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A179983 Positive integers m such that, if k appears in m's prime signature, k-1 appears at least as often as k (for any integer k > 1).

%I A179983 #24 Aug 04 2024 03:01:59
%S A179983 1,2,3,5,6,7,10,11,12,13,14,15,17,18,19,20,21,22,23,26,28,29,30,31,33,
%T A179983 34,35,37,38,39,41,42,43,44,45,46,47,50,51,52,53,55,57,58,59,60,61,62,
%U A179983 63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,82,83,84,85,86,87,89,90
%N A179983 Positive integers m such that, if k appears in m's prime signature, k-1 appears at least as often as k (for any integer k > 1).
%C A179983 Numbers m such that A181819(m) is a term of A025487.
%H A179983 Amiram Eldar, <a href="/A179983/b179983.txt">Table of n, a(n) for n = 1..10000</a>
%H A179983 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>.
%e A179983 The prime signature of 20 = 2^2*5 is (2,1). Since the largest number appearing in 20's prime signature is 2, and 1 appears as many times as 2, 20 is a member of this sequence.
%p A179983 isA179983 := proc(n)
%p A179983     local es,me,k ;
%p A179983     # list of exponents in prime signature
%p A179983     es := [seq(op(2,pe), pe =ifactors(n)[2])] ;
%p A179983     # maximum exponent
%p A179983     me := max(op(es)) ;
%p A179983     for k from me to 2 by -1 do
%p A179983         if numboccur(es,k-1) < numboccur(es,k) then
%p A179983             return false;
%p A179983         end if;
%p A179983     end do:
%p A179983     true ;
%p A179983 end proc:
%p A179983 for n from 1 to 100 do
%p A179983     if isA179983(n) then
%p A179983         printf("%d,",n) ;
%p A179983     end if;
%p A179983 end do: # _R. J. Mathar_, Mar 21 2023
%t A179983 q[n_] := Module[{t = SortBy[Tally[FactorInteger[n][[;; , 2]]], First], t1, t2}, t1 = t[[;; , 1]]; t2 = t[[;; , 2]]; Sort[t1] == Range[Length[t1]] && Max[Differences[t2]] < 1]; Select[Range[100], q] (* _Amiram Eldar_, Aug 04 2024 *)
%Y A179983 Includes all squarefree numbers (A005117); also includes all members of A054753, A085987, A163569, A182862, A182863.
%Y A179983 Cf. A025487, A181819.
%K A179983 nonn
%O A179983 1,2
%A A179983 _Matthew Vandermast_, Jan 15 2011