A069023 Define a subset of divisors of n to be a dedicated subset if the product of any two members is also a divisor of n. 1 is not allowed as a member as it gives trivially 1*d = d a divisor. a(n) is the number of dedicated subsets of divisors of n with at least two members.

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 3, 0, 1, 1, 2, 0, 3, 0, 3, 1, 1, 0, 9, 0, 1, 1, 3, 0, 7, 0, 5, 1, 1, 1, 9, 0, 1, 1, 9, 0, 7, 0, 3, 3, 1, 0, 17, 0, 3, 1, 3, 0, 9, 1, 9, 1, 1, 0, 20, 0, 1, 3, 8, 1, 7, 0, 3, 1, 7, 0, 28, 0, 1, 3, 3, 1, 7, 0, 17, 2, 1, 0, 20, 1, 1, 1, 9, 0, 20, 1, 3, 1, 1, 1, 35, 0, 3, 3, 9, 0, 7

Offset

1,12

Comments

a(n) is determined by the prime signature of n.

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Antti Karttunen, Scheme-program for computing this sequence

Index entries for sequences computed from exponents in factorization of n

Formula

It seems that for n >= 3, a(p^n) = A077866(n-3). - Antti Karttunen, Nov 24 2017

Example

a(12) = 3. The divisors of 12 are 1,2,3,4,6,12. The divisor subsets (2,3),(2,6) and (3,4) are such that their product is also a divisor of 12. a(24) = 9 and the dedicated divisor subsets are (2,3),(2,4),(2,6),(2,12),(3,4),(3,8),(4,6),(2,3,4),(2,4,6).

Prog

(PARI)
\\ The following program is very inefficient:
A069023(n) = { if(bigomega(n)<2, return(0)); my(pds=(divisors(n)[2..numdiv(n)]), subsets = select(v -> (length(v)>=2), powerset(pds)), pair_products = apply(ss -> podp(ss), subsets), prodsmodn = apply(pps -> vector(#pps, i, n%pps[i]), pair_products)); length(select(s -> 0==vecsum(s), prodsmodn)); };
powerset(v) = { my(siz=2^length(v), pv=vector(siz)); for(i=0, siz-1, pv[i+1] = choosebybits(v, i)); pv; };
choosebybits(v, m) = { my(s=vector(hammingweight(m)), i=j=1); while(m>0, if(m%2, s[j] = v[i]; j++); i++; m >>= 1); s; };
podp(v) = { my(siz=binomial(length(v), 2), rv=vector(siz), k=0); for(i=1, length(v)-1, for(j=i+1, length(v), k++; rv[k] = v[i]*v[j])); rv; }; \\ podp = product of distinct pairs
\\ Antti Karttunen, Nov 24 2017
(Scheme) ;; See in the links-section.

Crossrefs

Cf. A077866.

Sequence in context: A249442 A334739 A131802 * A275336 A373949 A091614

Adjacent sequences: A069020 A069021 A069022 * A069024 A069025 A069026

Keyword

nonn

Author

Amarnath Murthy, Apr 02 2002

Extensions

Edited by David Wasserman, Mar 26 2003

Status

approved