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A255231
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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.
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9
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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
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OFFSET
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1,4
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COMMENTS
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Not multiplicative: a(48) = a(2^4*3) = 5 <> a(2^4)*a(3) = 4*1 = 4. - R. J. Mathar, Nov 05 2016
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LINKS
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FORMULA
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EXAMPLE
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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)
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MAPLE
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# 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:
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:
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CROSSREFS
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Cf. A000688 (b_i not necessarily distinct).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Values corrected. Incorrect comments removed. - R. J. Mathar, Nov 05 2016
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STATUS
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approved
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