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

1,1

Comments

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.

Links

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

Example

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.

Mathematica

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[#] &]

Crossrefs

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

Sequence in context: A035527 A220437 A332465 * A173360 A085844 A325638

Adjacent sequences: A335287 A335288 A335289 * A335291 A335292 A335293

Keyword

nonn

Author

Amiram Eldar, May 30 2020

Status

approved