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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 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: A159631 A335428 A050377 * 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 April 18 17:25 EDT 2021. Contains 343089 sequences. (Running on oeis4.)