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 1-4 of 4 results.

A003337 Numbers n which are the sum of 3 nonzero 4th powers.

Original entry on oeis.org

3, 18, 33, 48, 83, 98, 113, 163, 178, 243, 258, 273, 288, 338, 353, 418, 513, 528, 593, 627, 642, 657, 707, 722, 768, 787, 882, 897, 962, 1137, 1251, 1266, 1298, 1313, 1328, 1331, 1378, 1393, 1458, 1506, 1553, 1568, 1633, 1808, 1875, 1922, 1937, 2002, 2177
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Numbers which are in this sequence but not in A047714 must also be the sum of 2 biquadrates, or equal to a fourth power. Among the first 1000 terms of this sequence, this is the case for 4802 = 2*7^4, 57122 = 2*13^4 and 76832 = 2*14^4. - M. F. Hasler, Dec 31 2012
The union of A047714, A336536, and fourth powers of A003294. - Robert Israel, Jul 24 2020
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)
194818 is in the sequence as 194818 = 3^4 + 4^4 + 21^4.
480113 is in the sequence as 480113 = 7^4 + 12^4 + 26^4.
693842 is in the sequence as 693842 = 13^4 + 15^4 + 28^4. (End)

Links

Crossrefs

Cf. A003294, A047714, A336536, A309762, A344188.
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

  • Python
    def aupto(lim):
  p1 = set(i**4 for i in range(1, int(lim**.25)+2) if i**4 <= lim)
  p2 = set(a+b for a in p1 for b in p1 if a+b <= lim)
  p3 = set(apb+c for apb in p2 for c in p1 if apb+c <= lim)
  return sorted(p3)
print(aupto(2400)) # Michael S. Branicky, Mar 18 2021

A344188 Numbers that are the sum of three fourth powers in exactly one way.

Original entry on oeis.org

3, 18, 33, 48, 83, 98, 113, 163, 178, 243, 258, 273, 288, 338, 353, 418, 513, 528, 593, 627, 642, 657, 707, 722, 768, 787, 882, 897, 962, 1137, 1251, 1266, 1298, 1313, 1328, 1331, 1378, 1393, 1458, 1506, 1553, 1568, 1633, 1808, 1875, 1922, 1937, 2002, 2177, 2403, 2418, 2433, 2483, 2498, 2546, 2563, 2593, 2608, 2658
Offset: 1

Views

Author

David Consiglio, Jr., May 11 2021

Keywords

Comments

Differs from A003337 and A047714 at term 60 because 2673 = 2^4 + 4^4 + 7^4 = 3^4 + 6^4 + 6^4, see A309762.

Examples

			33 is a member of this sequence because 33 = 1^4 + 2^4 + 2^4

Links

Crossrefs

Cf. A003337, A025395, A344187, A344189, A344192, A344641.

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(1,50)]
for pos in cwr(power_terms,3):
    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])

A336536 Numbers n that can be written as both the sum of two nonzero fourth powers and the sum of three nonzero fourth powers.

Original entry on oeis.org

4802, 57122, 76832, 260642, 388962, 617057, 913952, 1229312, 1847042, 1957682, 3001250, 3502322, 3748322, 3959297, 4170272, 4626882, 6223392, 6837602, 6959682, 9872912, 11529602, 14623232, 19668992, 21112002, 27691682, 29552672, 31322912, 31505922, 35701250, 40127377, 40302242, 46712801, 48020000, 48355137
Offset: 1

Views

Author

Robert Israel, Jul 24 2020

Keywords

Comments

The fourth powers are not necessarily distinct.
If n is in the sequence, then so is k^4*n for every k.
The sum of two nonzero fourth powers is never a fourth power (a case of Fermat's last theorem).

Examples

			a(3) = 76832 is in the sequence because 76832 = 14^4 + 14^4 = 6^4 + 10^4 + 16^4.
a(6) = 617057 is in the sequence because 617057 = 7^4 + 28^4 = 3^4 + 20^4 + 26^4.

Links

Crossrefs

Cf. A000583, A003294, A047714.
Intersection of A003336 and A003337.

Programs

  • Maple
    N:= 10^8: # for terms <= N
F1:= {seq(i^4,i=1..floor(N^(1/4)))}: n1:= nops(F1):
F2:= select(`<=`,{seq(seq(F1[i]+F1[j],i=1..j),j=1..nops(F1))},N):
F3:= select(`<=`,{seq(seq(s+t,s=F1),t=F2)},N):
sort(convert(F3 intersect F2,list));
  • Python
    def aupto(lim):
  p1 = set(i**4 for i in range(1, int(lim**.25)+2) if i**4 <= lim)
  p2 = set(a+b for a in p1 for b in p1 if a+b <= lim)
  p3 = set(apb+c for apb in p2 for c in p1 if apb+c <= lim)
  return sorted(p3 & p2)
print(aupto(5*10**7)) # Michael S. Branicky, Mar 18 2021

A193244 Numbers that are the sum of three biquadrates (fourth powers) in more than one way.

Original entry on oeis.org

16562, 28593, 35378, 43218, 54977, 94178, 106353, 122018, 134162, 137633, 149058, 181202, 195122, 198497, 235298, 235553, 264113, 264992, 300833, 318402, 324818, 364658, 384833, 439922, 457488, 462722, 514098, 522242, 566048, 611618, 624962
Offset: 1

Views

Author

M. F. Hasler, Jan 01 2013

Keywords

Comments

Erroneous version of A309762. - Ilya Gutkovskiy, Aug 15 2019
A subsequence of A003337. Lists the indices such that A193243(n) > 1.

Examples

			a(1) = 16562 = 1^4 + 9^4 + 10^4 = 5^4 + 6^4 + 11^4, and this is the least number having two such decompositions.

Crossrefs

Cf. A003337, A193243, A047714, A085559, A122540, A123075.

Programs

  • PARI
    is_A000404(n)={ for( i=1, #n=factor(n)~%4, n[1, i]==3 & n[2, i]%2 & return); n & ( vecmin(n[1, ])==1 || (n[1, 1]==2 & n[2, 1]%2))} \\ M. F. Hasler, Feb 07 2009
  • PARI
    for(n=1,9e9, is_A000404(n) && A193243(n)>1 && print1(n","))
Showing 1-4 of 4 results.

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

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