A006036 Primitive pseudoperfect numbers.

6, 20, 28, 88, 104, 272, 304, 350, 368, 464, 490, 496, 550, 572, 650, 748, 770, 910, 945, 1184, 1190, 1312, 1330, 1376, 1430, 1504, 1575, 1610, 1696, 1870, 1888, 1952, 2002, 2030, 2090, 2170, 2205, 2210, 2470, 2530, 2584, 2590, 2870, 2990, 3010, 3128, 3190, 3230, 3290, 3410, 3465, 3496, 3710, 3770, 3944, 4070, 4095, 4130, 4216, 4270, 4288, 4408, 4510, 4544, 4672, 4690, 4712, 4730, 4970

Offset: 1

Author

N. J. A. Sloane, R. K. Guy

Keywords

• nonn
• nice

Comments

A primitive pseudoperfect number is a pseudoperfect number that is not a multiple of any other pseudoperfect number.

The odd entries so far are identical to the odd primitive abundant A006038. - Walter Kehowski, Aug 12 2005

Zachariou and Zachariou (1972) called these numbers "irreducible semiperfect numbers". - Amiram Eldar, Dec 04 2020

References

• Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, Section B2, pp. 74-75.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Donovan Johnson, Table of n, a(n) for n = 1..10000
• Richard K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991
• Eric Weisstein's World of Mathematics, Primitive Pseudoperfect Number.
• Andreas Zachariou and Eleni Zachariou, Perfect, Semi-Perfect and Ore Numbers, Bull. Soc. Math. Grèce (New Ser.), Vol. 13, No. 13A (1972), pp. 12-22; alternative link.

Crossrefs

Cf. A005835.

Cf. A210455, A027751.

Programs

Haskell

a006036 n = a006036_list !! (n-1)
a006036_list = filter (all (== 0) . map a210455 . a027751_row) a005835_list
-- Reinhard Zumkeller, Jan 21 2013

Maple

with(numtheory): with(combinat): issemiperfect := proc(n) local b, S;
b:=false; S:=subsets(divisors(n) minus {n}); while not S[finished] do if
convert(S[nextvalue](),`+`)=n then b:=true; break fi od; return b end:
L:=remove(proc(z) isprime(z) end,[$1..5000]): PP:=[]: for zz from 1 to 1 do
for n in L do if issemiperfect(n) then PP:=[op(PP),n] fi od od;
sr := proc(l::list) local x, R, S, P, L; S:=sort(l); R:=[]; P:=S;
for x in S do
if not(x in R) then
L:=selectremove(proc(z) z>x and z mod x = 0 end, P);
R:=[op(R),op(L[1])]; P:=L[2];
fi; od; return P; end:
PPP:=sr(PP); # primitive pseudoperfect numbers less than 5000 # Walter Kehowski, Aug 12 2005

Mathematica

(* First run one of the programs for A005835 *) A006036 = A005835; curr = 1; max = A005835[[-1]]; While[curr < Length[A006036], currMult = A006036[[curr]]; A006036 = Complement[A006036, Range[2currMult, Ceiling[max/currMult] currMult, currMult]]; curr++]; A006036 (* Alonso del Arte, Sep 08 2012 *)

Extensions

More terms from Walter Kehowski, Aug 12 2005