A285325 - OEIS

%I #17 Jan 27 2021 22:36:21

%S 1,2,2,3,4,3,4,4,6,4,5,8,5,8,5,6,6,12,5,10,6,7,8,7,16,6,12,7,8,8,10,9,

%T 20,6,14,8,9,16,9,10,8,24,7,16,9,10,10,24,9,12,10,28,7,18,10,11,12,11,

%U 32,11,14,9,32,7,20,11,12,12,18,17,40,10,12,11,36,8,22,12,13,16,13,14,12,48,10,14,13,40,8,24,13

%N Square array A(n,k) = A048675(A285321(n,k)), read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

%H Antti Karttunen, <a href="/A285325/b285325.txt">Table of n, a(n) for n = 1..120; the first 15 antidiagonals of array</a>

%F A(n,k) = A048675(A285321(n,k)).

%e The top left 15x6 corner of the array:

%e   1,  2, 3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15

%e   2,  4, 4,  8,  6,  8,  8, 16, 10, 12, 12, 16, 14, 16, 16

%e   3,  6, 5, 12,  7, 10,  9, 24, 11, 18, 13, 20, 15, 18, 17

%e   4,  8, 5, 16,  9, 10,  9, 32, 17, 14, 13, 20, 17, 22, 17

%e   5, 10, 6, 20,  8, 12, 11, 40, 12, 20, 14, 24, 21, 18, 19

%e   6, 12, 6, 24, 10, 14, 10, 48, 18, 16, 19, 28, 16, 20, 18

%o (PARI)

%o A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from _M. F. Hasler_

%o A007947(n) = factorback(factorint(n)[, 1]); \\ _Andrew Lelechenko_, May 09 2014

%o A065642(n) = { my(r=A007947(n)); if(1==n,n,n = n+r; while(A007947(n) <> r, n = n+r); n); };

%o A285321bi(row,col) = if(1==row,A019565(col),A065642(A285321bi(row-1,col)));

%o A002260(n)= { n-binomial((sqrtint(8*n)+1)\2, 2); }; \\ _M. F. Hasler_, Mar 10 2014

%o A004736(n)= { 1 + binomial(1 + floor(1/2 + sqrt(2*n)), 2) - n; };

%o A285321(n) = A285321bi(A002260(n), A004736(n));

%o A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ _Michel Marcus_, Oct 10 2016

%o A285325(n) = A048675(A285321(n));

%o for(n=1, 120, write("b285325.txt", n, " ", A285325(n)));

%o (Scheme)

%o (define (A285325 n) (A285325bi (A002260 n) (A004736 n)))

%o (define (A285325bi row col) (A048675 (A285321bi row col)))

%Y Cf. A048675, A285321, A285333.

%Y Row 1 & column 1: A000027.

%Y Row 2: A285326, Row 3: A285327.

%K nonn,tabl

%O 1,2

%A _Antti Karttunen_, Apr 17 2017