A300716 - OEIS

%I #18 Mar 13 2018 22:11:12

%S 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,

%T 1000,1,546,4,4,4,21780,1,4,4,4620,1,1000,1,60,60,4,1,1021020,2,60,4,

%U 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

%N a(1) = 0; for n > 1, a(n) = Product_{d|n, 1<d<n} prime(A101296(d)-1).

%C a(n) = Product formed from the primes indexed with the prime signatures of proper divisors of n.

%C 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.

%C For all i, j: a(i) = a(j) <=> A101296(i) = A101296(j).

%H Antti Karttunen, <a href="/A300716/b300716.txt">Table of n, a(n) for n = 1..8192</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%e 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).

%t 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 *)

%o (PARI)

%o up_to = 8192;

%o 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; };

%o write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }

%o A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523

%o v101296 = rgs_transform(vector(up_to, n, A046523(n)));

%o A101296(n) = v101296[n];

%o A300716(n) = { my(m=1); if(1==n, 0, fordiv(n,d,if((d>1)&(d<n),m *= prime(A101296(d)-1))); (m)); };

%o for(n=1,up_to,write("b300716.txt", n, " ", A300716(n)));

%Y Cf. A046523, A101296, A297174, A300250.

%Y Cf. also A292258, A294897, A300827.

%K nonn

%O 1,4

%A _Antti Karttunen_, Mar 13 2018