A379154 -id:A379154 - cp's OEIS frontend

A378355 Numbers appearing exactly once in A378035 (greatest perfect power < prime(n)).

125, 216, 243, 64000, 1295029, 2535525316, 542939080312

Offset: 1

Gus Wiseman, Nov 26 2024

These are perfect-powers p such that the interval from p to the next perfect power contains a unique prime.

Is this sequence infinite? See A178700.

			We have 125 because 127 is the only prime between 125 and 128.

The next prime is A178700.
Singletons in A378035 (union A378253), restriction of A081676.
The next perfect power is A378374.
Swapping primes and perfect powers gives A379154, unique case of A377283.
A000040 lists the primes, differences A001223.
A001597 lists the perfect powers, differences A053289.
A007916 lists the not perfect powers, differences A375706.
A069623 counts perfect powers <= n.
A076411 counts perfect powers < n.
A377432 counts perfect powers between primes, see A377434, A377436, A377466.
A378249 gives least perfect power > prime(n) (run-lengths A378251), restrict of A377468.
Cf. A000961, A031218, A045542, A052410, A067871, A076412, A080769, A131605, A188951, A216765, A378250, A378356.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
y=Table[NestWhile[#-1&,Prime[n],radQ[#]&],{n,1000}];
Select[Union[y],Count[y,#]==1&]

A151800(a(n)) = A178700(n).

A378364 Prime numbers such that the interval from the previous prime number contains a unique perfect power.

Original entry on oeis.org

2, 5, 17, 53, 67, 83, 101, 131, 149, 173, 197, 223, 227, 251, 257, 293, 331, 347, 367, 401, 443, 487, 521, 541, 577, 631, 677, 733, 787, 853, 907, 967, 1009, 1031, 1091, 1163, 1229, 1297, 1361, 1373, 1447, 1523, 1601, 1693, 1733, 1777, 1861, 1949, 2027, 2053

Offset: 1

Gus Wiseman, Dec 16 2024

nonn

Perfect-powers (A001597) are 1 and numbers with a proper integer root.

			The prime before 17 is 13, and the interval (13,14,15,16,17) contains only one perfect power 16, so 17 is in the sequence.
The prime before 29 is 23, and the interval (23,24,25,26,27,28,29) contains two perfect powers 25 and 27, so 29 is not in the sequence.

For non prime powers we have A006512.
For zero instead of one perfect power we have the prime terms of A345531.
The indices of these primes are the positions of 1 in A377432.
The indices of these primes are 1 + A377434(n-1).
For more than one perfect power see A377466.
Swapping "prime" with "perfect power" gives A378374.
For next instead of previous prime we have A379154.
A000040 lists the primes, differences A001223.
A001597 lists the perfect powers, differences A053289.
A007916 lists the non perfect powers, differences A375706.
A081676 gives the greatest perfect power <= n.
A377468 gives the least perfect power > n.
Cf. A023055, A045542, A052410, A053607, A069623, A376560, A377287, A377436.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Select[Range[1000],PrimeQ[#]&&Length[Select[Range[NextPrime[#,-1],#],perpowQ]]==1&]

cp's OEIS Frontend

A378355 Numbers appearing exactly once in A378035 (greatest perfect power < prime(n)).

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