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A332065
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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.
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11
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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, 40, 10, 13, 15, 30, 28, 32, 43, 84, 11, 14, 16, 31, 29, 34, 44, 85, 47, 12, 15, 17, 33, 30, 35, 45, 86, 49, 63, 13, 16, 18, 35, 31, 36, 46, 87, 52, 64, 68
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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T(1,k) = 2 + k for all k. (Indeed, s^1 = (s-1)^1 + 1 and s-1 > 1 for s > 2.)
T(2,k) = 6 + k for all k >= 3. (Use s^2 = (s-1)^2 + 2*s-1 and A001422, A009003.)
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.
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EXAMPLE
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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.
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PROG
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(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)
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CROSSREFS
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Cf. A009003 (hypotenuse numbers; subsequence of row 2).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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