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Difference between prime(n) and the previous prime-power (exclusive).
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%I #12 Oct 26 2024 22:58:22

%S 1,1,1,2,2,2,1,2,4,2,2,5,4,2,4,4,6,2,3,4,2,6,2,6,8,4,2,4,2,4,2,3,6,2,

%T 10,2,6,6,4,4,6,2,10,2,4,2,12,12,4,2,4,6,2,8,1,6,6,2,6,4,2,4,14,4,2,4,

%U 14,6,4,2,4,6,6,6,6,4,6,8,4,8,10,2,10,2

%N Difference between prime(n) and the previous prime-power (exclusive).

%F a(n) = prime(n) - A031218(prime(n)-1).

%F a(n) = prime(n) - A065514(n).

%F a(n) = A276781(prime(n)-1).

%e The twelfth prime is 37, with previous prime-power 32, so a(12) = 5.

%t Table[Prime[n]-NestWhile[#-1&, Prime[n]-1,#>1&&!PrimePowerQ[#]&],{n,100}]

%o (Python)

%o from sympy import prime, factorint

%o def A377289(n): return (p:=prime(n))-next(filter(lambda m:len(factorint(m))<=1, range(p-1,0,-1))) # _Chai Wah Wu_, Oct 25 2024

%Y For powers of two see A013597, A014210, A014234, A244508, A304521.

%Y For prime instead of prime-power we have A075526.

%Y This is the restriction of A276781 (shifted right) to the primes.

%Y For next instead of previous prime-power we have A377281, restriction of A377282.

%Y A000015 gives the least prime-power >= n.

%Y A000040 lists the primes, differences A001223.

%Y A000961 lists the powers of primes, differences A057820, complement A361102.

%Y A031218 gives the greatest prime-power <= n.

%Y A065514 gives the greatest prime-power < prime(n).

%Y A080101 counts prime-powers between primes (exclusive), cf. A377286, A377287, A377288.

%Y A246655 lists the prime-powers not including 1.

%Y Cf. A002808, A024619, A053707, A059305, A064113, A065890, A366833, A376596, A376597, A377051, A377054.

%K nonn

%O 1,4

%A _Gus Wiseman_, Oct 23 2024