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

Offset: 1

Author

Gus Wiseman, Jul 22 2018

Keywords

• nonn

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

Examples

			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)
		

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

Crossrefs

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

Programs

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

Formula

a(prime^n) = A002846(n).

a(n) = A320105(A064988(n)). - Antti Karttunen, Oct 08 2018

Extensions

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