 1^3 + 2^3 + 2^3 + 2^3 + 2^3 + 4^3.

Links

Crossrefs

Cf. A003330, A345479, A345511, A345521, A345532, A345568, A345774.

Programs

  • Python
    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)]
for pos in cwr(power_terms, 7):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v >= 2])
    for x in range(len(rets)):
        print(rets[x])

A344806 Numbers that are the sum of six squares in two or more ways.

Original entry on oeis.org

21, 24, 29, 30, 33, 36, 38, 39, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98
Offset: 1

Views

Author

David Consiglio, Jr., Jun 20 2021

Keywords

Examples

			24 = 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 4^2
   = 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2
so 24 is a term.

Links

Crossrefs

Cf. A344795, A344805, A344807, A345479, A345511.

Programs

  • Python
    from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**2 for x in range(1, 1000)]
for pos in cwr(power_terms, 6):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v >= 2])
    for x in range(len(rets)):
        print(rets[x])
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.