A368950 Numbers with 11 odd divisors.

59049, 118098, 236196, 472392, 944784, 1889568, 3779136, 7558272, 9765625, 15116544, 19531250, 30233088, 39062500, 60466176, 78125000, 120932352, 156250000, 241864704, 282475249, 312500000, 483729408, 564950498, 625000000, 967458816, 1129900996, 1250000000, 1934917632

Offset: 1

Author

Hartmut F. W. Hoft, Jan 10 2024

Keywords

• nonn

Comments

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.

Examples

			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.
		

Crossrefs

Cf. A267983 (lists the sequences of numbers with 1 .. 10 odd divisors), A367370, A367377.

Cf. A235791, A237048, A237270, A237593, A241008, A241010, A249223, A341969.

Programs

Maple

N:= 10^10: # for terms <= N
R:= NULL: p:= 2:
do
  p:= nextprime(p);
  if p^10 > N then break fi;
  R:= R, seq(2^i*p^10, i = 0 .. floor(log[2](N/p^10)))
od:
sort([R]); # Robert Israel, Jan 16 2024

Mathematica

numL[p_, b_] := Map[2^# p^10&, Range[0, Floor[Log[2, b/p^10]]]]
primeL[b_] := Most[NestWhileList[NextPrime[#]&, 3, #^10<=b&]]
a368950[b_] := Union[Flatten[Map[numL[#, b]&, primeL[b]]]]
a368950[2 10^9]