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A369031
LCM-transform of permutation induced by partition conjugation via Heinz numbers (A122111).
4
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
1,2
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.
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.
PROG
(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));
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 12 2024
STATUS
approved