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A286249 Triangular table: T(n,k) = 0 if k does not divide n, otherwise T(n,k) = P(A046523(n/k), k), where P is sequence A000027 used as a pairing function N x N -> N. Table is read by rows as T(1,1), T(2,1), T(2,2), etc.

%I A286249 #31 Jun 10 2025 13:54:51
%S A286249 1,3,2,3,0,4,10,5,0,7,3,0,0,0,11,21,5,8,0,0,16,3,0,0,0,0,0,22,36,14,0,
%T A286249 12,0,0,0,29,10,0,8,0,0,0,0,0,37,21,5,0,0,17,0,0,0,0,46,3,0,0,0,0,0,0,
%U A286249 0,0,0,56,78,27,19,12,0,23,0,0,0,0,0,67,3,0,0,0,0,0,0,0,0,0,0,0,79,21,5,0,0,0,0,30,0,0,0,0,0,0,92,21,0,8,0,17,0,0,0,0,0,0,0,0,0,106
%N A286249 Triangular table: T(n,k) = 0 if k does not divide n, otherwise T(n,k) = P(A046523(n/k), k), where P is sequence A000027 used as a pairing function N x N -> N. Table is read by rows as T(1,1), T(2,1), T(2,2), etc.
%C A286249 This sequence packs the values of A046523(n/k) and k (whenever k divides n) to a single term with the pairing function A000027. The two "components" can be accessed with functions A002260 & A004736, which allows us to generate from this sequence (among other things) various sums related to the enumeration of aperiodic necklaces, because Moebius mu (A008683) obtains the same value on any representative of the same prime signature.
%C A286249 For example, we have:
%C A286249   Sum_{i=A000217(n-1) .. A000217(n)} [a(i) > 0] * mu(A002260(a(i))) * 2^(A004736(a(i))) = A027375(n).
%C A286249 and
%C A286249   Sum_{i=A000217(n-1) .. A000217(n)} [a(i) > 0] * mu(A002260(a(i))) * 3^(A004736(a(i))) = A054718(n).
%C A286249 Triangle A286247 has the same property.
%H A286249 Antti Karttunen, <a href="/A286249/b286249.txt">Table of n, a(n) for n = 1..10585; the first 145 rows of triangle/antidiagonals of array</a>
%H A286249 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%F A286249 As a triangle (with n >= 1, 1 <= k <= n):
%F A286249 T(n,k) = 0 if k does not divide n, otherwise T(n,k) = (1/2)*(2 + ((A046523(n/k)+k)^2) - A046523(n/k) - 3*k).
%e A286249 The first fifteen rows of triangle:
%e A286249    1,
%e A286249    3,  2,
%e A286249    3,  0,  4,
%e A286249   10,  5,  0,  7,
%e A286249    3,  0,  0,  0, 11,
%e A286249   21,  5,  8,  0,  0, 16,
%e A286249    3,  0,  0,  0,  0,  0, 22,
%e A286249   36, 14,  0, 12,  0,  0,  0, 29,
%e A286249   10,  0,  8,  0,  0,  0,  0,  0, 37,
%e A286249   21,  5,  0,  0, 17,  0,  0,  0,  0, 46,
%e A286249    3,  0,  0,  0,  0,  0,  0,  0,  0,  0, 56,
%e A286249   78, 27, 19, 12,  0, 23,  0,  0,  0,  0,  0, 67,
%e A286249    3,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, 79,
%e A286249   21,  5,  0,  0,  0,  0, 30,  0,  0,  0,  0,  0,  0, 92,
%e A286249   21,  0,  8,  0, 17,  0,  0,  0,  0,  0,  0,  0,  0,  0, 106
%e A286249   -------------------------------------------------------------
%e A286249 Note how triangle A286247 contains on each row the same numbers in the same "divisibility-allotted" positions, but in reverse order.
%o A286249 (Scheme)
%o A286249 (define (A286249 n) (A286249tr (A002024 n) (A002260 n)))
%o A286249 (define (A286249tr n k) (if (not (zero? (modulo n k))) 0 (let ((a (A046523 (/ n k))) (b k)) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2)))))
%o A286249 (Python)
%o A286249 from sympy import factorint
%o A286249 import math
%o A286249 def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
%o A286249 def P(n):
%o A286249     f = factorint(n)
%o A286249     return sorted([f[i] for i in f])
%o A286249 def a046523(n):
%o A286249     x=1
%o A286249     while True:
%o A286249         if P(n) == P(x): return x
%o A286249         else: x+=1
%o A286249 def t(n, k): return 0 if n%k!=0 else T(a046523(n//k), k)
%o A286249 for n in range(1, 21): print([t(n, k) for k in range(1, n + 1)]) # _Indranil Ghosh_, May 08 2017
%Y A286249 Transpose: A286248 (triangle reversed).
%Y A286249 Cf. A000027, A008683, A027375, A046523, A054718, A054525, A286156, A286247, A286239.
%Y A286249 Cf. A000124 (the right edge of the triangle).
%K A286249 nonn,tabl
%O A286249 1,2
%A A286249 _Antti Karttunen_, May 06 2017