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A368950 Numbers with 11 odd divisors.

%I A368950 #12 Jan 20 2024 09:31:55
%S A368950 59049,118098,236196,472392,944784,1889568,3779136,7558272,9765625,
%T A368950 15116544,19531250,30233088,39062500,60466176,78125000,120932352,
%U A368950 156250000,241864704,282475249,312500000,483729408,564950498,625000000,967458816,1129900996,1250000000,1934917632
%N A368950 Numbers with 11 odd divisors.
%C A368950 Every number in this sequence has the form 2^k * p^10, k >= 0, where p is an odd prime. Exactly 11 different width patterns (A341969) of the symmetric representation of sigma are instantiated by the numbers in this sequence. The width pattern becomes unimodal for k >= floor(log_2(p^10)), see A367370 and A367377.
%e A368950 a(1) = 59049 = 3^10, a(9) = 5^10 = 9765625 is the smallest number with prime factor 5, a(19) = 282475249 is the smallest number with prime factor 7 and a(27) = 2^floor(log_2(3^10)) * 3^10 = 32768 * 59049 = 1934917632 is the smallest whose width pattern of its symmetric representation of sigma is unimodal.
%p A368950 N:= 10^10: # for terms <= N
%p A368950 R:= NULL: p:= 2:
%p A368950 do
%p A368950   p:= nextprime(p);
%p A368950   if p^10 > N then break fi;
%p A368950   R:= R, seq(2^i*p^10, i = 0 .. floor(log[2](N/p^10)))
%p A368950 od:
%p A368950 sort([R]); # _Robert Israel_, Jan 16 2024
%t A368950 numL[p_, b_] := Map[2^# p^10&, Range[0, Floor[Log[2, b/p^10]]]]
%t A368950 primeL[b_] := Most[NestWhileList[NextPrime[#]&, 3, #^10<=b&]]
%t A368950 a368950[b_] := Union[Flatten[Map[numL[#, b]&, primeL[b]]]]
%t A368950 a368950[2 10^9]
%Y A368950 Cf. A267983 (lists the sequences of numbers with 1 .. 10 odd divisors), A367370, A367377.
%Y A368950 Cf. A235791, A237048, A237270, A237593, A241008, A241010, A249223, A341969.
%K A368950 nonn
%O A368950 1,1
%A A368950 _Hartmut F. W. Hoft_, Jan 10 2024