A300716 a(1) = 0; for n > 1, a(n) = Product_{d|n, 1A101296(d)-1).

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

Offset: 1

Author

Antti Karttunen, Mar 13 2018

Keywords

• nonn

Comments

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.

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

Examples

			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).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..8192
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A046523, A101296, A297174, A300250.

Cf. also A292258, A294897, A300827.

Programs

Mathematica

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 *)

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)));
A101296(n) = v101296[n];
A300716(n) = { my(m=1); if(1==n, 0, fordiv(n,d,if((d>1)&(dA101296(d)-1))); (m)); };
for(n=1,up_to,write("b300716.txt", n, " ", A300716(n)));