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A317145 Number of maximal chains of factorizations of n into factors > 1, ordered by refinement. 24
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 3, 1, 4, 1, 1, 1, 7, 1, 1, 1, 5, 1, 3, 1, 2, 2, 1, 1, 15, 1, 2, 1, 2, 1, 5, 1, 5, 1, 1, 1, 11, 1, 1, 2, 11, 1, 3, 1, 2, 1, 3, 1, 26, 1, 1, 2, 2, 1, 3, 1, 15, 2, 1, 1, 11, 1, 1, 1, 5, 1, 11, 1, 2, 1, 1, 1, 52, 1, 2, 2, 7, 1, 3, 1, 5, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,12
COMMENTS
If x and y are factorizations of the same integer and it is possible to produce x by further factoring the factors of y, flattening, and sorting, then x <= y.
a(n) depends only on prime signature of n (cf. A025487). - Antti Karttunen, Oct 08 2018
LINKS
FORMULA
a(prime^n) = A002846(n).
a(n) = A320105(A064988(n)). - Antti Karttunen, Oct 08 2018
EXAMPLE
The a(36) = 7 maximal chains:
(2*2*3*3) < (2*2*9) < (2*18) < (36)
(2*2*3*3) < (2*2*9) < (4*9) < (36)
(2*2*3*3) < (2*3*6) < (2*18) < (36)
(2*2*3*3) < (2*3*6) < (3*12) < (36)
(2*2*3*3) < (2*3*6) < (6*6) < (36)
(2*2*3*3) < (3*3*4) < (3*12) < (36)
(2*2*3*3) < (3*3*4) < (4*9) < (36)
PROG
(PARI)
A064988(n) = { my(f = factor(n)); for (k=1, #f~, f[k, 1] = prime(f[k, 1]); ); factorback(f); }; \\ From A064988
memoA320105 = Map();
A320105(n) = if(bigomega(n)<=2, 1, if(mapisdefined(memoA320105, n), mapget(memoA320105, n), my(f=factor(n), u = #f~, s = 0); for(i=1, u, for(j=i+(1==f[i, 2]), u, s += A320105(prime(primepi(f[i, 1])*primepi(f[j, 1]))*(n/(f[i, 1]*f[j, 1]))))); mapput(memoA320105, n, s); (s)));
A317145(n) = A320105(A064988(n)); \\ Antti Karttunen, Oct 08 2018
CROSSREFS
Sequence in context: A344770 A293895 A300385 * A330665 A345985 A103765
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 22 2018
EXTENSIONS
Data section extended to 105 terms by Antti Karttunen, Oct 08 2018
STATUS
approved

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Last modified March 28 13:41 EDT 2024. Contains 371254 sequences. (Running on oeis4.)