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 A212646 #11 Feb 16 2025 08:33:17 %S A212646 1,2,3,5,7,4,11,6,15,10,9,22,14,15,30,22,21,8,42,30,25,33,12,56,44,35, %T A212646 45,20,77,60,55,18,66,28,49,101,84,75,30,90,44,77,135,112,110,42,27, %U A212646 126,60,105,50,176,154,150,66,16,121,45,168,88,154,70,231,202 %N A212646 a(n) = number of Abelian groups of order A181800(n) (n-th powerful number that is the first integer of its prime signature). %C A212646 The number of Abelian groups of order n, or A000688(n), is a function of the second signature of n (cf. A212172). Since A181800 gives the first integer of each second signature, this sequence gives the value of A000688 for each second signature in order of its first appearance. %H A212646 Amiram Eldar, <a href="/A212646/b212646.txt">Table of n, a(n) for n = 1..10000</a> %H A212646 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AbelianGroup.html">Abelian Group</a> %F A212646 a(n) = A000688(A181800(n)). %e A212646 There are 6 Abelian groups of order 72, corresponding to the 6 factorizations of 72 into prime powers: 2^3*3^2, 2^3*3*3, 2^2*2*3^2, 2^2*2*3*3, 2*2*2*3^2, and 2*2*2*3*3. Since 72 = A181800(8), a(8) = 6. %Y A212646 Cf. A000688, A046054, A046055, A046056, A050360, A181800, A212172. %K A212646 nonn %O A212646 1,2 %A A212646 _Matthew Vandermast_, Jun 09 2012