A273673 - OEIS

%I #24 Jun 24 2017 05:38:05

%S 1,1,2,1,2,2,1,2,3,4,1,2,3,4,5,1,2,3,4,3,4,1,2,3,4,3,6,7,1,2,3,4,2,6,

%T 7,8,1,2,3,4,2,6,7,8,4,1,2,3,4,5,6,7,8,9,10,1,2,3,4,5,6,7,8,9,6,11,1,

%U 2,3,4,5,6,5,8,9,6,11,8,1,2,3,4,5,6,5,8,9,4,11,12,13,1,2,3,4,5,6,5,8,9,4,11,12,13,14,1,2,3,4,5,6,5,8,9,10,11,12,13,14,10

%N Square array A(n,k) = (n / prime(1+A084558(k))^e) * prime(1+A084558(k)-A099563(k))^e, where e = A249344((1+A084558(k)), n) = the exponent of the largest power of prime(1+A084558(k)) which divides n. Array is read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

%C Informally: "clear" the exponent of prime(1+A084558(k)) and add it (the old value of exponent) to the exponent of prime(1+A084558(k)-A099563(k)) in the prime factorization of n.

%C Auxiliary function for computing array A275723.

%H Antti Karttunen, <a href="/A273673/b273673.txt">Table of n, a(n) for n = 1..5050; the first 100 antidiagonals of array</a>

%H Indranil Ghosh, <a href="/A273673/a273673.txt">Python program for computing this sequence</a>

%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>

%F A(n,k) = (n / prime(1+A084558(k))^e) * prime(1+A084558(k)-A099563(k))^e, where e = A249344((1+A084558(k)), n), the exponent of the largest power prime(1+A084558(k)) which divides n.

%e The top left 6 x 15 corner of the array:

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

%e    2,  2,  2,  2,  2,  2

%e    2,  3,  3,  3,  3,  3

%e    4,  4,  4,  4,  4,  4

%e    5,  3,  3,  2,  2,  5

%e    4,  6,  6,  6,  6,  6

%e    7,  7,  7,  7,  7,  5

%e    8,  8,  8,  8,  8,  8

%e    4,  9,  9,  9,  9,  9

%e   10,  6,  6,  4,  4, 10

%e   11, 11, 11, 11, 11, 11

%e    8, 12, 12, 12, 12, 12

%e   13, 13, 13, 13, 13, 13

%e   14, 14, 14, 14, 14, 10

%e   10,  9,  9,  6,  6, 15

%o (Scheme)

%o (define (A273673 n) (A273673bi (A002260 n) (A004736 n)))

%o (define (A273673bi n c) (if (zero? c) n (* (/ n (expt (A000040 (+ 1 (A084558 c))) (A249344bi (+ 1 (A084558 c)) n))) (expt (A000040 (+ 1 (- (A084558 c) (A099563 c)))) (A249344bi (+ 1 (A084558 c)) n)))))

%o ;; Code for A249344bi given in A249344.

%Y Cf. A000040, A084558, A099563, A249344.

%Y Cf. also A007623, A275723.

%K nonn,base,tabl

%O 1,3

%A _Antti Karttunen_, Aug 09 2016