cp's OEIS Frontend

A351979 Numbers whose prime factorization has all odd prime indices and all even prime exponents.

%I A351979 #12 Sep 19 2022 07:23:31
%S A351979 1,4,16,25,64,100,121,256,289,400,484,529,625,961,1024,1156,1600,1681,
%T A351979 1936,2116,2209,2500,3025,3481,3844,4096,4489,4624,5329,6400,6724,
%U A351979 6889,7225,7744,8464,8836,9409,10000,10609,11881,12100,13225,13924,14641,15376
%N A351979 Numbers whose prime factorization has all odd prime indices and all even prime exponents.
%C A351979 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 A351979 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 A351979 Also Heinz numbers of integer partitions with all odd parts and all even multiplicities, counted by A035457 (see Emeric Deutsch's comment there).
%H A351979 Amiram Eldar, <a href="/A351979/b351979.txt">Table of n, a(n) for n = 1..10000</a>
%F A351979 Squares of elements of A066208.
%F A351979 Intersection of A066208 and A000290.
%F A351979 A257991(a(n)) = A001222(a(n)).
%F A351979 A162641(a(n)) = A001221(a(n)).
%F A351979 A162642(a(n)) = A257992(a(n)) = 0.
%F A351979 Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2*k-1)^2) = 1.4135142... . - _Amiram Eldar_, Sep 19 2022
%e A351979 The terms together with their prime indices begin:
%e A351979      1: 1
%e A351979      4: prime(1)^2
%e A351979     16: prime(1)^4
%e A351979     25: prime(3)^2
%e A351979     64: prime(1)^6
%e A351979    100: prime(1)^2 prime(3)^2
%e A351979    121: prime(5)^2
%e A351979    256: prime(1)^8
%e A351979    289: prime(7)^2
%e A351979    400: prime(1)^4 prime(3)^2
%e A351979    484: prime(1)^2 prime(5)^2
%e A351979    529: prime(9)^2
%e A351979    625: prime(3)^4
%e A351979    961: prime(11)^2
%e A351979   1024: prime(1)^10
%e A351979   1156: prime(1)^2 prime(7)^2
%e A351979   1600: prime(1)^6 prime(3)^2
%e A351979   1681: prime(13)^2
%e A351979   1936: prime(1)^4 prime(5)^2
%t A351979 Select[Range[1000],#==1||And@@OddQ/@PrimePi/@First/@FactorInteger[#]&&And@@EvenQ/@Last/@FactorInteger[#]&]
%o A351979 (Python)
%o A351979 from sympy import factorint, primepi
%o A351979 def ok(n):
%o A351979     return all(primepi(p)%2==1 and e%2==0 for p, e in factorint(n).items())
%o A351979 print([k for k in range(15500) if ok(k)]) # _Michael S. Branicky_, Mar 12 2022
%Y A351979 The second condition alone (exponents all even) is A000290, counted by A035363.
%Y A351979 The distinct prime factors of terms all come from A031368.
%Y A351979 These partitions are counted by A035457 or A000009 aerated.
%Y A351979 The first condition alone (indices all odd) is A066208, counted by A000009.
%Y A351979 The squarefree square roots are A258116, even A258117.
%Y A351979 A056166 = exponents all prime, counted by A055923.
%Y A351979 A066207 = indices all even, counted by complement of A086543.
%Y A351979 A076610 = indices all prime, counted by A000607.
%Y A351979 A109297 = same indices as exponents, counted by A114640.
%Y A351979 A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A351979 A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
%Y A351979 A162641 counts even exponents, odd A162642.
%Y A351979 A257991 counts odd indices, even A257992.
%Y A351979 A268335 = exponents all odd, counted by A055922.
%Y A351979 A325131 = disjoint indices from exponents, counted by A114639.
%Y A351979 A346068 = indices and exponents all prime, counted by A351982.
%Y A351979 A352140 = even indices with odd exponents, counted by A055922 (aerated).
%Y A351979 A352141 = even indices with even exponents, counted by A035444.
%Y A351979 A352142 = odd indices and odd multiplicities, counted by A117958.
%Y A351979 Cf. A000720, A028260, A045931, A055396, A061395, A106529, A181819, A195017, A276078, A324588, A325698, A325700.
%K A351979 nonn
%O A351979 1,2
%A A351979 _Gus Wiseman_, Mar 11 2022