A364061 Numbers whose exponent of 2 in their canonical prime factorization is smaller than all the other exponents.

2, 4, 8, 16, 18, 32, 50, 54, 64, 98, 108, 128, 162, 242, 250, 256, 324, 338, 450, 486, 500, 512, 578, 648, 686, 722, 882, 972, 1024, 1058, 1250, 1350, 1372, 1458, 1682, 1922, 1944, 2048, 2178, 2250, 2450, 2500, 2646, 2662, 2738, 2916, 3042, 3362, 3698, 3888

Offset

1,1

Comments

Also numbers whose multiset of prime factors has unique co-mode 2. Here, a co-mode in a multiset is an element that appears at most as many times as each of the other elements. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

Sum_{n>=1} 1/a(n) = 1 + Sum_{k>=2} (1-1/2^(k-1))*(s(k)-s(k+1)) = 1.16896822653093929144..., where s(k) = Product_{primes p >= 3} (1 + 1/(p^(k-1)*(p-1))) is the sum of reciprocals of the odd k-full numbers (numbers whose prime factorization has no exponent that is smaller than k). - Amiram Eldar, Aug 30 2024

Example

The terms together with their prime factors begin:
    2 = 2
    4 = 2*2
    8 = 2*2*2
   16 = 2*2*2*2
   18 = 2*3*3
   32 = 2*2*2*2*2
   50 = 2*5*5
   54 = 2*3*3*3
   64 = 2*2*2*2*2*2
   98 = 2*7*7
  108 = 2*2*3*3*3
  128 = 2*2*2*2*2*2*2

Maple

filter:= proc(n) local F, F2, Fo;
  F:= ifactors(n)[2];
  F2, Fo:= selectremove(t -> t[1]=2, F);
  Fo = [] or F2[1, 2] < min(Fo[.., 2])
end proc:
select(filter, 2*[$1..5000]); # Robert Israel, Apr 22 2024

Mathematica

prifacs[n_]:=If[n==1, {}, Flatten[ConstantArray@@@FactorInteger[n]]];
comodes[ms_]:=Select[Union[ms], Count[ms, #]<=Min@@Length/@Split[ms]&];
Select[Range[100], comodes[prifacs[#]]=={2}&]

Prog

(Python)
from sympy import factorint
from itertools import count, islice
def A364061_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:(l:=(~n&n-1).bit_length()) < min(factorint(m:=n>>l).values(), default=0) or m==1, count(max(startvalue+startvalue&1, 2), 2))
A364061_list = list(islice(A364061_gen(), 30)) # Chai Wah Wu, Jul 14 2023

Crossrefs

For any unique co-mode: A359178, counted by A362610, complement A362606.

For high mode: A360013, positions of 1's in A363487, counted by A241131.

For low mode: A360015, positions of 1's in A363486, counted by A241131.

Partitions of this type are counted by A364062.

For low co-mode: A364158, positions of 1's in A364192, counted by A364159.

Positions of 1's in A364191, high A364192.

A112798 lists prime indices, length A001222, sum A056239.

A356862 ranks partitions w/ unique mode, count A362608, complement A362605.

A362611 counts modes in prime indices, triangle A362614.

A362613 counts co-modes in prime indices, triangle A362615.

Cf. A000265, A007814, A327473, A327476, A362616, A360014, A363722, A363723, A363725, A363727, A363730.

Sequence in context: A379503 A316900 A076057 * A133809 A381070 A128700

Adjacent sequences: A364058 A364059 A364060 * A364062 A364063 A364064

Keyword

nonn

Author

Gus Wiseman, Jul 12 2023

Status

approved