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 A335742 Pseudoperfect (or semiperfect) numbers having more than one set of contiguous proper divisors whose sum equals n. 0

%I

%S 12978,13338,34920,41382,76626,176946,253422,455202,1336734,2410254,

%T 3187782,3214458,3277800,3347838,3387240,3427866,3507894,3587922,

%U 3614598,3694626,3747978,3774654,3908034,4094766,4148118,4174794,4228146,4414878,4494906,4628286

%N Pseudoperfect (or semiperfect) numbers having more than one set of contiguous proper divisors whose sum equals n.

%C Observation of some pseudoperfect numbers with an attribute similar to multiperfect numbers.

%C A total of 84 of the 96 terms (representing all terms less than 10^7) are equal to 0 (mod 13338).

%C 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 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 (Uni-)Perfect numbers cannot be in this sequence.

%e 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 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 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 #(python3)

%o # Pseudoperfect (or semiperfect) numbers having more than one set of contiguous proper divisors whose sum equals n.

%o import sympy

%o A335742_list = []

%o for n in range(1, (10**7)+1):

%o # create an ascending list of divisors of n.

%o n_divs = list(sympy.divisors(n))

%o # pop last divisor, which equals n, so only proper divisors are examined.

%o n_divs.pop()

%o # reset iterator for sets of contiguous proper divisors whose sum equals n.

%o itr = 0

%o # run the outer loop for each proper divisor of n.

%o for i in range(len(n_divs)+1):

%o # run the inner loop for each divisor >= i.

%o for j in range(i, len(n_divs)+1):

%o # if sum of divisors i:j is greater than n; continue to next n.

%o if sum(n_divs[i:j]) > n:

%o continue

%o # elif sum of divisors i:j equals n; increment itr; if itr > 1; append n to sequence.

%o elif sum(n_divs[i:j]) == n:

%o itr += 1

%o if itr > 1:

%o A335742_list.append(n)

%Y Subsequence of A005835 and A236359.

%K nonn

%O 1,1

%A _Matthew Schuster_, Jul 02 2020

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Last modified June 14 08:31 EDT 2021. Contains 345018 sequences. (Running on oeis4.)