cp's OEIS Frontend

A290532 Irregular triangle read by rows in which row n lists the number of divisors of each divisor of n.

%I A290532 #43 Dec 28 2021 04:18:02
%S A290532 1,1,2,1,2,1,2,3,1,2,1,2,2,4,1,2,1,2,3,4,1,2,3,1,2,2,4,1,2,1,2,2,3,4,
%T A290532 6,1,2,1,2,2,4,1,2,2,4,1,2,3,4,5,1,2,1,2,2,4,3,6,1,2,1,2,3,2,4,6,1,2,
%U A290532 2,4,1,2,2,4,1,2,1,2,2,3,4,4,6,8,1,2,3
%N A290532 Irregular triangle read by rows in which row n lists the number of divisors of each divisor of n.
%C A290532 Or, in the triangle A027750, replace each element with the number of its divisors.
%C A290532 The row of index n = p^m (p prime and m >= 0) is equal to (1, 2, ..., m + 1);
%C A290532 We observe an interesting property when the index n of the row n is the product of k distinct primes, k = 1,2,... For example:
%C A290532 The index n is prime => row n = (1, 2);
%C A290532 The index n equals the product of two distinct primes => row n = (1, 2, 2, 4);
%C A290532 The index n equals the product of three distinct primes => row n = (1, 2, 2, 2, 4, 4, 4, 8) or a permutation of the elements;
%C A290532 ...
%C A290532 Let us now consider Pascal's triangle (A007318(n) for n > 0):
%C A290532 1, 1;
%C A290532 1, 2, 1;
%C A290532 1, 3, 3, 1;
%C A290532 1, 4, 6, 4, 1;
%C A290532 ...
%C A290532 Row 1 of Pascal's triangle gives the number of "1" and the number of "2" respectively belonging to the row of index n = prime(m) of the sequence;
%C A290532 Row 2 of Pascal's triangle gives the number of "1", the number of "2" and the number of "4" respectively belonging to the row of index n = p*q of the sequence, where p and q are distinct primes;
%C A290532 Row 3 of Pascal's triangle gives the number of "1", the number of "2", the number of "4" and the number of "8" respectively belonging to the row of index n = p*q*r of the sequence, where p, q and r are distinct primes;
%C A290532 ...
%C A290532 It is now easy to generalize this process by the following proposition.
%C A290532 Proposition: binomial(m,k) is the number of terms of the form 2^k belonging to the row of index n in the sequence when n is the product of m distinct primes.
%H A290532 Robert Israel, <a href="/A290532/b290532.txt">Table of n, a(n) for n = 1..10006</a> (rows 1 to 1358, flattened)
%F A290532 T(n, k) = tau(A027750(n, k)).
%e A290532 Row 6 is (1, 2, 2, 4) because the 6th row of A027750 is [1, 2, 3, 6] and tau(1) = 1, tau(2) = 2, tau(3) = 2 and tau(6) = 4.
%e A290532 Triangle begins:
%e A290532   1;
%e A290532   1, 2;
%e A290532   1, 2;
%e A290532   1, 2, 3;
%e A290532   1, 2;
%e A290532   1, 2, 2, 4;
%e A290532   1, 2;
%e A290532   1, 2, 3, 4;
%e A290532   1, 2, 3;
%e A290532   1, 2, 2, 4;
%e A290532   ...
%p A290532 with(numtheory):nn:=100:
%p A290532 for n from 1 to nn do:
%p A290532   d1:=divisors(n):n1:=nops(d1):
%p A290532    for i from 1 to n1 do:
%p A290532      n2:=tau(d1[i]):
%p A290532      printf(`%d, `,n2):
%p A290532    od:
%p A290532 od:
%t A290532 Table[DivisorSigma[0, Divisors@ n], {n, 25}] // Flatten (* _Michael De Vlieger_, Aug 07 2017 *)
%o A290532 (PARI) row(n) = apply(numdiv, divisors(n)); \\ _Michel Marcus_, Dec 27 2021
%Y A290532 Cf. A000005, A007318, A027750, A084997, A290478.
%K A290532 nonn,tabf
%O A290532 1,3
%A A290532 _Michel Lagneau_, Aug 05 2017