A352140 - OEIS

%I #14 Mar 13 2022 19:01:19

%S 1,3,7,13,19,21,27,29,37,39,43,53,57,61,71,79,87,89,91,101,107,111,

%T 113,129,131,133,139,151,159,163,173,181,183,189,193,199,203,213,223,

%U 229,237,239,243,247,251,259,263,267,271,273,281,293,301,303,311,317

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

%C 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 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 Also Heinz numbers of integer partitions with all even parts and all odd multiplicities, counted by A055922 aerated.

%C All terms are odd. - _Michael S. Branicky_, Mar 12 2022

%F Intersection of A066207 and A268335.

%F A257991(a(n)) = A162641(a(n)) = 0.

%F A162642(a(n)) = A001221(a(n)).

%F A257992(a(n)) = A001222(a(n)).

%e The terms together with their prime indices begin:

%e       1 = 1

%e       3 = prime(2)^1

%e       7 = prime(4)^1

%e      13 = prime(6)^1

%e      19 = prime(8)^1

%e      21 = prime(4)^1 prime(2)^1

%e      27 = prime(2)^3

%e      29 = prime(10)^1

%e      37 = prime(12)^1

%e      39 = prime(6)^1 prime(2)^1

%e      43 = prime(14)^1

%e      53 = prime(16)^1

%e      57 = prime(8)^1 prime(2)^1

%e      61 = prime(18)^1

%e      71 = prime(20)^1

%t Select[Range[100],And@@EvenQ/@PrimePi/@First/@FactorInteger[#]&&And@@OddQ/@Last/@FactorInteger[#]&]

%o (Python)

%o from sympy import factorint, primepi

%o def ok(n):

%o     if n%2 == 0: return False

%o     return all(primepi(p)%2==0 and e%2==1 for p, e in factorint(n).items())

%o print([k for k in range(318) if ok(k)]) # _Michael S. Branicky_, Mar 12 2022

%Y The restriction to primes is A031215.

%Y These partitions are counted by A055922 (aerated).

%Y The first condition alone is A066207, counted by A035363.

%Y The squarefree case is A258117.

%Y The second condition alone is A268335, counted by A055922.

%Y A056166 = exponents all prime, counted by A055923.

%Y A066208 = prime indices all odd, counted by A000009.

%Y A109297 = same indices as exponents, counted by A114640.

%Y A112798 lists prime indices, reverse A296150, length A001222, sum A056239.

%Y A124010 gives prime signature, sorted A118914, length A001221, sum A001222.

%Y A162641 counts even prime exponents, odd A162642.

%Y A257991 counts odd prime indices, even A257992.

%Y A325131 = disjoint indices from exponents, counted by A114639.

%Y A346068 = indices and exponents all prime, counted by A351982.

%Y A351979 = odd indices with even exponents, counted by A035457.

%Y A352141 = even indices with even exponents, counted by A035444.

%Y A352142 = odd indices with odd exponents, counted by A117958.

%Y Cf. A000720, A028260, A055396, A061395, A181819, A195017, A241638, A276078, A324517, A324524, A324525, A325698.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 11 2022