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A359178 Numbers with a unique smallest prime exponent.

%I A359178 #52 May 15 2023 08:45:49
%S A359178 2,3,4,5,7,8,9,11,12,13,16,17,18,19,20,23,24,25,27,28,29,31,32,37,40,
%T A359178 41,43,44,45,47,48,49,50,52,53,54,56,59,61,63,64,67,68,71,72,73,75,76,
%U A359178 79,80,81,83,88,89,92,96,97,98,99,101,103,104,107,108,109,112,113,116,117
%N A359178 Numbers with a unique smallest prime exponent.
%C A359178 180 is the smallest number with a unique smallest prime exponent that is not a member of A130091.
%H A359178 Jens Ahlström, <a href="/A359178/b359178.txt">Table of n, a(n) for n = 1..9999</a>
%H A359178 Wikipedia, <a href="https://en.wikipedia.org/wiki/Mode_(statistics)">Mode (statistics)</a>.
%e A359178 2 = 2^1 is a term since it has 1 as a unique smallest exponent.
%e A359178 6 = 2^1 * 3^1 is not a term since it has two primes with the same smallest exponent.
%e A359178 180 = 2^2 * 3^2 * 5^1 is a term since it has 1 as a unique smallest exponent.
%t A359178 q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Count[e, Min[e]] == 1]; Select[Range[2, 200], q] (* _Amiram Eldar_, Jan 08 2023 *)
%o A359178 (Python)
%o A359178 from sympy import factorint
%o A359178 def ok(k):
%o A359178   c = sorted(factorint(k).values())
%o A359178   return len(c) == 1 or c[0] != c[1]
%o A359178 print([k for k in range(2, 118) if ok(k)])
%o A359178 (Python)
%o A359178 from itertools import count, islice
%o A359178 from sympy import factorint
%o A359178 def A359178_gen(startvalue=2): # generator of terms >= startvalue
%o A359178     return filter(lambda n:(f:=list(factorint(n).values())).count(min(f))==1,count(max(startvalue,2)))
%o A359178 A359178_list = list(islice(A359178_gen(),20)) # _Chai Wah Wu_, Feb 08 2023
%o A359178 (PARI) isok(n) = if (n>1, my(f=factor(n), e = vecmin(f[,2])); #select(x->(x==e), f[,2], 1) == 1); \\ _Michel Marcus_, Jan 27 2023
%Y A359178 For parts instead of multiplicities we have A247180, counted by A002865.
%Y A359178 For greatest instead of smallest we have A356862, counted by A362608.
%Y A359178 The complement is A362606, counted by A362609.
%Y A359178 Partitions of this type are counted by A362610.
%Y A359178 These are the positions of 1's in A362613, for modes A362611.
%Y A359178 A001221 counts prime exponents and A001222 adds them up.
%Y A359178 A027746 lists prime factors, A112798 indices, A124010 exponents.
%Y A359178 Cf. A102750, A130091, A327473, A327476, A359908, A362605, A362615.
%K A359178 nonn
%O A359178 1,1
%A A359178 _Jens Ahlström_, Jan 08 2023