

A071562


Numbers n such that the sum of the middle divisors of n (A071090) is not zero.


43



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, 56, 60, 63, 64, 66, 70, 72, 77, 80, 81, 84, 88, 90, 91, 96, 98, 99, 100, 104, 108, 110, 112, 117, 120, 121, 126, 128, 130, 132, 135, 140, 143, 144, 150, 153, 154, 156, 160
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OFFSET

1,2


COMMENTS

Numbers n such that A067742(n) is nonzero.
Numbers of the form m*k with m <= k <= 2m.  Vladeta Jovovic, May 07 2005
Numbers occurring in A100345 (except 0).  Franklin T. AdamsWatters, Apr 04 2010
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. AdamsWatters, Apr 07 2010
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
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
Records in A244367.  Omar E. Pol, Jul 27 2014
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
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
All hexagonal numbers A000384 > 0 are in the sequence.  Omar E. Pol, Aug 28 2018


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Hartmut F. W. Hoft, Proof of property for odd divisors
H. Maier and G. Tenenbaum, On the set of divisors of an integer, Invent. Math. 76:1 (1984), 121128.


EXAMPLE

From Hartmut F. W. Hoft, Feb 12 2018: (Start)
63 = 3^2*7 is in the sequence since 7*2^1 > r(63) = 10.
80 = 2^4*5 is in the sequence since 1*2^5 > r(80) = 12. (End)


MATHEMATICA

f[n_] := Plus @@ Select[ Divisors[n], Sqrt[n/2] <= # < Sqrt[n*2] &]; Select[ Range[175], f[ # ] != 0 &]
(* Related to the symmetric representation of sigma *)
(* subsequence of odd parts of number k for m <= k <= n *)
(* Function a237270[] is defined in A237270 *)
(* Using Wilson's Mathematica program (see above) I verified the equality of both for numbers k <= 10000 *)
a071562[m_, n_]:=Select[Range[m, n], OddQ[Length[a237270[#]]]&]
a071562[1, 160] (* data *)
(* Hartmut F. W. Hoft, Jun 23 2014 *)
(* implementation using the odd divisor property *)
evenExp[n_] := First[NestWhile[{#[[1]]+1, #[[2]]/2}&, {0, n}, EvenQ[Last[#]]&]]
oddSRQ[n_] := Module[{e=2^evenExp[n], Floor[(Sqrt[8n+1]1)/2]}, Select[Divisors[n/e], #<=r&&2 e #>r&]!={}]
a071562D[m_, n_] := Select[Range[m, n], oddSRQ]
a071562D[1, 160] (* data *) (* Hartmut F. W. Hoft, Feb 12 2018 *)


PROG

(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
(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
(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
(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


CROSSREFS

Cf. A067742.
The complement is A071561.
Cf. A000203, A028982, A071090, A100345, A175040, A176039 (primitive elements), A237270, A237271, A237593, A244367.
Sequence in context: A244799 A316350 A335127 * A241562 A100345 A261011
Adjacent sequences: A071559 A071560 A071561 * A071563 A071564 A071565


KEYWORD

nonn


AUTHOR

Robert G. Wilson v, May 30 2002


STATUS

approved



