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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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%I #25 Aug 05 2024 12:46:52

%S 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,

%T 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,

%U 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

%N 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.

%C 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.

%C Each (infinite) row is the lexicographically earliest with product n and terms that are a (2^k)-th power for all k.

%C For all k, column k is column k+1 of A060176 conjugated by A225546.

%F A(n,0) = A007913(n); for k > 0, A(n,k) = A(A000188(n), k-1)^2.

%F A(n,k) = A225546(A060176(A225546(n), k+1)).

%F A331591(A(n,k)) <= 1.

%e The top left corner of the array:

%e n/k | 0 1 2 3 4 5 6

%e ------+------------------------------

%e 1 | 1, 1, 1, 1, 1, 1, 1,

%e 2 | 2, 1, 1, 1, 1, 1, 1,

%e 3 | 3, 1, 1, 1, 1, 1, 1,

%e 4 | 1, 4, 1, 1, 1, 1, 1,

%e 5 | 5, 1, 1, 1, 1, 1, 1,

%e 6 | 6, 1, 1, 1, 1, 1, 1,

%e 7 | 7, 1, 1, 1, 1, 1, 1,

%e 8 | 2, 4, 1, 1, 1, 1, 1,

%e 9 | 1, 9, 1, 1, 1, 1, 1,

%e 10 | 10, 1, 1, 1, 1, 1, 1,

%e 11 | 11, 1, 1, 1, 1, 1, 1,

%e 12 | 3, 4, 1, 1, 1, 1, 1,

%e 13 | 13, 1, 1, 1, 1, 1, 1,

%e 14 | 14, 1, 1, 1, 1, 1, 1,

%e 15 | 15, 1, 1, 1, 1, 1, 1,

%e 16 | 1, 1, 16, 1, 1, 1, 1,

%e 17 | 17, 1, 1, 1, 1, 1, 1,

%e 18 | 2, 9, 1, 1, 1, 1, 1,

%e 19 | 19, 1, 1, 1, 1, 1, 1,

%e 20 | 5, 4, 1, 1, 1, 1, 1,

%o (PARI)

%o up_to = 105;

%o A352780sq(n, k) = if(k==0, core(n), A352780sq(core(n, 1)[2], k-1)^2);

%o 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); };

%o v352780 = A352780list(up_to);

%o A352780(n) = v352780[n];

%Y Sequences used in a formula defining this sequence: A000188, A007913, A060176, A225546.

%Y Cf. A007913 (column 0), A335324 (column 1).

%Y Range of values: {1} U A340682 (whole table), A005117 (column 0), A062503 (column 1), {1} U A113849 (column 2).

%Y Row numbers of rows:

%Y - with a 1 in column 0: A000290\{0};

%Y - with a 1 in column 1: A252895;

%Y - with a 1 in column 0, but not in column 1: A030140;

%Y - where every 1 is followed by another 1: A337533;

%Y - with 1's in all even columns: A366243;

%Y - with 1's in all odd columns: A366242;

%Y - where every term has an even number of distinct prime factors: A268390;

%Y - where every term is a power of a prime: A268375;

%Y - where the terms are pairwise coprime: A138302;

%Y - where the last nonunit term is coprime to the earlier terms: A369938;

%Y - where the last nonunit term is a power of 2: A335738.

%Y 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).

%K nonn,easy,tabl

%O 1,3

%A _Antti Karttunen_ and _Peter Munn_, Apr 02 2022