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A378250 Perfect-powers x > 1 such that it is not possible to choose a prime y and a perfect-power z satisfying x > y > z.

%I A378250 #5 Nov 21 2024 19:11:15
%S A378250 4,8,16,25,32,49,64,81,100,121,128,144,169,196,216,225,243,256,289,
%T A378250 324,343,361,400,441,484,512,529,576,625,676,729,784,841,900,961,1000,
%U A378250 1024,1089,1156,1225,1296,1331,1369,1444,1521,1600,1681,1728,1764,1849,1936
%N A378250 Perfect-powers x > 1 such that it is not possible to choose a prime y and a perfect-power z satisfying x > y > z.
%C A378250 Perfect-powers (A001597) are numbers with a proper integer root, complement A007916.
%e A378250 The first number line below shows the perfect-powers. The second shows the primes. The third is a(n).
%e A378250 -1-----4-------8-9------------16----------------25--27--------32------36----
%e A378250 ===2=3===5===7======11==13======17==19======23==========29==31==========37==
%e A378250        4       8              16                25            32
%e A378250 The terms together with their prime indices begin:
%e A378250      4: {1,1}
%e A378250      8: {1,1,1}
%e A378250     16: {1,1,1,1}
%e A378250     25: {3,3}
%e A378250     32: {1,1,1,1,1}
%e A378250     49: {4,4}
%e A378250     64: {1,1,1,1,1,1}
%e A378250     81: {2,2,2,2}
%e A378250    100: {1,1,3,3}
%e A378250    121: {5,5}
%e A378250    128: {1,1,1,1,1,1,1}
%e A378250    144: {1,1,1,1,2,2}
%e A378250    169: {6,6}
%e A378250    196: {1,1,4,4}
%e A378250    216: {1,1,1,2,2,2}
%e A378250    225: {2,2,3,3}
%e A378250    243: {2,2,2,2,2}
%e A378250    256: {1,1,1,1,1,1,1,1}
%t A378250 radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
%t A378250 Union[Table[NestWhile[#+1&,Prime[n],radQ[#]&],{n,100}]]
%Y A378250 A version for prime-powers (but starting with prime(k) + 1) is A345531.
%Y A378250 The opposite is union of A378035, restriction of A081676.
%Y A378250 Union of A378249, run-lengths are A378251.
%Y A378250 A000040 lists the primes, differences A001223.
%Y A378250 A000961 lists the powers of primes, differences A057820.
%Y A378250 A001597 lists the perfect-powers, differences A053289, seconds A376559.
%Y A378250 A007916 lists the non-perfect-powers, differences A375706, seconds A376562.
%Y A378250 A069623 counts perfect-powers <= n.
%Y A378250 A076411 counts perfect-powers < n.
%Y A378250 A131605 lists perfect-powers that are not prime-powers.
%Y A378250 A377432 counts perfect-powers between primes, zeros A377436, positive A377283, postpositive A377466.
%Y A378250 Cf. A000015, A007918, A023055, A045542, A052410, A076412, A188951, A216765, A377431, A377434, A377468.
%K A378250 nonn
%O A378250 1,1
%A A378250 _Gus Wiseman_, Nov 21 2024