A300827 Lexicographically earliest sequence such that a(i) = a(j) => A324193(i) = A324193(j) for all i, j >= 1.

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

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_{d|n, d<n} A297112(d), still a single 1-bit in binary expansion of A297168(n) might be formed as a sum of several terms of A297112(d), i.e., could be born of carries.

From Antti Karttunen, Feb 28 2019: (Start)

A297168(n) = Sum_{d|n, d<n} A297112(d) will not produce any carries (in base-2) if and only if n is a power of prime. Only in that case the number of summands (A000005(n)-1) is equal to the number of prime factors counted with multiplicity, A001222(n) = A000120(A156552(n)). (A notable subset of such numbers is A324201, numbers that are mapped to even perfect numbers by A156552). Precisely because there are so few points with duplicate values (apart from primes), this sequence is not particularly good for filtering other sequences, because the number of false positives is high. Any of the related sequences like A324203, A324196, A324197 or A324181 might work better in that respect. In any case, the following implications hold (see formula section of A324193 for the latter): (End)

For all i, j:

a(i) = a(j) => A297168(i) = A297168(j). (The same holds for A297169).

a(i) = a(j) => A324181(i) = A324181(j) => A324120(i) = A324120(j).

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_{d|n, 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, n-1, 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. A000120, A000961, A006094, A030078, A046660, A061395, A101296, A156552, A297112, A297167, A297168, A297169, A324193, A324120, A324201.

Cf. also A324181, A323914, A324196, A324197, A324203.

Sequence in context: A318835 A319353 A319343 * A144371 A323157 A348554

Adjacent sequences: A300824 A300825 A300826 * A300828 A300829 A300830

Keyword

nonn

Author

Antti Karttunen, Mar 13 2018

Extensions

Name changed by Antti Karttunen, Feb 21 2019

Status

approved