This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A378180 #5 Nov 24 2024 09:38:43
%S A378180 1,1,1,2,1,1,2,3,1,1,2,4,1,3,1,2,5,1,1,2,3,4,6,9,1,1,2,7,1,3,5,1,2,4,
%T A378180 8,1,1,2,3,4,6,9,1,1,2,4,5,10,25,1,3,7,1,2,11,1,1,2,3,4,6,8,9,12,18,
%U A378180 27,1,5,1,2,13,1,3,9,1,2,4,7,14,49,1,1,2,3,4,5,6,9,10,15,25
%N A378180 Irregular triangle where row n lists m such that rad(m) | n and bigomega(m) < bigomega(n), where rad = A007947 and bigomega = A001222.
%C A378180 Row n is a finite set of products of prime power factors p^k (i.e., p^k | n) such that Sum_{p|n} k < bigomega(n).
%C A378180 Row n contains numbers m such that rad(m) | n, where the number of prime factors of m with repetition is less than that of n.
%C A378180 Row 1 of this sequence is {}, hence offset of this sequence is set to 2.
%C A378180 For n = p^k (in A246655), row n contains p^j, j = 0..k-1.
%C A378180 For prime p, row p = {1}.
%C A378180 For n in A024619, row n of this sequence does not match row n of A162306, since the former contains gpf(n)^bigomega(n) = A006530(n)^A001222(n), which is larger than n, and since row n of A162306 contains n itself.
%H A378180 Michael De Vlieger, <a href="/A378180/b378180.txt">Table of n, a(n) for n = 2..10325</a>
%F A378180 Row n of this sequence is { m : rad(m) | n, bigomega(m) < bigomega(n) } = S \ T, where S is row n of A376248, and T is row n of A377070.
%F A378180 A378181(n) = binomial(bigomega(n) + omega(n) - 1, omega(n)) = Length of row n, where omega = A001221.
%F A378180 A378183(n) = rad(n)^binomial(omega(n) + bigomega(n) - 1, bigomega(n)-2) = A377073(n)/A377379(n) = product of row n.
%e A378180 Select rows n, showing nondivisors k parenthetically (i.e., k not in row n of A027750), and numbers k > n in brackets (i.e., k neither in row n of A162306 nor in row n of A027750):
%e A378180 n row n of this sequence:
%e A378180 -------------------------------------------
%e A378180 2: 1;
%e A378180 3: 1;
%e A378180 4: 1, 2;
%e A378180 6: 1, 2, 3;
%e A378180 8: 1, 2, 4;
%e A378180 9: 1, 3;
%e A378180 10: 1, 2, 5;
%e A378180 12: 1, 2, 3, 4, 6, (9);
%e A378180 18: 1, 2, 3, (4), 6, 9;
%e A378180 20: 1, 2, 4, 5, 10, [25];
%e A378180 24: 1, 2, 3, 4, 6, 8, (9), 12, (18), [27];
%e A378180 28: 1, 2, 4, 7, 14, [49];
%e A378180 30: 1, 2, 3, (4), 5, 6, (9), 10, 15, (25);
%e A378180 36: 1, 2, 3, 4, 6, 8, 9, 12, 18, (27);
%t A378180 Table[Clear[p]; MapIndexed[Set[p[First[#2]], #1] &, FactorInteger[n][[All, 1]]];
%t A378180 k = PrimeOmega[n]; w = PrimeNu[n];
%t A378180 Union@ Map[Times @@ MapIndexed[p[First[#2]]^#1 &, #] &,
%t A378180 Select[Tuples[Range[0, k], w], Total[#] < k &]], {n, 120}]
%Y A378180 Cf. A000961, A001221, A001222, A007947, A010846, A024619, A027750, A162306, A376567, A376248, A377070, A378181, A378183.
%K A378180 nonn,tabf,easy
%O A378180 2,4
%A A378180 _Michael De Vlieger_, Nov 19 2024