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.

Previous Showing 31-39 of 39 results.

A004804 Numbers that are the sum of 4 nonzero 10th powers.

Original entry on oeis.org

4, 1027, 2050, 3073, 4096, 59052, 60075, 61098, 62121, 118100, 119123, 120146, 177148, 178171, 236196, 1048579, 1049602, 1050625, 1051648, 1107627, 1108650, 1109673, 1166675, 1167698, 1225723, 2097154, 2098177, 2099200, 2156202, 2157225
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)
65969099123 is in the sequence as 65969099123 = 7^10 + 7^10 + 9^10 + 12^10.
1099804917226 is in the sequence as 1099804917226 = 4^10 + 5^10 + 7^10 + 16^10.
1164925542026 is in the sequence as 1164925542026 = 5^10 + 9^10 + 12^10 + 16^10. (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
    k = 4; p = 10; amax = 3*10^6; bmax = amax^(1/p) // Ceiling; Clear[b]; b[0] = 1; Select[Table[Total[Array[b, k]^p], {b[1], b[0], bmax}, Evaluate[ Sequence @@ Table[{b[j], b[j-1], bmax}, {j, 1, k}]]] // Flatten // Union, # <= amax&] (* Jean-François Alcover, Jul 19 2017 *)

Extensions

Removed incorrect program. - David A. Corneth, Aug 01 2020

A004805 Numbers that are the sum of 5 positive 10th powers.

Original entry on oeis.org

5, 1028, 2051, 3074, 4097, 5120, 59053, 60076, 61099, 62122, 63145, 118101, 119124, 120147, 121170, 177149, 178172, 179195, 236197, 237220, 295245, 1048580, 1049603, 1050626, 1051649, 1052672, 1107628, 1108651, 1109674, 1110697, 1166676, 1167699, 1168722, 1225724, 1226747
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)
10352707051 is in the sequence as 10352707051 = 1^10 + 5^10 + 6^10 + 7^10 + 10^10.
59130893253 is in the sequence as 59130893253 = 7^10 + 9^10 + 9^10 + 11^10 + 11^10.
69011865378 is in the sequence as 69011865378 = 6^10 + 6^10 + 9^10 + 9^10 + 12^10. (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
    k = 5; p = 10; amax = 2*10^6; bmax = amax^(1/p) // Ceiling; Clear[b]; b[0] = 1; Select[Table[Total[Array[b, k]^p], {b[1], b[0], bmax}, Evaluate[ Sequence @@ Table[{b[j], b[j - 1], bmax}, {j, 1, k}]]] //Flatten // Union, # <= amax&] (* Jean-François Alcover, Jul 19 2017 *)

Extensions

Removed incorrect program. - David A. Corneth, Aug 01 2020

A004814 Numbers that are the sum of 3 positive 11th powers.

Original entry on oeis.org

3, 2050, 4097, 6144, 177149, 179196, 181243, 354295, 356342, 531441, 4194306, 4196353, 4198400, 4371452, 4373499, 4548598, 8388609, 8390656, 8565755, 12582912, 48828127, 48830174, 48832221, 49005273, 49007320, 49182419, 53022430
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)
204800049005272 is in the sequence as 204800049005272 = 3^11 + 5^11 + 20^11.
2518268235958260 is in the sequence as 2518268235958260 = 16^11 + 19^11 + 25^11.
3786934745885995 is in the sequence as 3786934745885995 = 10^11 + 19^11 + 26^11. (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).

Extensions

Removed incorrect program. - David A. Corneth, Aug 01 2020

A186659 Total number of positive integers below 10^n requiring 7 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

1, 6, 83, 1101, 12813, 135273, 1242976, 10072614, 81753935, 720313380, 6785033130
Offset: 1

Views

Author

Martin Renner, Feb 25 2011

Keywords

Comments

A114322(n) + A186649(n) + A186651(n) + A186653(n) + A186655(n) + A186657(n) + a(n) + A186661(n) + A186663(n) + A186665(n) + A186667(n) + A186669(n) + A186671(n) + A186673(n) + A186675(n) + A186677(n) + A186680(n) + A186682(n) + A186684(n) = A002283(n).

