cp's OEIS Frontend

A352141 Numbers whose prime factorization has all even indices and all even exponents.

%I A352141 #11 Sep 19 2022 07:23:46
%S A352141 1,9,49,81,169,361,441,729,841,1369,1521,1849,2401,2809,3249,3721,
%T A352141 3969,5041,6241,6561,7569,7921,8281,10201,11449,12321,12769,13689,
%U A352141 16641,17161,17689,19321,21609,22801,25281,26569,28561,29241,29929,32761,33489,35721
%N A352141 Numbers whose prime factorization has all even indices and all even exponents.
%C A352141 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 A352141 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 A352141 These are the Heinz numbers of partitions with all even parts and all even multiplicities, counted by A035444.
%H A352141 Amiram Eldar, <a href="/A352141/b352141.txt">Table of n, a(n) for n = 1..10000</a>
%F A352141 Intersection of A000290 and A066207.
%F A352141 A257991(a(n)) = A162642(a(n)) = 0.
%F A352141 A257992(a(n)) = A001222(a(n)).
%F A352141 A162641(a(n)) = A001221(a(n)).
%F A352141 Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2*k)^2) = 1.163719... . - _Amiram Eldar_, Sep 19 2022
%e A352141 The terms together with their prime indices begin:
%e A352141      1 = 1
%e A352141      9 = prime(2)^2
%e A352141     49 = prime(4)^2
%e A352141     81 = prime(2)^4
%e A352141    169 = prime(6)^2
%e A352141    361 = prime(8)^2
%e A352141    441 = prime(2)^2 prime(4)^2
%e A352141    729 = prime(2)^6
%e A352141    841 = prime(10)^2
%e A352141   1369 = prime(12)^2
%e A352141   1521 = prime(2)^2 prime(6)^2
%e A352141   1849 = prime(14)^2
%e A352141   2401 = prime(4)^4
%e A352141   2809 = prime(16)^2
%e A352141   3249 = prime(2)^2 prime(8)^2
%e A352141   3721 = prime(18)^2
%e A352141   3969 = prime(2)^4 prime(4)^2
%t A352141 Select[Range[1000],#==1||And@@EvenQ/@PrimePi/@First/@FactorInteger[#]&&And@@EvenQ/@Last/@FactorInteger[#]&]
%o A352141 (Python)
%o A352141 from itertools import count, islice
%o A352141 from sympy import factorint, primepi
%o A352141 def A352141_gen(startvalue=1): # generator of terms >= startvalue
%o A352141     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 A352141_list = list(islice(A352141_gen(),30)) # _Chai Wah Wu_, Mar 18 2022
%Y A352141 The second condition alone (all even exponents) is A000290, counted by A035363.
%Y A352141 The restriction to primes is A031215.
%Y A352141 These partitions are counted by A035444.
%Y A352141 The first condition alone is A066207, counted by A035363, squarefree A258117.
%Y A352141 A056166 = exponents all prime, counted by A055923.
%Y A352141 A066208 = prime indices all odd, counted by A000009.
%Y A352141 A109297 = same indices as exponents, counted by A114640.
%Y A352141 A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A352141 A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
%Y A352141 A162641 counts even exponents, odd A162642.
%Y A352141 A257991 counts odd indices, even A257992.
%Y A352141 A325131 = disjoint indices from exponents, counted by A114639.
%Y A352141 A346068 = indices and exponents all prime, counted by A351982.
%Y A352141 A351979 = odd indices with even exponents, counted by A035457.
%Y A352141 A352140 = even indices with odd exponents, counted by A055922 aerated.
%Y A352141 A352142 = odd indices with odd exponents, counted by A117958.
%Y A352141 Cf. A000720, A028260, A055396, A061395, A181819, A195017, A241638, A268335, A276078, A324524, A324525, A324588, A325698, A325700, A352143.
%K A352141 nonn
%O A352141 1,2
%A A352141 _Gus Wiseman_, Mar 18 2022