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A006036 Primitive pseudoperfect numbers.
(Formerly M4133)
10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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
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.
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 *)
PROG
(Haskell)
a006036 n = a006036_list !! (n-1)
a006036_list = filter (all (== 0) . map a210455 . a027751_row) a005835_list
-- Reinhard Zumkeller, Jan 21 2013
CROSSREFS
Cf. A005835.
Sequence in context: A180332 A338133 A064771 * A308710 A242341 A140738
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Walter Kehowski, Aug 12 2005
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)