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A369031 LCM-transform of permutation induced by partition conjugation via Heinz numbers (A122111).

%I A369031 #21 Jan 13 2024 16:15:23
%S A369031 1,2,2,3,2,1,2,5,3,1,2,1,2,1,1,7,2,1,2,1,1,1,2,1,3,1,5,1,2,1,2,11,1,1,
%T A369031 1,1,2,1,1,1,2,1,2,1,1,1,2,1,3,1,1,1,2,1,1,1,1,1,2,1,2,1,1,13,1,1,2,1,
%U A369031 1,1,2,1,2,1,1,1,1,1,2,1,7,1,2,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,2,1,2,1,1
%N A369031 LCM-transform of permutation induced by partition conjugation via Heinz numbers (A122111).
%C A369031 See discussion at A368900.
%C A369031 From the reduced formula it follows that for all i, j >= 1: A101296(i) = A101296(j) => a(i) = a(j), that is, the value of each a(n) is completely determined by its prime signature. Note that the same does not hold for related A369032.
%H A369031 Antti Karttunen, <a href="/A369031/b369031.txt">Table of n, a(n) for n = 1..65537</a>
%H A369031 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A369031 a(n) = lcm {1..A122111(n)} / lcm {1..A122111(n-1)}.
%F A369031 a(n) = A014963(A122111(n)). [A122111 satisfies the property S given in A368900]
%F A369031 If n = p^k, p prime, k >= 1, then a(n) = A000040(k), otherwise a(n) = 1.
%o A369031 (PARI)
%o A369031 up_to = 2^18;
%o A369031 LCMtransform(v) = { my(len = length(v), b = vector(len), g = vector(len)); b[1] = g[1] = 1; for(n=2,len, g[n] = lcm(g[n-1],v[n]); b[n] = g[n]/g[n-1]); (b); };
%o A369031 A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
%o A369031 v369031 = LCMtransform(vector(up_to,i,A122111(i)));
%o A369031 A369031(n) = v369031[n];
%o A369031 (PARI) A369031(n) = if(isprime(n),2, my(e=ispower(n,,&n)); if(e && isprime(n), prime(e), 1));
%Y A369031 Cf. A014963, A101296, A122111, A368900, A369032, A369033.
%K A369031 nonn
%O A369031 1,2
%A A369031 _Antti Karttunen_, Jan 12 2024