cp's OEIS Frontend

A378456 Number of composite numbers between consecutive nonprime prime powers (exclusive).

%I A378456 #8 Dec 02 2024 15:38:39
%S A378456 1,0,4,5,1,2,12,11,12,31,3,1,32,59,11,25,46,13,125,14,80,88,94,103,52,
%T A378456 261,35,267,147,172,120,9,9,163,355,279,313,207,329,347,376,108,257,
%U A378456 805,283,262,25,917,242,1081,702,365,752,389,251,535,1679,877,447
%N A378456 Number of composite numbers between consecutive nonprime prime powers (exclusive).
%C A378456 The inclusive version is a(n) + 2.
%C A378456 Nonprime prime powers (A246547) begin: 4, 8, 9, 16, 25, 27, 32, 49, ...
%e A378456 The initial terms count the following composite numbers:
%e A378456   {6}, {}, {10,12,14,15}, {18,20,21,22,24}, {26}, {28,30}, ...
%e A378456 The composite numbers for a(77) = 6 together with their prime indices are the following. We have also shown the nonprime prime powers before and after:
%e A378456   32761: {42,42}
%e A378456   32762: {1,1900}
%e A378456   32763: {2,19,38}
%e A378456   32764: {1,1,1028}
%e A378456   32765: {3,847}
%e A378456   32766: {1,2,14,31}
%e A378456   32767: {4,11,36}
%e A378456   32768: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
%t A378456 nn=1000;
%t A378456 v=Select[Range[nn],PrimePowerQ[#]&&!PrimeQ[#]&];
%t A378456 Table[Length[Select[Range[v[[i]]+1,v[[i+1]]-1],CompositeQ]],{i,Length[v]-1}]
%Y A378456 For prime instead of composite we have A067871.
%Y A378456 For nonsquarefree numbers we have A378373, for primes A236575.
%Y A378456 A000015 gives the least prime-power >= n.
%Y A378456 A000040 lists the primes, differences A001223.
%Y A378456 A000961 lists the powers of primes, differences A057820.
%Y A378456 A002808 lists the composite numbers.
%Y A378456 A031218 gives the greatest prime-power <= n.
%Y A378456 A046933 counts composite numbers between primes.
%Y A378456 A053707 gives first differences of nonprime prime powers.
%Y A378456 A080101 = A366833 - 1 counts prime powers between primes.
%Y A378456 A246655 lists the prime-powers not including 1, complement A361102.
%Y A378456 A345531 gives the nearest prime power after prime(n) + 1, difference A377281.
%Y A378456 Cf. A377286, A377287, A377288 (primes A053706).
%Y A378456 Cf. A024619, A053607, A065890, A076259, A078147, A243348, A276781, A377057, A377282.
%K A378456 nonn
%O A378456 1,3
%A A378456 _Gus Wiseman_, Nov 30 2024