cp's OEIS Frontend

A352142 Numbers whose prime factorization has all odd indices and all odd exponents.

%I A352142 #14 Mar 19 2022 02:33:03
%S A352142 1,2,5,8,10,11,17,22,23,31,32,34,40,41,46,47,55,59,62,67,73,82,83,85,
%T A352142 88,94,97,103,109,110,115,118,125,127,128,134,136,137,146,149,155,157,
%U A352142 160,166,167,170,179,184,187,191,194,197,205,206,211,218,227,230
%N A352142 Numbers whose prime factorization has all odd indices and all odd exponents.
%C A352142 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 A352142 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 A352142 These are the Heinz numbers of integer partitions with all odd parts and all odd multiplicities, counted by A117958.
%H A352142 David A. Corneth, <a href="/A352142/b352142.txt">Table of n, a(n) for n = 1..10000</a>
%F A352142 Intersection of A066208 and A268335.
%F A352142 A257991(a(n)) = A001222(a(n)).
%F A352142 A162642(a(n)) = A001221(a(n)).
%F A352142 A257992(a(n)) = A162641(a(n)) = 0.
%e A352142 The terms together with their prime indices begin:
%e A352142    1 = 1
%e A352142    2 = prime(1)
%e A352142    5 = prime(3)
%e A352142    8 = prime(1)^3
%e A352142   10 = prime(1) prime(3)
%e A352142   11 = prime(5)
%e A352142   17 = prime(7)
%e A352142   22 = prime(1) prime(5)
%e A352142   23 = prime(9)
%e A352142   31 = prime(11)
%e A352142   32 = prime(1)^5
%e A352142   34 = prime(1) prime(7)
%e A352142   40 = prime(1)^3 prime(3)
%t A352142 Select[Range[100],#==1||And@@OddQ/@PrimePi/@First/@FactorInteger[#]&&And@@OddQ/@Last/@FactorInteger[#]&]
%o A352142 (Python)
%o A352142 from itertools import count, islice
%o A352142 from sympy import primepi, factorint
%o A352142 def A352142_gen(startvalue=1): # generator of terms >= startvalue
%o A352142     return filter(lambda k:all(map(lambda x:x[1]%2 and primepi(x[0])%2, factorint(k).items())),count(max(startvalue,1)))
%o A352142 A352142_list = list(islice(A352142_gen(),30)) # _Chai Wah Wu_, Mar 18 2022
%Y A352142 The restriction to primes is A031368.
%Y A352142 The first condition alone is A066208, counted by A000009.
%Y A352142 These partitions are counted by A117958.
%Y A352142 The squarefree case is A258116, even A258117.
%Y A352142 The second condition alone is A268335, counted by A055922.
%Y A352142 The even-even version is A352141 counted by A035444.
%Y A352142 A000290 = exponents all even, counted by A035363.
%Y A352142 A056166 = exponents all prime, counted by A055923.
%Y A352142 A066207 = indices all even, counted by A035363 (complement A086543).
%Y A352142 A109297 = same indices as exponents, counted by A114640.
%Y A352142 A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A352142 A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
%Y A352142 A162641 counts even prime exponents, odd A162642.
%Y A352142 A257991 counts odd prime indices, even A257992.
%Y A352142 A325131 = disjoint indices from exponents, counted by A114639.
%Y A352142 A346068 = indices and exponents all prime, counted by A351982.
%Y A352142 A351979 = odd indices with even exponents, counted by A035457.
%Y A352142 A352140 = even indices with odd exponents, counted by A055922 aerated.
%Y A352142 A352143 = odd indices with odd conjugate indices, counted by A053253 aerated.
%Y A352142 Cf. A000720, A028260, A055396, A061395, A106529, A181819, A195017, A241638, A276078, A324517, A324524, A324525, A325698, A325700.
%K A352142 nonn
%O A352142 1,2
%A A352142 _Gus Wiseman_, Mar 18 2022