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Numbers whose prime factorization has all even indices and all even exponents.
7

%I #11 Sep 19 2022 07:23:46

%S 1,9,49,81,169,361,441,729,841,1369,1521,1849,2401,2809,3249,3721,

%T 3969,5041,6241,6561,7569,7921,8281,10201,11449,12321,12769,13689,

%U 16641,17161,17689,19321,21609,22801,25281,26569,28561,29241,29929,32761,33489,35721

%N Numbers whose prime factorization has all even indices and all even 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 These are the Heinz numbers of partitions with all even parts and all even multiplicities, counted by A035444.

%H Amiram Eldar, <a href="/A352141/b352141.txt">Table of n, a(n) for n = 1..10000</a>

%F Intersection of A000290 and A066207.

%F A257991(a(n)) = A162642(a(n)) = 0.

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

%F A162641(a(n)) = A001221(a(n)).

%F Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2*k)^2) = 1.163719... . - _Amiram Eldar_, Sep 19 2022

%e The terms together with their prime indices begin:

%e 1 = 1

%e 9 = prime(2)^2

%e 49 = prime(4)^2

%e 81 = prime(2)^4

%e 169 = prime(6)^2

%e 361 = prime(8)^2

%e 441 = prime(2)^2 prime(4)^2

%e 729 = prime(2)^6

%e 841 = prime(10)^2

%e 1369 = prime(12)^2

%e 1521 = prime(2)^2 prime(6)^2

%e 1849 = prime(14)^2

%e 2401 = prime(4)^4

%e 2809 = prime(16)^2

%e 3249 = prime(2)^2 prime(8)^2

%e 3721 = prime(18)^2

%e 3969 = prime(2)^4 prime(4)^2

%t Select[Range[1000],#==1||And@@EvenQ/@PrimePi/@First/@FactorInteger[#]&&And@@EvenQ/@Last/@FactorInteger[#]&]

%o (Python)

%o from itertools import count, islice

%o from sympy import factorint, primepi

%o def A352141_gen(startvalue=1): # generator of terms >= startvalue

%o return filter(lambda k:all(map(lambda x: not (x[1]%2 or primepi(x[0])%2), factorint(k).items())),count(max(startvalue,1)))

%o A352141_list = list(islice(A352141_gen(),30)) # _Chai Wah Wu_, Mar 18 2022

%Y The second condition alone (all even exponents) is A000290, counted by A035363.

%Y The restriction to primes is A031215.

%Y These partitions are counted by A035444.

%Y The first condition alone is A066207, counted by A035363, squarefree A258117.

%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 exponents, odd A162642.

%Y A257991 counts odd 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 A352140 = even indices with odd exponents, counted by A055922 aerated.

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

%Y Cf. A000720, A028260, A055396, A061395, A181819, A195017, A241638, A268335, A276078, A324524, A324525, A324588, A325698, A325700, A352143.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 18 2022