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A255231 The number of factorizations n = Product_i b_i^e_i, where all bases b_i are distinct, and all exponents e_i are distinct >=1. 9

%I #53 May 24 2017 14:40:50

%S 1,1,1,2,1,1,1,2,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,3,2,1,2,2,1,1,1,5,1,1,

%T 1,4,1,1,1,3,1,1,1,2,2,1,1,5,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,7,1,1,1,2,

%U 1,1,1,6,1,1,2,2,1,1,1,5,4,1,1,2,1,1,1,3,1,2,1,2,1,1,1,6,1,2,2,4,1,1,1,3,1,1,1,6,1,1,1,5,1,1,1,2,2,1,1,3

%N The number of factorizations n = Product_i b_i^e_i, where all bases b_i are distinct, and all exponents e_i are distinct >=1.

%C Not multiplicative: a(48) = a(2^4*3) = 5 <> a(2^4)*a(3) = 4*1 = 4. - _R. J. Mathar_, Nov 05 2016

%H R. J. Mathar, <a href="/A255231/b255231.txt">Table of n, a(n) for n = 1..419</a>

%H R. J. Mathar, <a href="/A255231/a255231.pdf">Factorizations of integers into factors with distinct bases and exponents</a>

%H <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>

%F a(n)=1 for all n in A005117. a(n)=2 for all n in A001248 and for all n in A054753 and for all n in A085987 and for all n in A030078. a(n)=3 for all n in A065036. a(n)=4 for all n in A085986 and for all n in A030514. a(n)=5 for all n in A178739, all n in A179644 and for all n in A050997. a(n)=6 for all n in A143610, all n in A162142 and all n in A178740. a(n)=7 for all n in A030516. a(n)=9 for all n in A189988 and all n in A189987. a(n)=10 for all n in A092759. a(n) = 11 for all n in A179664. a(n)=12 for all n in A179646. - _R. J. Mathar_, Nov 05 2016, May 20 2017

%e From _R. J. Mathar_, Nov 05 2016: (Start)

%e a(4)=2: 4^1 = 2^2.

%e a(8)=2: 8^1 = 2^3.

%e a(9)=2: 9^1 = 3^2.

%e a(12)=2: 12^1 = 2^2*3^1.

%e a(16)=4: 16^1 = 4^2 = 2^2*4^1 = 2^4.

%e a(18)=2: 18^1 = 2*3^2.

%e a(20)=2: 20^1 = 2^2*5^1.

%e a(24)=3: 24^1 = 2^2*6^1 = 2^3*3^1.

%e a(32)=5: 32^1 = 2^1*4^2 = 2^2*8^1 = 2^3*4^1 = 2^5.

%e a(36)=4: 36^1 = 6^2 = 3^2*4^1 = 2^2*9^1.

%e a(48)=5: 48^1 = 3^1*4^2 = 2^2*12^1 = 2^3*6^1 = 2^4*3^1.

%e a(60)=2 : 60^1 = 2^2*15^1.

%e a(64)=7: 64^1 = 8^2 = 4^3 = 2^2*16^1 = 2^3*8^1 = 2^4*4^1 = 2^6.

%e a(72)=6 : 72^1 = 3^2*8^1 = 2^1*6^2 = 2^2*18^1 = 2^3*9^1 = 2^3*3^2.

%e (End)

%p # Count solutions for products if n = dvs_i^exps(i) where i=1..pividx are fixed

%p Apiv := proc(n,dvs,exps,pividx)

%p local dvscnt, expscopy,i,a,expsrt,e ;

%p dvscnt := nops(dvs) ;

%p a := 0 ;

%p if pividx > dvscnt then

%p # have exhausted the exponent list: leave of the recursion

%p # check that dvs_i^exps(i) is a representation

%p if n = mul( op(i,dvs)^op(i,exps),i=1..dvscnt) then

%p # construct list of non-0 exponents

%p expsrt := [];

%p for i from 1 to dvscnt do

%p if op(i,exps) > 0 then

%p expsrt := [op(expsrt),op(i,exps)] ;

%p end if;

%p end do;

%p # check that list is duplicate-free

%p if nops(expsrt) = nops( convert(expsrt,set)) then

%p return 1;

%p else

%p return 0;

%p end if;

%p else

%p return 0 ;

%p end if;

%p end if;

%p # need a local copy of the list to modify it

%p expscopy := [] ;

%p for i from 1 to nops(exps) do

%p expscopy := [op(expscopy),op(i,exps)] ;

%p end do:

%p # loop over all exponents assigned to the next base in the list.

%p for e from 0 do

%p candf := op(pividx,dvs)^e ;

%p if modp(n,candf) <> 0 then

%p break;

%p end if;

%p # assign e to the local copy of exponents

%p expscopy := subsop(pividx=e,expscopy) ;

%p a := a+procname(n,dvs,expscopy,pividx+1) ;

%p end do:

%p return a;

%p end proc:

%p A255231 := proc(n)

%p local dvs,dvscnt,exps ;

%p if n = 1 then

%p return 1;

%p end if;

%p # candidates for the bases are all divisors except 1

%p dvs := convert(numtheory[divisors](n) minus {1},list) ;

%p dvscnt := nops(dvs) ;

%p # list of exponents starts at all-0 and is

%p # increased recursively

%p exps := [seq(0,e=1..dvscnt)] ;

%p # take any subset of dvs for the bases, i.e. exponents 0 upwards

%p Apiv(n,dvs,exps,1) ;

%p end proc:

%p seq(A255231(n),n=1..120) ; # _R. J. Mathar_, Nov 05 2016

%Y Cf. A000688 (b_i not necessarily distinct).

%Y Cf. A001248, A005117, A030078, A030514, A054753, A065036, A085986, A085987, A143610, A178739.

%K nonn

%O 1,4

%A _Saverio Picozzi_, Feb 18 2015

%E Values corrected. Incorrect comments removed. - _R. J. Mathar_, Nov 05 2016

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Last modified March 29 10:44 EDT 2024. Contains 371268 sequences. (Running on oeis4.)