A300716 - OEIS

A300716

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

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OFFSET

1,4

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).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..8192

Index entries for sequences computed from exponents in factorization of n

EXAMPLE

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).

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 *)

PROG

(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)&(d<n), m *= prime(A101296(d)-1))); (m)); };

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

CROSSREFS

Cf. A046523, A101296, A297174, A300250.

Cf. also A292258, A294897, A300827.

Sequence in context: A300509 A074643 A060794 * A074919 A362232 A138009

Adjacent sequences: A300713 A300714 A300715 * A300717 A300718 A300719

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 13 2018

STATUS

approved