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

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, 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, 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

Offset: 1

Author

Antti Karttunen, Jan 12 2024

Keywords

• nonn

Comments

See discussion at A368900.

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.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

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

Programs

PARI

up_to = 2^18;
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); };
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));
v369031 = LCMtransform(vector(up_to,i,A122111(i)));
A369031(n) = v369031[n];

PARI

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

Formula

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

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

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