A369031 - OEIS

%I #21 Jan 13 2024 16:15:23

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

%C See discussion at A368900.

%C 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 Antti Karttunen, <a href="/A369031/b369031.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(n) = lcm {1..A122111(n)} / lcm {1..A122111(n-1)}.

%F a(n) = A014963(A122111(n)). [A122111 satisfies the property S given in A368900]

%F If n = p^k, p prime, k >= 1, then a(n) = A000040(k), otherwise a(n) = 1.

%o (PARI)

%o up_to = 2^18;

%o 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 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 v369031 = LCMtransform(vector(up_to,i,A122111(i)));

%o A369031(n) = v369031[n];

%o (PARI) A369031(n) = if(isprime(n),2, my(e=ispower(n,,&n)); if(e && isprime(n), prime(e), 1));

%Y Cf. A014963, A101296, A122111, A368900, A369032, A369033.

%K nonn

%O 1,2

%A _Antti Karttunen_, Jan 12 2024