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 1-3 of 3 results.

A130012 Smallest natural number whose cube is the sum of n cubes of distinct natural numbers, or 0 if no such number exists.

Original entry on oeis.org

1, 0, 6, 13, 9, 13, 14, 16, 18, 19, 21, 22, 24, 27, 28, 31, 33, 36, 38, 40, 42, 44, 45, 49, 52, 56, 58, 59, 62, 63, 67, 69, 71, 75, 79, 79, 83, 87, 89, 92, 95, 99, 102, 105, 107, 109, 114, 116, 117, 120, 126, 129, 131, 135, 138, 140, 145, 147, 150, 153, 158, 161, 165, 168
Offset: 1

Views

Author

J. Lowell, Jun 15 2007

Keywords

Comments

a(2)=0 is a special case of Fermat's Last Theorem. - Martin Fuller, Jul 06 2007

Examples

			a(3) = 6 because 3^3 + 4^3 + 5^3 = 6^3.

Links

Crossrefs

Cf. A130022 (for 4th powers), A007666 (for n-th powers).

Extensions

More terms from Martin Fuller, Jul 06 2007

A347773 Square array read by antidiagonals downwards: T(n,k) is the smallest positive integer whose n-th power is the sum of k n-th powers of positive integers, or 0 if no such number exists.

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 4, 3, 0, 1, 5, 2, 6, 0, 1, 6, 4, 7, 422481, 0, 1, 7, 3, 4, 353
Offset: 1

Views

Author

Eric Chen, Sep 15 2021

Keywords

Comments

a(26) = T(5,3) is conjectured to be 0, but this has not been proved.
By Fermat's last theorem, T(n,2) = 0 for n > 2.
Euler's sum of powers conjecture is that T(n,k) = 0 for n > k > 1, but this conjecture is not true: T(4,3) = 422481, T(5,4) = 144, there are no known counterexamples for n >= 6.
There are no known 0s for k > 2.
Conjecture: If T(n,k) = 0, then T(r,k) = T(n,s) = T(r,s) = 0 for all r >= n, 2 <= s <= k.

Examples

			Table begins:
  n\k |  1   2       3    4   5   6     7     8
  ----+----------------------------------------
   1  |  1   2       3    4   5   6     7     8
   2  |  1   5       3    2   4   3     4     4
   3  |  1   0       6    7   4   3     5     2
   4  |  1   0  422481  353   5   3     9    13
   5  |  1   0       ?  144  72  12    23    14
   6  |  1   0       ?    ?   ?   ?  1141   251
   7  |  1   0       ?    ?   ?   ?   568   102
   8  |  1   0       ?    ?   ?   ?     ?  1409
T(2,5) = 4 because 4^2 = 1^2 + 1^2 + 1^2 + 2^2 + 3^2 and there is no smaller square that is the sum of 5 positive squares.
T(4,3) = 422481 because 422481^4 = 95800^4 + 217519^4 + 414560^4 and there is no smaller 4th power that is the sum of 3 positive 4th powers.
T(7,7) = 568 because 568^7 = 127^7 + 258^7 + 266^7 + 413^7 + 430^7 + 439^7 + 525^7 and there is no smaller 7th power that is the sum of 7 positive 7th powers.

Links

Crossrefs

Cf. A007666 (main diagonal), A264764 (subdiagonal for k = n-1).
Cf. A175610 and A003828 (both for a(19) = T(4,3) = 422481).
Cf. A003294 and A039664 (both for a(25) = T(4,4) = 353).
Cf. A134341 (for a(33) = T(5,4) = 144).
Cf. A063922 and A063923 (both for a(41) = T(5,5) = 72).
Cf. A130012, A130022 (these two sequences are not rows of this table, since they require DISTINCT n-th powers, but this table does not have that requirement).

