This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A048109 Numbers having equally many squarefree and nonsquarefree divisors; number of unitary divisors of n (A034444) = number of non-unitary divisors of n (A048105). 9
 8, 24, 27, 40, 54, 56, 88, 104, 120, 125, 135, 136, 152, 168, 184, 189, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 375, 376, 378, 408, 424, 440, 456, 459, 472, 488, 513, 520, 536, 552, 568, 584, 594, 616, 621, 632, 664, 680, 686, 696 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS For these terms the number of divisors should be a special power of two because ud(n)=2^r and nud(n)=ud(n). In particular the exponent of 2 is 1+A001221(n), the number of distinct prime factors + 1. Thus this is a subsequence of A036537 where A000005(A036537(n)) = 2^s; here s=1+A001221(n). Let us introduce a function D(n)=sigma_0(n)/(2^(alpha(1)+...+alpha(r)), sigma_0(n) number of divisors of n (A000005), prime factorization of n=p(1)^alpha(1) * ... * p(r)^alpha(r), alpha(1)+...+alpha(r) is sequence (A086436). This function splits the set of positive integers into subsets, according to the value of D(n). Squarefree numbers (A005117) has D(n)=1, other numbers are "deviated" from the squarefree ideal and have 0 < D(n) < 1. So for D(n)=1/2 we have A048109, D(n)=3/4 we have A067295. - Ctibor O. Zizka, Sep 21 2008 Integers n such that there are exactly 3 Abelian groups of order n. That is, n such that A000688(n)=3. In other words, in the prime factorization of n there is exactly one prime with exponent of 3 and the others have exponent of 1. - Geoffrey Critzer, Jun 09 2015 LINKS Robert Israel, Table of n, a(n) for n = 1..10000 FORMULA d[ x ] = 2^(r[ x ]+1) or A000005[ x ]=2^(A(001221[ x ])+1)=2*A034444[ x ]. EXAMPLE n = 88 = 2*2*2*11 has 8 divisors, of which 4 are unitary divisors (because of 2 distinct prime factors) and 4 are nonunitary divisors: U={1,88,11,8} and NU = {2,44,4,22}. MAPLE filter:= proc(n) local F;   F:= ifactors(n)[2];   mul(t[2]+1, t=F) = 2^(1+nops(F)) end proc; select(filter, [\$1..1000]); # Robert Israel, Jun 09 2015 MATHEMATICA Position[Table[FiniteAbelianGroupCount[n], {n, 1, 1000}], 3] // Flatten (* Geoffrey Critzer, Jun 09 2015 *) PROG (PARI) is(n)=select(e->e>1, factor(n)[, 2])==[3]~ \\ Charles R Greathouse IV, Jun 10 2015 (PARI) isok(n) = sumdiv(n, d, issquarefree(d)) == sumdiv(n, d, !issquarefree(d)); \\ Michel Marcus, Jun 24 2015 CROSSREFS Cf. A000005, A001221, A034444, A036537, A048106, A048107. Sequence in context: A195086 A176297 A175496 * A068781 A212861 A038524 Adjacent sequences:  A048106 A048107 A048108 * A048110 A048111 A048112 KEYWORD nonn AUTHOR EXTENSIONS New name based on comment by Ivan Neretin, Jun 19 2015 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 7 22:29 EST 2019. Contains 329850 sequences. (Running on oeis4.)