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%I A352780 #29 Jan 15 2025 16:43:14
%S A352780 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 A352780 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 A352780 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 A352780 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 A352780 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 A352780 Each (infinite) row is the lexicographically earliest with product n and terms that are a (2^k)-th power for all k.
%C A352780 For all k, column k is column k+1 of A060176 conjugated by A225546.
%H A352780 Antti Karttunen, <a href="/A352780/b352780.txt">Table of n, a(n) for n = 1..33153; the first 257 antidiagonals</a>
%F A352780 A(n,0) = A007913(n); for k > 0, A(n,k) = A(A000188(n), k-1)^2.
%F A352780 A(n,k) = A225546(A060176(A225546(n), k+1)).
%F A352780 A331591(A(n,k)) <= 1.
%e A352780 The top left corner of the array:
%e A352780 n/k | 0 1 2 3 4 5 6
%e A352780 ------+------------------------------
%e A352780 1 | 1, 1, 1, 1, 1, 1, 1,
%e A352780 2 | 2, 1, 1, 1, 1, 1, 1,
%e A352780 3 | 3, 1, 1, 1, 1, 1, 1,
%e A352780 4 | 1, 4, 1, 1, 1, 1, 1,
%e A352780 5 | 5, 1, 1, 1, 1, 1, 1,
%e A352780 6 | 6, 1, 1, 1, 1, 1, 1,
%e A352780 7 | 7, 1, 1, 1, 1, 1, 1,
%e A352780 8 | 2, 4, 1, 1, 1, 1, 1,
%e A352780 9 | 1, 9, 1, 1, 1, 1, 1,
%e A352780 10 | 10, 1, 1, 1, 1, 1, 1,
%e A352780 11 | 11, 1, 1, 1, 1, 1, 1,
%e A352780 12 | 3, 4, 1, 1, 1, 1, 1,
%e A352780 13 | 13, 1, 1, 1, 1, 1, 1,
%e A352780 14 | 14, 1, 1, 1, 1, 1, 1,
%e A352780 15 | 15, 1, 1, 1, 1, 1, 1,
%e A352780 16 | 1, 1, 16, 1, 1, 1, 1,
%e A352780 17 | 17, 1, 1, 1, 1, 1, 1,
%e A352780 18 | 2, 9, 1, 1, 1, 1, 1,
%e A352780 19 | 19, 1, 1, 1, 1, 1, 1,
%e A352780 20 | 5, 4, 1, 1, 1, 1, 1,
%o A352780 (PARI)
%o A352780 up_to = 105;
%o A352780 A352780sq(n, k) = if(k==0, core(n), A352780sq(core(n, 1)[2], k-1)^2);
%o A352780 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 A352780 v352780 = A352780list(up_to);
%o A352780 A352780(n) = v352780[n];
%Y A352780 Sequences used in a formula defining this sequence: A000188, A007913, A060176, A225546.
%Y A352780 Cf. A007913 (column 0), A335324 (column 1).
%Y A352780 Range of values: {1} U A340682 (whole table), A005117 (column 0), A062503 (column 1), {1} U A113849 (column 2).
%Y A352780 Row numbers of rows:
%Y A352780 - with a 1 in column 0: A000290\{0};
%Y A352780 - with a 1 in column 1: A252895;
%Y A352780 - with a 1 in column 0, but not in column 1: A030140;
%Y A352780 - where every 1 is followed by another 1: A337533;
%Y A352780 - with 1's in all even columns: A366243;
%Y A352780 - with 1's in all odd columns: A366242;
%Y A352780 - where every term has an even number of distinct prime factors: A268390;
%Y A352780 - where every term is a power of a prime: A268375;
%Y A352780 - where the terms are pairwise coprime: A138302;
%Y A352780 - where the last nonunit term is coprime to the earlier terms: A369938;
%Y A352780 - where the last nonunit term is a power of 2: A335738.
%Y A352780 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 A352780 nonn,easy,tabl
%O A352780 1,3
%A A352780 _Antti Karttunen_ and _Peter Munn_, Apr 02 2022