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A335742 Pseudoperfect (or semiperfect) numbers k having more than one set of contiguous proper divisors whose sum equals k.

%I A335742 #24 Aug 14 2021 18:55:38
%S A335742 12978,13338,34920,41382,76626,176946,253422,455202,1336734,2410254,
%T A335742 3187782,3214458,3277800,3347838,3387240,3427866,3507894,3587922,
%U A335742 3614598,3694626,3747978,3774654,3908034,4094766,4148118,4174794,4228146,4414878,4494906,4628286
%N A335742 Pseudoperfect (or semiperfect) numbers k having more than one set of contiguous proper divisors whose sum equals k.
%C A335742 Observation of some pseudoperfect numbers with an attribute similar to multiperfect numbers.
%C A335742 A total of 84 of the 96 terms (representing all terms less than 10^7) are equal to 0 (mod 13338).
%C A335742 Many of the terms greater than (13338*239)-1 are in the form of 13338*p where p>=239. Prime(52)*1338 through Prime(50188)*1338 were tested and are all terms in this sequence.
%C A335742 There are numbers greater than (13338*239)-1 in this sequence that do not have 13338 as a divisor, for example; 3277800, 3387240, 5007222 and 9233154.
%C A335742 (Uni-)Perfect numbers cannot be in this sequence.
%e A335742 The proper divisors of 12978 are (1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 103, 126, 206, 309, 618, 721, 927, 1442, 1854, 2163, 4326, 6489).
%e A335742 The contiguous divisor lists of (3+6+7+9+14+18+21+42+63+103+126+206+309+618+721+927+1442+1854+2163+4326) and (2163+4326+6489) equals 12978.
%t A335742 pspQ[n_] := Module[{d = Divisors[n]}, c = Accumulate[d]; Length @ Intersection[c, c + n] > 2]; Select[Range[10^6], pspQ] (* _Amiram Eldar_, Jul 02 2020 *)
%o A335742 (Python)
%o A335742 # Pseudoperfect (or semiperfect) numbers having more than one set of contiguous proper divisors whose sum equals n.
%o A335742 import sympy
%o A335742 A335742_list = []
%o A335742 for n in range(1, (10**7)+1):
%o A335742     # create an ascending list of divisors of n.
%o A335742     n_divs = list(sympy.divisors(n))
%o A335742     # pop last divisor, which equals n, so only proper divisors are examined.
%o A335742     n_divs.pop()
%o A335742     # reset iterator for sets of contiguous proper divisors whose sum equals n.
%o A335742     itr = 0
%o A335742     # run the outer loop for each proper divisor of n.
%o A335742     for i in range(len(n_divs)+1):
%o A335742         # run the inner loop for each divisor >= i.
%o A335742         for j in range(i, len(n_divs)+1):
%o A335742             # if sum of divisors i:j is greater than n; continue to next n.
%o A335742             if sum(n_divs[i:j]) > n:
%o A335742                 continue
%o A335742             # elif sum of divisors i:j equals n; increment itr; if itr > 1; append n to sequence.
%o A335742             elif sum(n_divs[i:j]) == n:
%o A335742                 itr += 1
%o A335742                 if itr > 1:
%o A335742                     A335742_list.append(n)
%Y A335742 Subsequence of A005835 and A236359.
%K A335742 nonn
%O A335742 1,1
%A A335742 _Matthew Schuster_, Jul 02 2020