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A289484 Semigroup of numbers under multiplication, distinct from mutinous sequence.

%I A289484 #36 Oct 20 2017 05:03:11
%S A289484 12,24,30,36,40,45,48,56,60,63,70,72,80,84,90,96,105,108,112,120,126,
%T A289484 132,135,140,144,150,154,156,160,165,168,175,176,180,182,189,192,195,
%U A289484 198,200,204,208,210,216,220,224,225,228,231,234,240,252,260,264,270,273,275,276,280
%N A289484 Semigroup of numbers under multiplication, distinct from mutinous sequence.
%C A289484 Definition: Set S(S numbers) of all numbers n whose prime factorization contains at least one initial product greater than a later prime. That is, write n as n=p1^e1*p2^e2*...pm^em, with the ei>0 and p1<p2<..<pm. Then if p1^e1*p2^e2..pk^ek is greater than p(k+1) for some prime p(k+1) in the factorization, then n is in S.
%C A289484 Examples: 156=2^2*3*13 is in S, since 2^2>3. Another example is 200=2^3*5^2, since 8>5, so 200 is in S.
%C A289484 Counterexamples: 20=2^2*5 and 42=2*3*7 are not S numbers because 2^2<5 and 2*3<7.
%C A289484 Properties: No primes or prime powers are in S, nor are any numbers pq with p and q prime. S is closed under multiplication, so it is a semigroup. In fact, any positive multiple of an S number is also an S number.
%C A289484 Subset of generators: The numbers 12,30,40,45,56,63,.., belong to an infinite subset of S that could be called "S primes" because no proper factor of an S prime is an S number, and because every S number is a positive multiple of at least one of the S primes.
%C A289484 Algebra: If one adjoins the numbers 0 and all the negatives of numbers in S into S and call the result S#, then S# remains a semigroup and is the set union of infinitely many principal ideals:S#=(12)U(30)U(40)U(45)U...U.(note presence of S primes). But S# itself is not an ideal, because it is not closed under addition.
%C A289484 Density: In the integers from 1 to 500, about 19% are in S. Using Wolfram Alpha, about 63% of the integers from 10^40+1 to 10^40+62 were found to be S numbers.
%H A289484 Charles R Greathouse IV, <a href="/A289484/b289484.txt">Table of n, a(n) for n = 1..10000</a>
%p A289484 isA289484 := proc(n)
%p A289484         local pset,p,pprodidx,pprod,nu ;
%p A289484         pset := sort(convert(numtheory[factorset](n),list)) ;
%p A289484         pprod := 1;
%p A289484         for pprodidx from 1 to nops(pset)-1 do
%p A289484                 p := pset[pprodidx] ;
%p A289484                 nu := padic[ordp](n,p) ;
%p A289484                 pprod := pprod*p^nu ;
%p A289484                 if pprod > pset[pprodidx+1] then
%p A289484                         return true;
%p A289484                 end if;
%p A289484         end do:
%p A289484         return false ;
%p A289484 end proc:
%p A289484 for n from 1 to 300 do
%p A289484         if isA289484(n) then
%p A289484                 printf("%d,",n) ;
%p A289484         end if;
%p A289484 end do: # _R. J. Mathar_, Oct 20 2017
%t A289484 Select[Range@ 280, Function[f, AnyTrue[Range[Length@ f - 1], Times @@ Map[#1^#2 & @@ # &, #1] > #2[[1, 1]] & @@ TakeDrop[f, #] &]]@ FactorInteger@ # &] (* _Michael De Vlieger_, Aug 17 2017 *)
%o A289484 (PARI) is(n)=my(f=factor(n),t=1); for(i=1,#f~, if(t>f[i,1], return(1)); t*=f[i,1]^f[i,2]); 0 \\ _Charles R Greathouse IV_, Jul 10 2017
%o A289484 (PARI) has(f)=my(t=1); for(i=1,#f~, if(t>f[i,1], return(1)); t*=f[i,1]^f[i,2]); 0
%o A289484 list(lim)=my(m=Map(),v=List()); forfactored(n=12,lim\=1, if(mapisdefined(m,n), next); if(has(n[2]), forstep(k=n[1],lim,n[1], mapput(m,k,0)))); for(n=12,lim, if(mapisdefined(m,n), listput(v,n))); m=0; Vec(v) \\ _Charles R Greathouse IV_, Jul 12 2017
%Y A289484 Cf. A027854, mutinous numbers. Contained in S, differs in that for a mutinous number it must be the greatest prime in the factorization that is exceeded by the initial product, while in S, the prime that is exceeded can be any of the primes later than the initial primes. For example 156 is an S number but not a mutinous number.
%K A289484 nonn
%O A289484 1,1
%A A289484 _Richard Locke Peterson_, Jul 06 2017