OFFSET
1,1
COMMENTS
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.
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.
LINKS
R. Sprague, Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen, Math. Z. 51 (1948) 466-468.
Various authors, Decomposition of T(n,1)^n = A030052(n)^n.
FORMULA
T(1,k) = 2 + k for all k. (Indeed, s^1 = (s-1)^1 + 1 and s-1 > 1 for s > 2.)
T(3,k) = 9 + k for all k >= 3. (Use max A001476 = 12758 < 24^3.)
T(4,k) = 32 + k for all k >= 13. (Use max A046039 < 48^4.)
T(5,k) = 24 + k for all k >= 4. (Use max(N \ A194768) < 37^5.)
T(6,k) = 30 + k for all k >= 4. (Use max(N \ A194769) < 48^6.)
T(7,k) = 41 + k for all k >= 2.
T(9,k) = 49 + k for all k >= 3.
EXAMPLE
The table reads: (Entries from where on T(n,k+1) = T(n,k)+1 are marked by *.)
n | k=1 2 3 4 5 6 7 8 9 10 11 12 13 ...
---+---------------------------------------------------------------------
1 | 3* 4 5 6 7 8 9 10 11 12 13 14 15 ...
2 | 5 7 9* 10 11 12 13 14 15 16 17 18 19 ...
3 | 6 9 12* 13 14 15 16 17 18 19 20 21 22 ...
4 | 15 25 27 29 30 31 33 35 37 39 41 43 45* ...
5 | 12 23 24 28* 29 30 31 32 33 34 35 36 37 ...
6 | 25 28 32 34* 35 36 37 38 39 40 41 42 43 ...
7 | 40 43* 44 45 46 47 48 49 50 51 52 53 54 ...
8 | 84* 85 86 87 88 89 90 91 92 93 94 95 96 ...
9 | 47 49 52* 53 54 55 56 57 58 59 60 61 62 ...
10 | 63* 64 65 66 67 68 69 70 71 72 73 74 75 ...
11 | 68 73* 74 75 76 77 78 79 80 81 82 83 84 ...
...| ...
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, ...
Row 2: 5^2 = 4^2 + 3^2, 7^2 = 6^2 + 3^2 + 2^2, 9^2 = 8^2 + 4^2 + 1^2, ...
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, ...
Row 4: 15^4 = Sum {14, 9, 8, 6, 4}^4, 25^4 = Sum {21, 20, 12, 10, 8, 6, 2}^4, ...
See the link for other rows.
PROG
(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)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
M. F. Hasler, Mar 31 2020
EXTENSIONS
More terms from M. F. Hasler, Jul 19 2020
STATUS
approved