A334406
Unitary pseudoperfect numbers k such that there is a subset of unitary divisors of k whose sum is 2*k and for each d in this subset k/d is also in it.
Original entry on oeis.org
6, 60, 90, 210, 330, 546, 660, 714, 1770, 2310, 2730, 3198, 3486, 3570, 3990, 4290, 4620, 4830, 5460, 5610, 6006, 6090, 6270, 6510, 6630, 6930, 7140, 7410, 7590, 7770, 7854, 7980, 8190, 8580, 8610, 8778, 8970, 9030, 9240, 9570, 9660, 9690, 9870, 10374, 10626, 10710
Offset: 1
210 is a term since {1, 2, 3, 14, 15, 70, 105, 210} is a subset of its unitary divisors whose sum is 420 = 2 * 210, and for each divisor d in this subset 210/d is also in it: 1 * 210 = 2 * 105 = 3 * 70 = 14 * 15 = 210.
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seqQ[n_] := Module[{d = Select[Divisors[n], CoprimeQ[#, n/#] &]}, nd = Length[d]; divpairs = d[[1 ;; nd/2]] + d[[-1 ;; nd/2 + 1 ;; -1]]; SeriesCoefficient[Series[Product[1 + x^divpairs[[i]], {i, Length[divpairs]}], {x, 0, 2*n}], 2*n] > 0]; Select[Range[2, 1000], seqQ]
A334409
Numbers m such that the sum of the first k divisors and the last k divisors of m is equal to 2*m for some k that is smaller than half of the number of divisors of m.
Original entry on oeis.org
36, 152812, 6112576, 72702928, 154286848, 397955025, 15356519488, 23003680492, 35755623784, 93789539668, 302122464256, 351155553970, 1081806148665, 1090488143872, 1663167899025, 2233955122576
Offset: 1
36 is a term since its divisors are {1, 2, 3, 4, 6, 9, 12, 18, 36} and the sum of the first 3 and last 3 divisors is (1 + 2 + 3) + (12 + 18 + 36) = 72 = 2 * 36.
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seqQ[n_] := Module[{d = Divisors[n]}, nd = Length[d]; nd2 = Ceiling[nd/2] - 1; s = Accumulate[d[[1 ;; nd2]] + n/d[[1 ;; nd2]]]; MemberQ[s, 2*n]]; Select[Range[10^6], seqQ]
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from itertools import count, islice, accumulate
from sympy import divisors
def A334409_gen(startvalue=1): # generator of terms >= startvalue
for n in count(max(startvalue,1)):
ds = divisors(n)
if any(s==2*n for s in accumulate(ds[i]+ds[-1-i] for i in range((len(ds)-1)//2))):
yield n
A334409_list = list(islice(A334409_gen(),2)) # Chai Wah Wu, Feb 19 2022
A366172
Strongly 2-near perfect numbers.
Original entry on oeis.org
156, 352, 6832, 60976, 91648, 152812, 260865, 2834572, 3335968, 3532096, 4077388, 5044725, 5725504, 6112576, 8102656, 10557148, 19762876, 39411712, 50718016, 66965104, 111372508, 232774912, 483879808, 2045453824, 6849461025, 7904670976, 8521265152, 11818720108, 13112466688, 13714642432
Offset: 1
156 is strongly 2-near perfect since sigma(156) = 392, 2*78 = 156, and 392-2-78 = 2*156.
- Vedant Aryan, Dev Madhavani, Savan Parikh, Ingrid Slattery, and Joshua Zelinsky, On 2-Near Perfect Numbers, arXiv:2310.01305 [math.NT], 2023. See p. 13.
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fQ[n_]:=AnyTrue[Table[DivisorSigma[1,n]-Divisors[n][[i]]-n/Divisors[n][[i]],{i,DivisorSigma[0,n]}],#==2*n&]; Select[Range[61000],fQ[#]&] (* Ivan N. Ianakiev, Oct 04 2023 *)
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isok(k) = my(s=sigma(k)); fordiv(k, d, if (s-d-k/d == 2*k, return(1)));
Showing 1-3 of 3 results.
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