A345775 -id:A345775 - cp's OEIS frontend

A048931 Numbers that are the sum of 6 positive cubes in exactly 3 ways.

221, 254, 369, 411, 443, 469, 495, 502, 576, 595, 600, 648, 658, 684, 704, 711, 720, 739, 746, 753, 760, 765, 767, 772, 774, 779, 786, 793, 811, 818, 828, 835, 844, 854, 863, 866, 874, 880, 884, 886, 892, 893, 899, 905, 910, 919, 928, 929, 935, 936, 937

Offset: 1

Eric W. Weisstein

It appears that this sequence has 1141 terms, the last of which is 26132. - Donovan Johnson, Jan 09 2013

			221 is in the sequence since 221 = 216+1+1+1+1+1 = 125+64+8+8+8+8 = 64+64+64+27+1+1.

Donovan Johnson, Table of n, a(n) for n = 1..1141 (terms < 10^7)
Eric Weisstein's World of Mathematics, Cubic Number.

Cf. A003329, A048929, A048930, A343705, A345512, A345766, A345775, A345815.

Reap[For[n = 1, n <= 1000, n++, pr = Select[ PowersRepresentations[n, 6, 3], Times @@ # != 0 &]; If[pr != {} && Length[pr] == 3, Print[n, pr]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jul 31 2013 *)

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]==3, n++; write("b048931.txt", n " " i))) /* Donovan Johnson, Jan 09 2013 */

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Oct 02 2000

A345521 Numbers that are the sum of seven cubes in three or more ways.

Original entry on oeis.org

222, 229, 248, 255, 262, 281, 283, 285, 318, 346, 370, 374, 377, 379, 381, 396, 400, 407, 412, 419, 426, 433, 437, 438, 444, 451, 463, 470, 472, 475, 477, 489, 494, 496, 501, 503, 505, 507, 510, 522, 529, 533, 536, 559, 564, 566, 568, 570, 577, 578, 584, 585

Offset: 1

David Consiglio, Jr., Jun 20 2021

nonn

			229 is a term because 229 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 2^3 + 5^3 = 1^3 + 1^3 + 2^3 + 3^3 + 3^3 + 3^3 + 3^3 = 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 3^3 + 4^3.

Sean A. Irvine, Table of n, a(n) for n = 1..10000

Cf. A345480, A345512, A345520, A345522, A345533, A345569, A345775.

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 >= 3])
    for x in range(len(rets)):
        print(rets[x])

A345825 Numbers that are the sum of seven fourth powers in exactly three ways.

Original entry on oeis.org

2677, 2692, 2757, 2852, 2867, 2917, 2997, 3107, 3172, 3301, 3476, 3541, 3972, 4132, 4227, 4242, 4257, 4307, 4322, 4372, 4437, 4452, 4482, 4497, 4562, 4627, 4737, 4756, 4851, 4866, 4867, 4931, 4996, 5077, 5106, 5107, 5122, 5187, 5252, 5282, 5317, 5347, 5362

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A345569 at term 7 because 2932 = 1^4 + 1^4 + 1^4 + 2^4 + 4^4 + 4^4 + 7^4 = 1^4 + 1^4 + 1^4 + 3^4 + 4^4 + 6^4 + 6^4 = 2^4 + 2^4 + 3^4 + 3^4 + 3^4 + 4^4 + 7^4 = 2^4 + 3^4 + 3^4 + 3^4 + 3^4 + 6^4 + 6^4.

			2692 is a term because 2692 = 1^4 + 1^4 + 1^4 + 2^4 + 2^4 + 4^4 + 7^4 = 1^4 + 1^4 + 1^4 + 2^4 + 3^4 + 6^4 + 6^4 = 2^4 + 2^4 + 2^4 + 3^4 + 3^4 + 3^4 + 7^4.

Sean A. Irvine, Table of n, a(n) for n = 1..10000

Cf. A345569, A345775, A345815, A345824, A345826, A345835, A346280.

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, 7):
    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])

A345774 Numbers that are the sum of seven cubes in exactly two ways.

