

A300827


Filter sequence formed from the multiset of values A297167(d), where d ranges over all proper divisors > 1 of n.


5



1, 2, 2, 3, 2, 4, 2, 4, 5, 6, 2, 7, 2, 8, 9, 10, 2, 11, 2, 12, 13, 14, 2, 15, 16, 17, 9, 18, 2, 19, 2, 20, 21, 22, 23, 24, 2, 25, 26, 27, 2, 28, 2, 29, 30, 31, 2, 32, 33, 34, 35, 36, 2, 37, 38, 39, 40, 41, 2, 42, 2, 43, 44, 45, 46, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 55, 56, 2, 57, 58, 59, 2, 60, 61, 62, 63, 64, 2, 65, 66, 67
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OFFSET

1,2


COMMENTS

Apart from primes, the sequence contains duplicate values at points p*q and p^3, where p*q are the product of two successive primes, with p < q (sequences A006094, A030078). Question: are there any other cases where a(x) = a(y), with x < y ?
The reason why this is not equal to A297169: Even though A297112 contains only powers of two after the initial zero, as A297112(n) = 2^A033265(A156552(d)) for n > 1, and A297168(n) is computed as Sum_{dn, d<n} A297112(d), still a single 1bit in binary expansion of A297168(n) might be formed as a sum of several terms of A297112(d), i.e., be a result of carries.
For all i, j: a(i) = a(j) => A297168(i) = A297168(j). (The same holds for A297169).


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization


FORMULA

Restricted growth sequence transform of sequence f, defined as f(1) = 0, f(2) = 1, and for n > 2, f(n) = Product_{dn, 1<d<n} prime(1+A297167(d)).
a(p) = 2 for all primes p.
a(A006094(n)) = a(A030078(n)), for all n >= 1.


EXAMPLE

For n = 15, with proper divisors 3 and 5, we have f(n) = prime(1+A297167(3)) * prime(1+A297167(5)) = prime(2)*prime(3) = 3*5 = 15.
For n = 27, with proper divisors 3 and 9, we have f(n) = prime(1+A297167(3)) * prime(1+A297167(9)) = prime(2)*prime(3) = 3*5 = 15.
Because f(15) = f(27), the restricted growth sequence transform allots the same number (in this case 9) for both, so a(15) = a(27) = 9.


PROG

(PARI)
up_to = 65537;
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])); }
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1]))); \\ After M. F. Hasler's code for A006530.
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)omega(n)) 1));
Aux300827(n) = { my(m=1); if(n<=2, n1, fordiv(n, d, if((d>1)&(d<n), m *= prime(1+A297167(d)))); (m)); };
write_to_bfile(1, rgs_transform(vector(up_to, n, Aux300827(n))), "b300827.txt");


CROSSREFS

Cf. A006094, A030078, A046660, A061395, A101296, A156552, A297112, A297167, A297168, A297169.
Sequence in context: A318835 A319353 A319343 * A144371 A323157 A305899
Adjacent sequences: A300824 A300825 A300826 * A300828 A300829 A300830


KEYWORD

nonn


AUTHOR

Antti Karttunen, Mar 13 2018


STATUS

approved



