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A300716
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a(1) = 0; for n > 1, a(n) = Product_{d|n, 1<d<n} prime(A101296(d)-1).
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2
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0, 1, 1, 2, 1, 4, 1, 6, 2, 4, 1, 60, 1, 4, 4, 42, 1, 60, 1, 60, 4, 4, 1, 4620, 2, 4, 6, 60, 1, 1000, 1, 546, 4, 4, 4, 21780, 1, 4, 4, 4620, 1, 1000, 1, 60, 60, 4, 1, 1021020, 2, 60, 4, 60, 1, 4620, 4, 4620, 4, 4, 1, 6897000, 1, 4, 60, 12558, 4, 1000, 1, 60, 4, 1000, 1, 75162780, 1, 4, 60, 60, 4, 1000, 1, 1021020, 42, 4, 1
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OFFSET
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1,4
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COMMENTS
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a(n) = Product formed from the primes indexed with the prime signatures of proper divisors of n.
The restricted growth sequence transform of this sequence is A101296 because from the set of prime signatures of the proper divisors of n it is always possible to determine the prime signature of n itself, and vice versa, from the prime signature of n, we can form the set of prime signatures of all its proper divisors.
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LINKS
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EXAMPLE
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For n = 12, whose proper divisors > 1 are 2, 3, 4, 6, their prime signature ranks from A101296 are: 2, 2, 3, 4. We subtract one from each, to form product prime(1)*prime(1)*prime(2)*prime(3) = 2*2*3*5 = 60, which is thus value of a(12).
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MATHEMATICA
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Block[{nn = 83, s}, s = Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Table[Times @@ MapIndexed[Prime[First@#2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]] - Boole[n == 1], {n, nn}]; Table[If[n == 1, 0, Times @@ Map[Prime[FirstPosition[Keys@ s, #][[1]] - 1] &, Most@ Rest@ Divisors@ n]], {n, nn}]] (* Michael De Vlieger, Mar 13 2018 *)
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PROG
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(PARI)
up_to = 8192;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset, vec, bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
v101296 = rgs_transform(vector(up_to, n, A046523(n)));
A300716(n) = { my(m=1); if(1==n, 0, fordiv(n, d, if((d>1)&(d<n), m *= prime(A101296(d)-1))); (m)); };
for(n=1, up_to, write("b300716.txt", n, " ", A300716(n)));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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