A378033 -id:A378033 - cp's OEIS frontend

A378616 Greatest non prime power <= prime(n).

1, 1, 1, 6, 10, 12, 15, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 255, 262, 268, 270

Offset: 1

Gus Wiseman, Dec 06 2024

nonn

Conjecture: Equal to A006093(n) = prime(n) - 1 except at terms of A159611.

			The first number line below shows the non prime powers. The second shows the primes:
--1-------------6----------10----12----14-15-------18----20-21-22----24--
=====2==3====5=====7==========11====13==========17====19==========23=====

For nonprime instead of non prime power we have A156037.
Restriction of A378367.
Lengths are A378615.
For nonsquarefree: A378032 (diffs A378034), restriction of A378033 (diffs A378036).
A000040 lists the primes, differences A001223
A000961 and A246655 list the prime powers, differences A057820.
A024619 lists the non prime powers, differences A375735, seconds A376599.
A080101 counts prime powers between primes (exclusive), inclusive A366833.
A361102 lists the non powers of primes, differences A375708.
Prime powers between primes:
- A377057 positive
- A377286 zero
- A377287 one
- A377288 two
Cf. A006093, A053607, A143731, A159611, A304521, A343249, A345531, A356068, A368748, A377281, A377289, A377703, A377781.

Table[Max[Select[Range[Prime[n]],Not@*PrimePowerQ]],{n,100}]

A380413 Terms appearing twice in A378086 (number of nonsquarefree numbers < prime(n)).

Original entry on oeis.org

0, 1, 11, 14, 39, 53, 109, 179, 222, 240, 251, 319, 337, 481, 505, 508, 578, 664, 674, 738, 818, 835, 877, 905, 933, 1041, 1069, 1098, 1325, 1352, 1392, 1535, 1539, 1567, 1652, 1663, 1732, 1817, 1849, 1960, 2134, 2148, 2158, 2220, 2387, 2428, 2457, 2622, 2625

Offset: 1

Gus Wiseman, Feb 06 2025

nonn

A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.
A061399 counts nonsquarefree integers between primes, see A068361, A061398, A068360, A377783, A378086.
A070321 gives the greatest squarefree number up to n.
A071403 counts squarefree numbers < prime(n), see A373198, A337030.
A112925 gives the greatest squarefree number between primes, least A112926.
Cf. A057627, A065890, A378032 (differences A378034), A378033 (differences A378036).
Cf. A072284, A120327, A224363, A378040, A378082, A378084.

y=Table[Length[Select[Range[Prime[n]],!SquareFreeQ[#]&]],{n,100}];
Select[Most[Union[y]],Count[y,#]==2&]

a(n) = A378086(A068361(n)) = A378086(A068361(n)+1).

cp's OEIS Frontend

A378616 Greatest non prime power <= prime(n).

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