Programs

  • PARI
    /* return 0 instead of 1 for n=1, and oo loop when T(n, k)=0 */ A347773(p, n, s, m)={ /* Check whether s can be written as sum of n positive p-th powers not larger than m^p. If so, return the base a of the largest term a^p. */ s>n*m^p && return; n==1&&return(ispower(s, p, &n)*n); /* if s and m are not given, s>=n and m are arbitrary. */ !s&&for(m=round(sqrtn(n, p)), 9e9, A347773(p, n, m^p, m-1)&&return(m)); for(a=ceil(sqrtn(s\n, p)), min(sqrtn(max(0, s-n+1), p), m), A347773(p, n-1, s-a^p, a)&&return(a)); } /* after M. F. Hasler in A007666 */ /* Just enter "A347773(n, k)" to get T(n, k) */

Formula

T(n,1) = 1.
T(1,k) = k.
T(n,2) = 0 for n > 2.
T(n,n) = A007666(n).
T(n,n-1) = A264764(n).
T(3,k) <= A130012(k).
T(4,k) <= A130022(k).

A239247 Numbers n such that n^4 can be written as a sum of five distinct positive 4th powers.

Original entry on oeis.org

15, 30, 35, 45, 55, 60, 65, 70, 75, 85, 89, 90, 95, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 165, 170, 175, 178, 180, 185, 190, 195, 205, 210, 215, 220, 225, 230, 233, 235, 240, 245, 250, 255, 260, 265, 267, 270, 275, 280, 285, 290, 295, 300
Offset: 1

Views

Author

Michel Marcus, Mar 13 2014

Keywords

Comments

Every multiple of a term is a term.

Examples

			15^4 = 4^4 + 6^4 + 8^4 + 9^4 + 14^4.
35^4 = 4^4 + 21^4 + 22^4 + 26^4 + 28^4.
55^4 = 2^4 + 13^4 + 16^4 + 44^4 + 48^4.
65^4 = 1^4 + 8^4 + 12^4 + 32^4 + 64^4.
85^4 = 2^4 + 13^4 + 32^4 + 34^4 + 84^4.
89^4 = 10^4 + 35^4 + 52^4 + 60^4 + 80^4.
95^4 = 6^4 + 48^4 + 66^4 + 67^4 + 78^4.
115^4 = 4^4 + 31^4 + 48^4 + 58^4 + 112^4.
125^4 = 8^4 + 11^4 + 26^4 + 84^4 + 118^4.
145^4 = 2^4 + 23^4 + 46^4 + 52^4 + 144^4.
155^4 = 6^4 + 39^4 + 88^4 + 96^4 + 144^4.
185^4 = 2^4 + 38^4 + 62^4 + 87^4 + 182^4.
205^4 = 4^4 + 133^4 + 142^4 + 146^4 + 156^4.
215^4 = 4^4 + 26^4 + 127^4 + 174^4 + 176^4.
233^4 = 40^4 + 65^4 + 94^4 + 150^4 + 220^4.
235^4 = 9^4 + 52^4 + 148^4 + 184^4 + 194^4.

Links

Crossrefs

Cf. A130022, A003828 (three 4th powers), A096739 (four 4th powers).

Programs

  • PARI
    isok(n) = {ret = 0; for (x=1, sqrtnint(n^4\5, 4), for (y=x+1, sqrtnint((n^4 - x^4)\4, 4), for (z=y+1, sqrtnint((n^4 - x^4 - y^4)\3, 4), for (t=z+1, sqrtnint((n^4 - x^4 - y^4 - z^4)\2, 4), for (u=t+1, sqrtnint((n^4 - x^4 - y^4 - z^4 - t^4), 4), if (x^4+y^4+z^4+t^4+u^4 == n^4, print(n, ": ", x, ", ", y, ", ",z ,", ",t, ", ",u); ret = 1;);););););); return (ret);}

Formula

a(1) = A130022(4).

Extensions

Missing terms 15 and its multiples found by Alois P. Heinz, Mar 14 2014
More examples from Michel Marcus, Mar 18 2014
More terms from Lars Blomberg, Apr 05 2014
Showing 1-3 of 3 results.

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