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A376519 Positions of terms not appearing for the first time in the first differences (A053289) of perfect-powers (A001597).

%I A376519 #6 Sep 30 2024 09:15:40
%S A376519 8,14,15,20,22,25,26,31,40,46,52,59,68,75,88,96,102,110,111,112,114,
%T A376519 128,136,144,145,162,180,188,198,216,226,235,246,264,265,275,285,295,
%U A376519 305,316,317,325,328,338,350,360,367,373,385,406,416,417,419,431,443
%N A376519 Positions of terms not appearing for the first time in the first differences (A053289) of perfect-powers (A001597).
%e A376519 The perfect powers (A001597) are:
%e A376519   1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
%e A376519 with first differences (A053289):
%e A376519   3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, ...
%e A376519 with positions of latter appearances (A376519):
%e A376519   8, 14, 15, 20, 22, 25, 26, 31, 40, 46, 52, 59, 68, 75, 88, 96, 102, 110, 111, ...
%t A376519 perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
%t A376519 q=Differences[Select[Range[1000],perpowQ]];
%t A376519 Select[Range[Length[q]],MemberQ[Take[q,#-1],q[[#]]]&]
%Y A376519 These are the sorted positions of latter appearances in A053289 (union A023055).
%Y A376519 The complement is A376268.
%Y A376519 A053707 lists first differences of consecutive prime-powers.
%Y A376519 A333254 lists run-lengths of differences between consecutive primes.
%Y A376519 Other families of numbers and their first differences:
%Y A376519 For prime numbers (A000040) we have A001223.
%Y A376519 For composite numbers (A002808) we have A073783.
%Y A376519 For nonprime numbers (A018252) we have A065310.
%Y A376519 For perfect powers (A001597) we have A053289.
%Y A376519 For non-perfect-powers (A007916) we have A375706.
%Y A376519 For squarefree numbers (A005117) we have A076259.
%Y A376519 For nonsquarefree numbers (A013929) we have A078147.
%Y A376519 For prime-powers inclusive (A000961) we have A057820.
%Y A376519 For prime-powers exclusive (A246655) we have A057820(>1).
%Y A376519 For non-prime-powers inclusive (A024619) we have A375735.
%Y A376519 For non-prime-powers exclusive (A361102) we have A375708.
%Y A376519 Cf. A025475, A045542, A046933, A052410, A069623, A174965, A216765, A303707, A305630, A305631, A375736.
%K A376519 nonn
%O A376519 1,1
%A A376519 _Gus Wiseman_, Sep 28 2024