Links

Crossrefs

Cf. A003341, A186660.

Extensions

a(5)-a(8) from Lars Blomberg, May 08 2011
a(9) from Hiroaki Yamanouchi, Oct 13 2014
a(10)-a(11) from Giovanni Resta, Apr 26 2016

A186660 Total number of n-digit numbers requiring 7 positive biquadrates in their representation as sum of biquadrates.

Original entry on oeis.org

1, 5, 77, 1018, 11712, 122460, 1107703, 8829638, 71681321, 638559445, 6064719750
Offset: 1

Views

Author

Martin Renner, Feb 25 2011

Keywords

Comments

A102831(n) + A186650(n) + A186652(n) + A186654(n) + A186656(n) + A186658(n) + a(n) + A186662(n) + A186664(n) + A186666(n) + A186668(n) + A186670(n) + A186672(n) + A186674(n) + A186676(n) + A186678(n) + A186681(n) + A186683(n) + A186685(n) = A052268(n), for n>1.

Links

Crossrefs

Cf. A003341, A186659.

Formula

a(n) = A186659(n) - A186659(n-1).

Extensions

a(5)-a(11) from Giovanni Resta, Apr 29 2016

A345568 Numbers that are the sum of seven fourth powers in two or more 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

Views

Author

David Consiglio, Jr., Jun 20 2021

Keywords

Examples

			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.

Links

Crossrefs

Cf. A003341, A345520, A345559, A345569, A345577, A345605, A345824.

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, 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])

A345823 Numbers that are the sum of seven fourth powers in exactly one ways.

Original entry on oeis.org

7, 22, 37, 52, 67, 82, 87, 97, 102, 112, 117, 132, 147, 162, 167, 177, 182, 197, 212, 227, 242, 247, 322, 327, 337, 352, 387, 402, 407, 417, 452, 467, 482, 487, 562, 567, 577, 582, 592, 627, 631, 642, 646, 657, 661, 662, 676, 691, 692, 706, 707, 711, 721, 722
Offset: 1

Views

Author

David Consiglio, Jr., Jun 26 2021

Keywords

Comments

Differs from A003341 at term 23 because 262 = 1^4 + 1^4 + 1^4 + 2^4 + 3^4 + 3^4 + 3^4 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 4^4.

Examples

			22 is a term because 22 = 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 1^4 + 2^4.

Links

Crossrefs

Cf. A003341, A345773, A345813, A345824, A345833, A346278.

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, 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])

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

A047718 Numbers that are the sum of 7 but no fewer nonzero fourth powers.

Original entry on oeis.org

7, 22, 37, 52, 67, 87, 102, 112, 117, 132, 147, 167, 177, 182, 197, 212, 227, 242, 247, 262, 277, 292, 307, 322, 327, 342, 352, 357, 372, 387, 402, 407, 417, 422, 437, 452, 467, 482, 487, 502, 517, 532, 547, 562, 567, 577, 582, 592, 597, 612, 631, 646, 662
Offset: 1

Views

Author

Arlin Anderson (starship1(AT)gmail.com)

Keywords

Links

Crossrefs

Cf. A000583, A002377, A003341.

Programs

  • PARI
    upto(n)={my(e=7); my(s=sum(k=1, sqrtint(sqrtint(n)), x^(k^4)) + O(x*x^n)); my(p=s^e, q=(1 + s)^(e-1)); select(k->polcoeff(p,k) && !polcoeff(q,k), [1..n])} \\ Andrew Howroyd, Jul 06 2018
  • Python
    from itertools import combinations_with_replacement as mc
def aupto(limit):
    qd = [k**4 for k in range(1, int(limit**.25)+2) if k**4 + 6 <= limit]
    ss = [set(sum(c) for c in mc(qd, i)) for i in range(8)]
    s7nf = ss[7] - ss[6] - ss[5] - ss[4] - ss[3] - ss[2] - ss[1]
    return sorted(s for s in s7nf if s <= limit)
print(aupto(663)) # Michael S. Branicky, Jul 22 2021
Previous Showing 31-39 of 39 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.