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A352780
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Square array A(n,k), n >= 1, k >= 0, read by descending antidiagonals, such that the row product is n and column k contains only (2^k)-th powers of squarefree numbers.
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5
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1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 4, 5, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 14
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OFFSET
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1,3
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COMMENTS
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This is well-defined because positive integers have a unique factorization into powers of nonunit squarefree numbers with distinct exponents that are powers of 2.
Each (infinite) row is the lexicographically earliest with product n and terms that are a (2^k)-th power for all k.
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LINKS
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FORMULA
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EXAMPLE
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The top left corner of the array:
n/k | 0 1 2 3 4 5 6
------+------------------------------
1 | 1, 1, 1, 1, 1, 1, 1,
2 | 2, 1, 1, 1, 1, 1, 1,
3 | 3, 1, 1, 1, 1, 1, 1,
4 | 1, 4, 1, 1, 1, 1, 1,
5 | 5, 1, 1, 1, 1, 1, 1,
6 | 6, 1, 1, 1, 1, 1, 1,
7 | 7, 1, 1, 1, 1, 1, 1,
8 | 2, 4, 1, 1, 1, 1, 1,
9 | 1, 9, 1, 1, 1, 1, 1,
10 | 10, 1, 1, 1, 1, 1, 1,
11 | 11, 1, 1, 1, 1, 1, 1,
12 | 3, 4, 1, 1, 1, 1, 1,
13 | 13, 1, 1, 1, 1, 1, 1,
14 | 14, 1, 1, 1, 1, 1, 1,
15 | 15, 1, 1, 1, 1, 1, 1,
16 | 1, 1, 16, 1, 1, 1, 1,
17 | 17, 1, 1, 1, 1, 1, 1,
18 | 2, 9, 1, 1, 1, 1, 1,
19 | 19, 1, 1, 1, 1, 1, 1,
20 | 5, 4, 1, 1, 1, 1, 1,
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PROG
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(PARI)
up_to = 105;
A352780sq(n, k) = if(k==0, core(n), A352780sq(core(n, 1)[2], k-1)^2);
A352780list(up_to) = { my(v = vector(up_to), i=0); for(a=1, oo, forstep(col=a-1, 0, -1, i++; if(i > up_to, return(v)); v[i] = A352780sq(a-col, col))); (v); };
v352780 = A352780list(up_to);
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CROSSREFS
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Row numbers of rows:
- with a 1 in column 0: A000290\{0};
- with a 1 in column 0, but not in column 1: A030140;
- where every 1 is followed by another 1: A337533;
- where the last nonunit term is a power of 2: A335738.
Number of nonunit terms in row n is A331591(n); their positions are given (in reversed binary) by A267116(n); the first nonunit is in column A352080(n)-1 and the infinite run of 1's starts in column A299090(n).
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KEYWORD
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AUTHOR
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STATUS
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approved
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