A236359 Pseudoperfect (or semiperfect) numbers in which a sum of contiguous proper divisors of n equals n.

6, 18, 24, 28, 36, 42, 54, 66, 78, 102, 108, 114, 126, 132, 138, 162, 174, 186, 196, 198, 222, 234, 246, 258, 282, 288, 294, 306, 318, 324, 342, 354, 360, 366, 378, 402, 414, 426, 432, 438, 462, 474, 486, 496, 498, 504, 522, 534, 540, 546, 558, 582, 594, 600, 606, 618, 642, 654, 666, 678, 684, 690, 696, 702, 714, 726

Offset

1,1

Comments

Also includes perfect numbers.

Are there numbers that contain multiple contiguous divisor sums?

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from Matthew Schuster)

Matthew Schuster, A236359.cpp; source file

Example

The proper divisors of 132 are [1,2,3,4,6,11,12,22,33,44,66]; the contiguous divisor set 4,6,11,12,22,33,44 sums to 132.

Mathematica

aQ[n_] := Catch@Block[{d = Most@Divisors@n, s, i=1}, s = Accumulate@d; While[s != {}, If[MemberQ[s, n], Throw@True, s = Rest[s - d[[i++]]]]]; False]; Select[ Range@ 726, aQ] (* Giovanni Resta, Jan 23 2014 *)
Select[Range[800], MemberQ[Flatten[Table[Total/@Partition[Most[Divisors[ #]], n, 1], {n, DivisorSigma[0, #]-1}]], #]&] (* Harvey P. Dale, Apr 25 2015 *)

Prog

(PARI) is(n)=my(d=divisors(n), i=1, j=1, s=1); while(i<#d, s+=d[i++]; while(s>n, s-=d[j]; j++); if(s==n, return(i<#d))); 0 \\ Charles R Greathouse IV, Jan 23 2014
(Python)
from sympy import divisors
A236359_list = []
for n in range(1, 10**3):
....d = divisors(n)
....d.pop()
....ld = len(d)
....if sum(d) >= n:
........s, j = d[0], 1
........for i in range(ld-1):
............while s < n and j < ld:
................s += d[j]
................j += 1
............if s == n:
................A236359_list.append(n)
................break
............j -= 1
............s -= d[i]+d[j] # Chai Wah Wu, Sep 16 2014

Crossrefs

Subsequence of A005835.

Sequence in context: A358748 A081318 A275168 * A011775 A015707 A236864

Adjacent sequences: A236356 A236357 A236358 * A236360 A236361 A236362

Keyword

nonn

Author

Matthew Schuster, Jan 23 2014

Status

approved