cp's OEIS Frontend

A376268 Sorted positions of first appearances in the first differences (A053289) of perfect-powers (A001597).

%I A376268 #7 Sep 30 2024 09:15:35
%S A376268 1,2,3,4,5,6,7,9,10,11,12,13,16,17,18,19,21,23,24,27,28,29,30,32,33,
%T A376268 34,35,36,37,38,39,41,42,43,44,45,47,48,49,50,51,53,54,55,56,57,58,60,
%U A376268 61,62,63,64,65,66,67,69,70,71,72,73,74,76,77,78,79,80,81
%N A376268 Sorted positions of first appearances in the first differences (A053289) of perfect-powers (A001597).
%e A376268 The perfect powers (A001597) are:
%e A376268   1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
%e A376268 with first differences (A053289):
%e A376268   3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, ...
%e A376268 with positions of first appearances (A376268):
%e A376268   1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 16, 17, 18, 19, 21, 23, 24, 27, 28, 29, ...
%t A376268 perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
%t A376268 q=Differences[Select[Range[1000],perpowQ]];
%t A376268 Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]
%Y A376268 These are the sorted positions of first appearances in A053289 (union A023055).
%Y A376268 The complement is A376519.
%Y A376268 A053707 lists first differences of consecutive prime-powers.
%Y A376268 A333254 lists run-lengths of differences between consecutive primes.
%Y A376268 Other families of numbers and their first differences:
%Y A376268 For prime numbers (A000040) we have A001223.
%Y A376268 For composite numbers (A002808) we have A073783.
%Y A376268 For nonprime numbers (A018252) we have A065310.
%Y A376268 For perfect powers (A001597) we have A053289.
%Y A376268 For non-perfect-powers (A007916) we have A375706.
%Y A376268 For squarefree numbers (A005117) we have A076259.
%Y A376268 For nonsquarefree numbers (A013929) we have A078147.
%Y A376268 For prime-powers inclusive (A000961) we have A057820.
%Y A376268 For prime-powers exclusive (A246655) we have A057820(>1).
%Y A376268 For non-prime-powers inclusive (A024619) we have A375735.
%Y A376268 For non-prime-powers exclusive (A361102) we have A375708.
%Y A376268 Cf. A025475, A045542, A052410, A069623, A093555, A174965, A216765, A303707, A305630, A305631, A375736.
%K A376268 nonn
%O A376268 1,2
%A A376268 _Gus Wiseman_, Sep 28 2024