A345774 -id:A345774 - cp's OEIS frontend

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

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

Eric W. Weisstein

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

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

Donovan Johnson, Table of n, a(n) for n = 1..1094 (terms < 10^7)
Eric Weisstein's World of Mathematics, Cubic Number.

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

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 *)

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 */

Terms corrected by Donovan Johnson, Jan 09 2013

A345824 Numbers that are the sum of seven fourth powers in exactly two ways.

Original entry on oeis.org

262, 277, 292, 307, 342, 357, 372, 422, 437, 502, 517, 532, 547, 597, 612, 677, 772, 787, 852, 886, 901, 916, 966, 981, 1027, 1046, 1141, 1156, 1221, 1362, 1377, 1396, 1442, 1510, 1525, 1557, 1572, 1587, 1590, 1617, 1637, 1652, 1717, 1765, 1812, 1827, 1892

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A345568 at term 61.

			277 is a term because 277 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 2^4 + 4^4 = 1^4 + 1^4 + 2^4 + 2^4 + 3^4 + 3^4 + 3^4.

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

Cf. A345568, A345774, A345814, A345823, A345825, A345834, A346279.

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 == 2])
    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

David Consiglio, Jr., Jun 20 2021

nonn

			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..10000

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

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])

A345775 Numbers that are the sum of seven cubes in exactly three 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, 472, 475, 477, 489, 494, 501, 505, 507, 510, 522, 529, 533, 536, 559, 564, 566, 568, 570, 577, 578, 584, 585, 592, 594, 596

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A345521 at term 28 because 470 = 1^3 + 1^3 + 1^3 + 1^3 + 5^3 + 5^3 + 6^3 = 1^3 + 1^3 + 1^3 + 2^3 + 3^3 + 6^3 + 6^3 = 1^3 + 3^3 + 4^3 + 4^3 + 4^3 + 5^3 + 5^3 = 2^3 + 3^3 + 3^3 + 4^3 + 4^3 + 4^3 + 6^3.

Likely finite.

			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..390

Cf. A048931, A345521, A345774, A345776, A345785, A345825.

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])

A345784 Numbers that are the sum of eight cubes in exactly two ways.

Original entry on oeis.org

132, 139, 158, 160, 167, 174, 181, 186, 193, 195, 197, 200, 212, 216, 219, 238, 244, 251, 258, 265, 272, 277, 288, 296, 298, 300, 301, 303, 307, 314, 315, 317, 321, 322, 327, 328, 329, 333, 334, 336, 338, 340, 341, 348, 350, 352, 356, 359, 360, 361, 363, 366

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A345532 at term 16 because 223 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 6^3 = 1^3 + 1^3 + 1^3 + 1^3 + 3^3 + 4^3 + 4^3 + 4^3 = 1^3 + 1^3 + 2^3 + 2^3 + 2^3 + 2^3 + 4^3 + 5^3.

Likely finite.

			139 is a term because 139 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 2^3 + 4^3 = 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 3^3 + 3^3.

Sean A. Irvine, Table of n, a(n) for n = 1..173

Cf. A345532, A345774, A345783, A345785, A345794, A345834.

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 == 2])
    for x in range(len(rets)):
        print(rets[x])

A345773 Numbers that are the sum of seven cubes in exactly one way.

Original entry on oeis.org

7, 14, 21, 28, 33, 35, 40, 42, 47, 49, 54, 56, 59, 61, 66, 68, 70, 73, 75, 77, 80, 84, 85, 87, 91, 92, 94, 96, 98, 99, 103, 105, 106, 110, 111, 112, 113, 117, 118, 122, 124, 125, 129, 132, 133, 136, 137, 138, 140, 143, 144, 145, 147, 148, 150, 151, 152, 154

Offset: 1

David Consiglio, Jr., Jun 26 2021

nonn

Differs from A003330 at term 44 because 131 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 5^3 = 2^3 + 2^3 + 2^3 + 2^3 + 2^3 + 3^3 + 4^3.

Likely finite.

			14 is a term because 14 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 2^3.

Sean A. Irvine, Table of n, a(n) for n = 1..324

Cf. A003330, A048929, A345774, A345783, A345823.

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 == 1])
    for x in range(len(rets)):
        print(rets[x])

cp's OEIS Frontend

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

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A345824 Numbers that are the sum of seven fourth powers in exactly two ways.

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

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A345775 Numbers that are the sum of seven cubes in exactly three ways.

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A345784 Numbers that are the sum of eight cubes in exactly two ways.

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A345773 Numbers that are the sum of seven cubes in exactly one way.

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