A306987 -id:A306987 - cp's OEIS frontend

A335290 Primitive pseudoperfect numbers (A006036) that are not primitive abundant (A071395).

6, 28, 350, 490, 496, 770, 910, 1190, 1330, 1610, 2030, 2170, 2590, 2870, 3010, 3290, 3710, 4130, 4270, 4690, 4970, 5110, 5530, 5810, 6230, 6790, 7070, 7210, 7490, 7630, 7910, 8128, 8890, 9170, 9196, 9590, 9730, 15884, 19228, 24244, 25916, 30932, 34276, 35948

Offset: 1

Amiram Eldar, May 30 2020

nonn

Includes all the perfect numbers (A000396). The nonperfect terms have an abundant proper divisor which is not pseudoperfect, i.e., a proper divisor which is a weird number (A006037).
The first term with one weird divisor is a(3) = 350, having the weird divisor 70.
The first term with 2 weird divisors is a(202) = 658312, having the 2 weird divisors 9272 and 10792.
The first term with 3 weird divisors is a(353) = 1574930, having the 3 weird divisors 70, 10430 and 10570.

			350 is a term since it is pseudoperfect: 1 + 5 + 14 + 35 + 50 + 70 + 175 = 350. All of its proper divisors, {1, 2, 5, 7, 10, 14, 25, 35, 50, 70, 175} are not pseudoperfect, and it is not primitive abundant, since its divisor 70 is abundant.

Amiram Eldar, Table of n, a(n) for n = 1..2500

Cf. A000396, A006036, A006037, A071395, A306987.

pspQ[n_] := Module[{d = Most @ Divisors[n], x}, Plus @@d >= n && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] > 0]; primPspQ[n_] := pspQ[n] && AllTrue[Most @ Divisors[n], !pspQ[#] &]; primAbQ[n_] := DivisorSigma[1, n] > 2*n && AllTrue[Most @ Divisors[n], DivisorSigma[1, #] < 2*# &]; Select[Range[1000], primPspQ[#] && !primAbQ[#] &]

cp's OEIS Frontend

A335290 Primitive pseudoperfect numbers (A006036) that are not primitive abundant (A071395).

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