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 A074736 #25 Nov 04 2024 22:07:18
%S A074736 1,4,8,36,32,108,128,900,216,972,2048,4500,8192,8748,1944,44100,
%T A074736 131072,13500,524288,112500,17496,708588,8388608,308700,7776,6377292,
%U A074736 27000,2812500,536870912,337500,2147483648,5336100,1417176,516560652,69984,1543500,137438953472
%N A074736 Goedel encoding of the prime factors of n, in increasing order and repeated according to multiplicity.
%C A074736 For irregular triangle T(n,k) at A027746, a(n) = Product_{1..A001222(n)} pi(k)^T(n,k). - _Michael De Vlieger_, May 04 2020.
%D A074736 K. Gödel, "On Formally Undecidable Propositions of Principia Mathematica and Related Systems", Dover Publications, 1992.
%H A074736 Michael De Vlieger, <a href="/A074736/b074736.txt">Table of n, a(n) for n = 1..3322</a>
%H A074736 Wikipedia, <a href="https://en.wikipedia.org/wiki/G%C3%B6del_numbering#G%C3%B6del's_encoding">Gödel's encoding</a>
%F A074736 a(n) = prime(1)^p_1 * prime(2)^p_2 * ... * prime(k)^p_k, where p_1 <= ... <= p_k are the prime factors of n, repeated according to multiplicity.
%e A074736 The prime factors of 12 in increasing order and repeated according to multiplicity are 2, 2, 3. Hence a(12) = 2^2 * 3^2 * 5^3 = 4500.
%p A074736 a:= n-> (l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
%p A074736 sort(map(i-> i[1]$i[2], ifactors(n)[2]))):
%p A074736 seq(a(n), n=1..40);
%t A074736 Array[Times @@ MapIndexed[Prime[First[#2]]^#1 &, Apply[Join, ConstantArray[#1, #2] & @@@ FactorInteger[#]]] &, 34, 2] (* _Michael De Vlieger_, May 04 2020 *)
%o A074736 (PARI) for(n=2,50,m=factor(n):s=1:c=1:for(k=1,matsize(m)[1], for(l=1,m[k,2],s=s*prime(c)^m[k,1]:c=c+1)):print1(s",")) [Does not compile. - _Robert C. Lyons_, Nov 04 2024]
%o A074736 (Python)
%o A074736 from math import prod
%o A074736 from sympy import prime, factorint
%o A074736 def A074736(n): return prod(prime(i)**j for i, j in enumerate(factorint(n,multiple=True),1)) # _Chai Wah Wu_, Nov 04 2024
%Y A074736 Cf. A001222, A027746.
%K A074736 nonn
%O A074736 1,2
%A A074736 _Joseph L. Pe_, Sep 28 2002
%E A074736 More terms from _Ralf Stephan_, Mar 22 2003
%E A074736 a(1)=1 prepended by _Alois P. Heinz_, Nov 04 2024