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A144096 A positive integer n is included if at least one of the exponents of the prime-factorization of n does not occur anywhere in n when the exponents and n are represented in base 2.

%I A144096 #14 Dec 06 2023 18:41:45
%S A144096 8,32,40,63,64,72,96,128,136,168,224,243,264,288,296,297,320,328,384,
%T A144096 480,486,512,513,520,544,552,576,584,594,608,640,648,680,800,891,972,
%U A144096 992,1024,1026,1029,1032,1056,1064,1088,1096,1120,1152,1160,1161,1188
%N A144096 A positive integer n is included if at least one of the exponents of the prime-factorization of n does not occur anywhere in n when the exponents and n are represented in base 2.
%C A144096 A144095(n) = A001221(n) if n is not included in this sequence.
%H A144096 Harvey P. Dale, <a href="/A144096/b144096.txt">Table of n, a(n) for n = 1..1000</a>
%e A144096 40 has the prime-factorization 2^3 * 5^1. So the exponents are 3 and 1. 40 in binary is 101000. 3 = 11 in binary. 11 does not occur anywhere in 101000. 1 is 1 in binary. 1 does occur (twice) in 101000. At least one exponent (3 = 11 in binary) does not occur in 101000 (= 40 in decimal), so 40 is in the sequence.
%p A144096 isA144096 := proc(n) local n2,a,ifa,e2,p ; n2 := convert(n,base,2) ; ifa := ifactors(n)[2] ; for p in ifa do e2 := convert( op(2,p),base,2) ; if not verify(n2,e2,'superlist') then RETURN(true) ; fi; od: RETURN(false) ; end: for n from 1 to 2000 do if isA144096(n) then printf("%d,",n) ; fi; od: # _R. J. Mathar_, Sep 17 2008
%t A144096 noexQ[n_]:=Min[SequenceCount[IntegerDigits[n,2],#]&/@(IntegerDigits[#,2]&/@(FactorInteger[ n][[;;,2]]))]==0; Select[Range[1200],noexQ] (* _Harvey P. Dale_, Dec 06 2023 *)
%Y A144096 Cf. A144095.
%K A144096 base,nonn
%O A144096 1,1
%A A144096 _Leroy Quet_, Sep 10 2008
%E A144096 63 and 64 inserted and extended by _R. J. Mathar_, Sep 17 2008