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A006036 Primitive pseudoperfect numbers.

%I A006036 M4133 #43 Feb 16 2025 08:32:29
%S A006036 6,20,28,88,104,272,304,350,368,464,490,496,550,572,650,748,770,910,
%T A006036 945,1184,1190,1312,1330,1376,1430,1504,1575,1610,1696,1870,1888,1952,
%U A006036 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
%N A006036 Primitive pseudoperfect numbers.
%C A006036 A primitive pseudoperfect number is a pseudoperfect number that is not a multiple of any other pseudoperfect number.
%C A006036 The odd entries so far are identical to the odd primitive abundant A006038. - _Walter Kehowski_, Aug 12 2005
%C A006036 Zachariou and Zachariou (1972) called these numbers "irreducible semiperfect numbers". - _Amiram Eldar_, Dec 04 2020
%D A006036 Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, Section B2, pp. 74-75.
%D A006036 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006036 Donovan Johnson, <a href="/A006036/b006036.txt">Table of n, a(n) for n = 1..10000</a>
%H A006036 Richard K. Guy, <a href="/A001599/a001599_1.pdf">Letter to N. J. A. Sloane with attachment, Jun. 1991</a>
%H A006036 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimitivePseudoperfectNumber.html">Primitive Pseudoperfect Number</a>.
%H A006036 Andreas Zachariou and Eleni Zachariou, <a href="http://www.hms.gr/apothema/?s=sap&amp;i=261">Perfect, Semi-Perfect and Ore Numbers</a>, Bull. Soc. Math. Grèce (New Ser.), Vol. 13, No. 13A (1972), pp. 12-22; <a href="https://eudml.org/doc/238923">alternative link</a>.
%p A006036 with(numtheory): with(combinat): issemiperfect := proc(n) local b, S;
%p A006036 b:=false; S:=subsets(divisors(n) minus {n}); while not S[finished] do if
%p A006036 convert(S[nextvalue](),`+`)=n then b:=true; break fi od; return b end:
%p A006036 L:=remove(proc(z) isprime(z) end,[$1..5000]): PP:=[]: for zz from 1 to 1 do
%p A006036 for n in L do if issemiperfect(n) then PP:=[op(PP),n] fi od od;
%p A006036 sr := proc(l::list) local x, R, S, P, L; S:=sort(l); R:=[]; P:=S;
%p A006036 for x in S do
%p A006036 if not(x in R) then
%p A006036 L:=selectremove(proc(z) z>x and z mod x = 0 end, P);
%p A006036 R:=[op(R),op(L[1])]; P:=L[2];
%p A006036 fi; od; return P; end:
%p A006036 PPP:=sr(PP); # primitive pseudoperfect numbers less than 5000 # _Walter Kehowski_, Aug 12 2005
%t A006036 (* 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 *)
%o A006036 (Haskell)
%o A006036 a006036 n = a006036_list !! (n-1)
%o A006036 a006036_list = filter (all (== 0) . map a210455 . a027751_row) a005835_list
%o A006036 -- _Reinhard Zumkeller_, Jan 21 2013
%Y A006036 Cf. A005835.
%Y A006036 Cf. A210455, A027751.
%K A006036 nonn,nice
%O A006036 1,1
%A A006036 _N. J. A. Sloane_, _R. K. Guy_
%E A006036 More terms from _Walter Kehowski_, Aug 12 2005