Original entry on oeis.org

131, 159, 166, 173, 185, 192, 211, 236, 243, 257, 264, 269, 274, 276, 288, 290, 292, 295, 299, 300, 302, 307, 309, 311, 314, 320, 321, 325, 332, 333, 337, 339, 340, 344, 348, 351, 353, 355, 358, 359, 360, 363, 372, 384, 385, 386, 388, 389, 393, 395, 398, 403

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A345520 at term 8 because 222 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 6^3 = 1^3 + 1^3 + 1^3 + 3^3 + 4^3 + 4^3 + 4^3 = 1^3 + 2^3 + 2^3 + 2^3 + 2^3 + 4^3 + 5^3.

Likely finite.

			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.

Sean A. Irvine, Table of n, a(n) for n = 1..355

Cf. A048930, A345520, A345773, A345775, A345784, A345824.

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])

A345776 Numbers that are the sum of seven cubes in exactly four ways.

Original entry on oeis.org

470, 496, 503, 603, 634, 653, 659, 685, 690, 692, 711, 712, 747, 751, 754, 761, 766, 773, 775, 777, 780, 783, 787, 792, 794, 812, 813, 829, 831, 836, 842, 843, 859, 867, 871, 875, 883, 885, 890, 892, 899, 901, 904, 906, 907, 911, 913, 918, 919, 927, 930, 936

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A345522 at term 5 because 627 = 1^3 + 1^3 + 1^3 + 1^3 + 4^3 + 6^3 + 7^3 = 1^3 + 1^3 + 5^3 + 5^3 + 5^3 + 5^3 + 5^3 = 1^3 + 2^3 + 3^3 + 5^3 + 5^3 + 5^3 + 6^3 = 1^3 + 3^3 + 4^3 + 4^3 + 4^3 + 4^3 + 7^3 = 2^3 + 2^3 + 3^3 + 3^3 + 5^3 + 6^3 + 6^3.

Likely finite.

			496 is a term because 496 = 1^3 + 1^3 + 1^3 + 3^3 + 4^3 + 4^3 + 5^3 = 1^3 + 1^3 + 2^3 + 3^3 + 3^3 + 5^3 + 5^3 = 1^3 + 2^3 + 2^3 + 2^3 + 3^3 + 3^3 + 6^3 = 3^3 + 3^3 + 3^3 + 3^3 + 3^3 + 4^3 + 4^3.

Sean A. Irvine, Table of n, a(n) for n = 1..360

Cf. A345522, A345766, A345775, A345777, A345786, A345826.

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 == 4])
    for x in range(len(rets)):
        print(rets[x])

A345785 Numbers that are the sum of eight cubes in exactly three ways.

Original entry on oeis.org

223, 230, 237, 249, 263, 270, 275, 282, 284, 286, 289, 291, 293, 308, 310, 312, 319, 326, 345, 349, 354, 364, 371, 373, 375, 378, 380, 385, 386, 387, 389, 397, 399, 404, 406, 410, 412, 413, 415, 420, 423, 439, 441, 443, 446, 449, 452, 453, 459, 460, 465, 473

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A345533 at term 5 because 256 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 5^3 + 5^3 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 2^3 + 3^3 + 6^3 = 1^3 + 1^3 + 2^3 + 3^3 + 3^3 + 4^3 + 4^3 + 4^3 = 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 3^3 + 4^3 + 5^3.

Likely finite.

			230 is a term because 230 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 2^3 + 5^3 = 1^3 + 1^3 + 1^3 + 2^3 + 3^3 + 3^3 + 3^3 + 3^3 = 1^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 3^3 + 4^3.

Sean A. Irvine, Table of n, a(n) for n = 1..198

Cf. A345533, A345775, A345784, A345786, A345795, A345835.

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, 8):
    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])

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A048931 Numbers that are the sum of 6 positive cubes in exactly 3 ways.

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A345521 Numbers that are the sum of seven cubes in three or more ways.

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A345825 Numbers that are the sum of seven fourth powers in exactly three ways.

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A345774 Numbers that are the sum of seven cubes in exactly two ways.

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A345776 Numbers that are the sum of seven cubes in exactly four ways.

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A345785 Numbers that are the sum of eight cubes in exactly three ways.

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