A317145 - OEIS

A317145

Number of maximal chains of factorizations of n into factors > 1, ordered by refinement.

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

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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

Antti Karttunen, Table of n, a(n) for n = 1..11520

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Index entries for sequences computed from exponents in factorization of n

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

Cf. A001055, A002846, A007716, A045778, A064988, A162247, A213427, A275024, A281113, A299202, A300385, A317144, A317146, A320105.

Sequence in context: A344770 A293895 A300385 * A330665 A345985 A103765

Adjacent sequences: A317142 A317143 A317144 * A317146 A317147 A317148

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 22 2018

EXTENSIONS

Data section extended to 105 terms by Antti Karttunen, Oct 08 2018

STATUS

approved