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A368748 a(n) is the number of numbers between prime(n) and prime(n+1) that are not prime powers.

%I A368748 #35 Jan 17 2024 09:20:23
%S A368748 0,0,1,1,1,2,1,3,3,1,4,3,1,3,4,5,1,4,3,1,5,2,5,7,3,1,3,1,3,11,2,5,1,9,
%T A368748 1,5,5,3,4,5,1,9,1,3,1,11,11,3,1,3,5,1,8,4,5,5,1,5,3,1,8,13,3,1,3,13,
%U A368748 5,8,1,3,5,6,5,5,3,5,7,3,7,9,1,9,1,5,3,5,7,3,1,3
%N A368748 a(n) is the number of numbers between prime(n) and prime(n+1) that are not prime powers.
%C A368748 Here "between" refers to numbers in the range [prime(n) + 1, prime(n+1) - 1], all of which are composite, and the sequence counts the numbers in each such range which are not prime powers. Whereas the corresponding number of prime powers seems bounded (see A080101), the number of numbers which are not prime powers is unbounded (see A014963). Conjecture: Every nonnegative integer appears in this sequence (at least once).
%H A368748 Michael De Vlieger, <a href="/A368748/b368748.txt">Table of n, a(n) for n = 1..10000</a>
%H A368748 Michael De Vlieger, <a href="/A368748/a368748.png">Annotated log scatterplot of a(n)</a>, n = 1..2^20, showing records in red.
%F A368748 a(n) = A001223(n) - A080101(n) - 1. - _Michael De Vlieger_, Jan 04 2024
%e A368748 Between 2 and 3 there are no other numbers so a(1) = 0.
%e A368748 Between 3 and 5 there is only one number (4) and it is a prime power, so a(2) = 0.
%e A368748 Between 5 and 7 the only number is 6 and it is not a prime power, so a(3) = 1.
%e A368748 Between 47 and 53 there are 5 composite numbers, but one of them (49) is a prime power, so since 47 = prime(15), a(15) = 4.
%p A368748 N:= 101: # for a(1) .. a(N-1)
%p A368748 P:= [seq(ithprime(i),i=1..N)]:
%p A368748 PP:= {seq(seq(P[i]^j, j = 2 .. ilog[P[i]](P[N])),i=1..N)}:
%p A368748 seq(nops({$P[i]+1 .. P[i+1]-1} minus PP), i=1 .. N-1); # _Robert Israel_, Jan 04 2024
%t A368748 Map[Count[Range[#1, #2 - 1], _?(Not@*PrimePowerQ)] & @@ # &, Partition[Prime@ Range[120], 2, 1]] (* _Michael De Vlieger_, Jan 04 2024 *)
%o A368748 (PARI) a(n) = sum(k=prime(n)+1, prime(n+1)-1, !isprimepower(k)); \\ _Michel Marcus_, Jan 04 2024
%Y A368748 Cf. A000040, A000961, A001223, A014963, A024619, A080101, A085970, A093555, A368749.
%K A368748 nonn
%O A368748 1,6
%A A368748 _David James Sycamore_, Jan 04 2024
%E A368748 More terms from _Michel Marcus_, Jan 04 2024