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.
%I A332065 #58 Dec 20 2024 02:08:50
%S A332065 3,4,5,5,7,6,6,9,9,15,7,10,12,25,12,8,11,13,27,23,25,9,12,14,29,24,28,
%T A332065 40,10,13,15,30,28,32,43,84,11,14,16,31,29,34,44,85,47,12,15,17,33,30,
%U A332065 35,45,86,49,63,13,16,18,35,31,36,46,87,52,64,68
%N A332065 Infinite square array where row n lists the integers whose n-th power is the sum of distinct n-th powers of positive integers; read by falling antidiagonals.
%C A332065 Each row contains all sufficiently large integers (Sprague). Sequences A001422, A001476, A046039, A194768, A194769, ... mention the largest number which can't be written as sum of distinct n-th powers for n = 2, 3, 4, 5, 6, ...; see also A001661. Sequence A332066 gives the number of positive integers not in row n.
%C A332065 All positive multiples of any T(n,k) appear later in that row (because if s^n = Sum_{x in S} x^n, then (k*s)^n = Sum_{x in k*S} x^n).
%H A332065 R. Sprague, <a href="https://doi.org/10.1007/BF01185779">Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen</a>, Math. Z. 51 (1948) 466-468.
%H A332065 Various authors, <a href="/A030052/a030052.txt">Decomposition of T(n,1)^n = A030052(n)^n</a>.
%F A332065 T(1,k) = 2 + k for all k. (Indeed, s^1 = (s-1)^1 + 1 and s-1 > 1 for s > 2.)
%F A332065 T(2,k) = 6 + k for all k >= 3. (Use s^2 = (s-1)^2 + 2*s-1 and A001422, A009003.)
%F A332065 T(3,k) = 9 + k for all k >= 3. (Use max A001476 = 12758 < 24^3.)
%F A332065 T(4,k) = 32 + k for all k >= 13. (Use max A046039 < 48^4.)
%F A332065 T(5,k) = 24 + k for all k >= 4. (Use max(N \ A194768) < 37^5.)
%F A332065 T(6,k) = 30 + k for all k >= 4. (Use max(N \ A194769) < 48^6.)
%F A332065 T(7,k) = 41 + k for all k >= 2.
%F A332065 T(9,k) = 49 + k for all k >= 3.
%e A332065 The table reads: (Entries from where on T(n,k+1) = T(n,k)+1 are marked by *.)
%e A332065 n | k=1 2 3 4 5 6 7 8 9 10 11 12 13 ...
%e A332065 ---+---------------------------------------------------------------------
%e A332065 1 | 3* 4 5 6 7 8 9 10 11 12 13 14 15 ...
%e A332065 2 | 5 7 9* 10 11 12 13 14 15 16 17 18 19 ...
%e A332065 3 | 6 9 12* 13 14 15 16 17 18 19 20 21 22 ...
%e A332065 4 | 15 25 27 29 30 31 33 35 37 39 41 43 45* ...
%e A332065 5 | 12 23 24 28* 29 30 31 32 33 34 35 36 37 ...
%e A332065 6 | 25 28 32 34* 35 36 37 38 39 40 41 42 43 ...
%e A332065 7 | 40 43* 44 45 46 47 48 49 50 51 52 53 54 ...
%e A332065 8 | 84* 85 86 87 88 89 90 91 92 93 94 95 96 ...
%e A332065 9 | 47 49 52* 53 54 55 56 57 58 59 60 61 62 ...
%e A332065 10 | 63* 64 65 66 67 68 69 70 71 72 73 74 75 ...
%e A332065 11 | 68 73* 74 75 76 77 78 79 80 81 82 83 84 ...
%e A332065 ...| ...
%e A332065 Row 1: 3^1 = 2^1 + 1^1, 4^1 = 3^1 + 1^1, 5^1 = 4^1 + 1^1, 6^1 = 5^1 + 1^1, ...
%e A332065 Row 2: 5^2 = 4^2 + 3^2, 7^2 = 6^2 + 3^2 + 2^2, 9^2 = 8^2 + 4^2 + 1^2, ...
%e A332065 Row 3: 6^3 = 5^3 + 4^3 + 3^3, 9^3 = 8^3 + 6^3 + 1, 12^3 = 10^3 + 8^3 + 6^3, ...
%e A332065 Row 4: 15^4 = Sum {14, 9, 8, 6, 4}^4, 25^4 = Sum {21, 20, 12, 10, 8, 6, 2}^4, ...
%e A332065 See the link for other rows.
%o A332065 (PARI) M332065=Map(); A332065(n,m,r)={if(r, if( m<2^n||m>r^n*(r+n+1)\(n+1), m<2, r=min(sqrtnint(m,n),r), m==r^n || while( !A332065(n,m-r^n,r-=1) && (m<r^n*(r+n+1)\(n+1) || r=0),); r), m||[m=A004736(n),n=A002260(n)]; mapisdefined(M332065,[n,m],&r), r, n<2, m+2, r=if(m>1,A332065(n,m-1),n+2); until(A332065(n, (r+=1)^n, r-1),); mapput(M332065,[n,m],r); r)} \\ Calls itself with nonzero (optional) 3rd argument to find by exhaustive search whether r can be written as sum of distinct powers <= m^n. (Comment added by _M. F. Hasler_, May 25 2020)
%Y A332065 Cf. A030052 (first column), A001661.
%Y A332065 Cf. A009003 (hypotenuse numbers; subsequence of row 2).
%Y A332065 Cf. A001422, A001476, A046039, A194768, A194769.
%Y A332065 Cf. A332066.
%K A332065 nonn,tabl
%O A332065 1,1
%A A332065 _M. F. Hasler_, Mar 31 2020
%E A332065 More terms from _M. F. Hasler_, Jul 19 2020