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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
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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

LINKS

R. J. Mathar, Table of n, a(n) for n = 1..419

R. J. Mathar, Factorizations of integers into factors with distinct bases and exponents

Index to sequences related to prime signature

FORMULA

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

EXAMPLE

From R. J. Mathar, Nov 05 2016: (Start)

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

(End)

MAPLE

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

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

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

    dvscnt := nops(dvs) ;

    a := 0 ;

    if pividx > dvscnt then

        # have exhausted the exponent list: leave of the recursion

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

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

            # construct list of non-0 exponents

            expsrt := [];

            for i from 1 to dvscnt do

                if op(i, exps) > 0 then

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

                end if;

            end do;

            # check that list is duplicate-free

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

                return 1;

            else

                return 0;

            end if;

        else

            return 0 ;

        end if;

    end if;

    # need a local copy of the list to modify it

    expscopy := [] ;

    for i from 1 to nops(exps) do

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

    end do:

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

    for e from 0 do

        candf := op(pividx, dvs)^e ;

        if modp(n, candf) <> 0 then

            break;

        end if;

        # assign e to the local copy of exponents

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

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

    end do:

    return a;

end proc:

A255231 := proc(n)

    local dvs, dvscnt, exps ;

    if n = 1 then

        return 1;

    end if;

    # candidates for the bases are all divisors except 1

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

    dvscnt := nops(dvs) ;

    # list of exponents starts at all-0 and is

    # increased recursively

    exps := [seq(0, e=1..dvscnt)] ;

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

    Apiv(n, dvs, exps, 1) ;

end proc:

seq(A255231(n), n=1..120) ; # R. J. Mathar, Nov 05 2016

CROSSREFS

Cf. A000688 (b_i not necessarily distinct).

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

Sequence in context: A335428 A050377 A344417 * A294874 A318324 A317934

Adjacent sequences:  A255228 A255229 A255230 * A255232 A255233 A255234

KEYWORD

nonn

AUTHOR

Saverio Picozzi, Feb 18 2015

EXTENSIONS

Values corrected. Incorrect comments removed. - R. J. Mathar, Nov 05 2016

STATUS

approved

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Last modified July 29 17:41 EDT 2021. Contains 346346 sequences. (Running on oeis4.)