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%I #93 Aug 28 2018 21:16:13
%S 1,2,4,6,8,9,12,15,16,18,20,24,25,28,30,32,35,36,40,42,45,48,49,50,54,
%T 56,60,63,64,66,70,72,77,80,81,84,88,90,91,96,98,99,100,104,108,110,
%U 112,117,120,121,126,128,130,132,135,140,143,144,150,153,154,156,160
%N Numbers n such that the sum of the middle divisors of n (A071090) is not zero.
%C Numbers n such that A067742(n) is nonzero.
%C Numbers of the form m*k with m <= k <= 2m. - _Vladeta Jovovic_, May 07 2005
%C Numbers occurring in A100345 (except 0). - _Franklin T. Adams-Watters_, Apr 04 2010
%C This sequence is closed under multiplication. If n = a*b with a <= b <= 2a, and m = c*d with c <= d <= 2c, then min(a*d,b*c)*max(a*d,b*c) is a factorization of m*n with the specified property. - _Franklin T. Adams-Watters_, Apr 07 2010
%C Also numbers n with the property that the number of parts in the symmetric representation of sigma(n) is odd. - _Michel Marcus_ and _Omar E. Pol_, Apr 25 2014. (For a proof see the link in A071561.) - _Hartmut F. W. Hoft_, Sep 09 2015
%C Among these numbers, those with sigma(n) also odd are 1, 2, 4, 8, 9, 16, ..., that is, probably A028982 (squares and twice squares). - _Michel Marcus_, Jun 21 2014
%C Records in A244367. - _Omar E. Pol_, Jul 27 2014
%C Starting from a(5), the sequence is a subset of a sequence formed out of the prime factorization of A129912(m), m >= 5; see associated PARI code in Prog section. - _Bill McEachen_, Jan 25 2018
%C For numbers n = 2^m * q, m >= 0, q odd, and where r(n) = floor( (sqrt(8n+1) - 1)/2 ), the symmetric representation of sigma(n) has an odd number of parts precisely when there exists an odd divisor d of n satisfying d <= r(n) and d*2^(m+1) > r(n); see the link for a proof and see the associated Mathematica code. - _Hartmut F. W. Hoft_, Feb 12 2018
%C All hexagonal numbers A000384 > 0 are in the sequence. - _Omar E. Pol_, Aug 28 2018
%H Charles R Greathouse IV, <a href="/A071562/b071562.txt">Table of n, a(n) for n = 1..10000</a>
%H Hartmut F. W. Hoft, <a href="/A071562/a071562.pdf">Proof of property for odd divisors</a>
%H H. Maier and G. Tenenbaum, <a href="http://deepblue.lib.umich.edu/bitstream/handle/2027.42/46612/222_2005_Article_BF01388495.pdf">On the set of divisors of an integer</a>, Invent. Math. 76:1 (1984), 121-128.
%e From _Hartmut F. W. Hoft_, Feb 12 2018: (Start)
%e 63 = 3^2*7 is in the sequence since 7*2^1 > r(63) = 10.
%e 80 = 2^4*5 is in the sequence since 1*2^5 > r(80) = 12. (End)
%t f[n_] := Plus @@ Select[ Divisors[n], Sqrt[n/2] <= # < Sqrt[n*2] &]; Select[ Range[175], f[ # ] != 0 &]
%t (* Related to the symmetric representation of sigma *)
%t (* subsequence of odd parts of number k for m <= k <= n *)
%t (* Function a237270[] is defined in A237270 *)
%t (* Using Wilson's Mathematica program (see above) I verified the equality of both for numbers k <= 10000 *)
%t a071562[m_,n_]:=Select[Range[m,n],OddQ[Length[a237270[#]]]&]
%t a071562[1,160] (* data *)
%t (* _Hartmut F. W. Hoft_, Jun 23 2014 *)
%t (* implementation using the odd divisor property *)
%t evenExp[n_] := First[NestWhile[{#[[1]]+1, #[[2]]/2}&, {0, n}, EvenQ[Last[#]]&]]
%t oddSRQ[n_] := Module[{e=2^evenExp[n], Floor[(Sqrt[8n+1]-1)/2]}, Select[Divisors[n/e], #<=r&&2 e #>r&]!={}]
%t a071562D[m_, n_] := Select[Range[m, n], oddSRQ]
%t a071562D[1, 160] (* data *) (* _Hartmut F. W. Hoft_, Feb 12 2018 *)
%o (PARI) is(n)=fordiv(n,d, if(d^2>=n/2 && d^2<2*n, return(1))); 0 \\ _Charles R Greathouse IV_, Aug 01 2016
%o (PARI) is(n,f=factor(n))=my(t=(n+1)\2); fordiv(f,d, if(d^2>=t, return(d^2<2*n))); 0 \\ _Charles R Greathouse IV_, Jan 22 2018
%o (PARI) list(lim)=my(v=List(),t); forfactored(n=1,lim\1, t=(n[1]+1)\2; fordiv(n[2],d, if(d^2>=t, if(d^2<2*n[1], listput(v,n[1])); break))); Vec(v) \\ _Charles R Greathouse IV_, Jan 22 2018
%o (PARI) /* functional code associated to the A129912 comment above */ for(j5=5, length(A129912), a=Mat(); a=factor(A129912[j5]); sum2=0; for(i5=1, length(a[,2]), sum2=sum2+a[i5,2]); listput(final,length(a[,1])*sum2)); v=Set(final); \\ _Bill McEachen_, Jan 25 2018
%Y Cf. A067742.
%Y The complement is A071561.
%Y Cf. A000203, A028982, A071090, A100345, A175040, A176039 (primitive elements), A237270, A237271, A237593, A244367.
%K nonn
%O 1,2
%A _Robert G. Wilson v_, May 30 2002
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