cp's OEIS Frontend

A352140 Numbers whose prime factorization has all even prime indices and all odd exponents.

%I A352140 #14 Mar 13 2022 19:01:19
%S A352140 1,3,7,13,19,21,27,29,37,39,43,53,57,61,71,79,87,89,91,101,107,111,
%T A352140 113,129,131,133,139,151,159,163,173,181,183,189,193,199,203,213,223,
%U A352140 229,237,239,243,247,251,259,263,267,271,273,281,293,301,303,311,317
%N A352140 Numbers whose prime factorization has all even prime indices and all odd exponents.
%C A352140 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239, length A001222.
%C A352140 A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
%C A352140 Also Heinz numbers of integer partitions with all even parts and all odd multiplicities, counted by A055922 aerated.
%C A352140 All terms are odd. - _Michael S. Branicky_, Mar 12 2022
%F A352140 Intersection of A066207 and A268335.
%F A352140 A257991(a(n)) = A162641(a(n)) = 0.
%F A352140 A162642(a(n)) = A001221(a(n)).
%F A352140 A257992(a(n)) = A001222(a(n)).
%e A352140 The terms together with their prime indices begin:
%e A352140       1 = 1
%e A352140       3 = prime(2)^1
%e A352140       7 = prime(4)^1
%e A352140      13 = prime(6)^1
%e A352140      19 = prime(8)^1
%e A352140      21 = prime(4)^1 prime(2)^1
%e A352140      27 = prime(2)^3
%e A352140      29 = prime(10)^1
%e A352140      37 = prime(12)^1
%e A352140      39 = prime(6)^1 prime(2)^1
%e A352140      43 = prime(14)^1
%e A352140      53 = prime(16)^1
%e A352140      57 = prime(8)^1 prime(2)^1
%e A352140      61 = prime(18)^1
%e A352140      71 = prime(20)^1
%t A352140 Select[Range[100],And@@EvenQ/@PrimePi/@First/@FactorInteger[#]&&And@@OddQ/@Last/@FactorInteger[#]&]
%o A352140 (Python)
%o A352140 from sympy import factorint, primepi
%o A352140 def ok(n):
%o A352140     if n%2 == 0: return False
%o A352140     return all(primepi(p)%2==0 and e%2==1 for p, e in factorint(n).items())
%o A352140 print([k for k in range(318) if ok(k)]) # _Michael S. Branicky_, Mar 12 2022
%Y A352140 The restriction to primes is A031215.
%Y A352140 These partitions are counted by A055922 (aerated).
%Y A352140 The first condition alone is A066207, counted by A035363.
%Y A352140 The squarefree case is A258117.
%Y A352140 The second condition alone is A268335, counted by A055922.
%Y A352140 A056166 = exponents all prime, counted by A055923.
%Y A352140 A066208 = prime indices all odd, counted by A000009.
%Y A352140 A109297 = same indices as exponents, counted by A114640.
%Y A352140 A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A352140 A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
%Y A352140 A162641 counts even prime exponents, odd A162642.
%Y A352140 A257991 counts odd prime indices, even A257992.
%Y A352140 A325131 = disjoint indices from exponents, counted by A114639.
%Y A352140 A346068 = indices and exponents all prime, counted by A351982.
%Y A352140 A351979 = odd indices with even exponents, counted by A035457.
%Y A352140 A352141 = even indices with even exponents, counted by A035444.
%Y A352140 A352142 = odd indices with odd exponents, counted by A117958.
%Y A352140 Cf. A000720, A028260, A055396, A061395, A181819, A195017, A241638, A276078, A324517, A324524, A324525, A325698.
%K A352140 nonn
%O A352140 1,2
%A A352140 _Gus Wiseman_, Mar 11 2022