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A376653 Sorted positions of first appearances in the second differences of consecutive prime-powers inclusive (A000961).

%I A376653 #5 Oct 07 2024 14:06:16
%S A376653 1,4,5,10,12,18,25,45,47,48,60,68,69,71,80,118,121,178,179,199,206,
%T A376653 207,216,244,245,304,325,327,402,466,484,605,801,880,939,1033,1055,
%U A376653 1077,1234,1281,1721,1890,1891,1906,1940,1960,1962,2257,2290,2410,2880,3150
%N A376653 Sorted positions of first appearances in the second differences of consecutive prime-powers inclusive (A000961).
%e A376653 The prime-powers inclusive (A000961) are:
%e A376653   1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, ...
%e A376653 with first differences (A057820):
%e A376653   1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, ...
%e A376653 with first differences (A376596):
%e A376653   0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, ...
%e A376653 with first appearances (A376653):
%e A376653   1, 4, 5, 10, 12, 18, 25, 45, 47, 48, 60, 68, 69, 71, 80, 118, 121, 178, 179, 199, ...
%t A376653 q=Differences[Select[Range[100],#==1||PrimePowerQ[#]&],2];
%t A376653 Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]
%Y A376653 For first differences we had A057820, sorted firsts A376340(n)+1 (except first term).
%Y A376653 These are the sorted positions of first appearances in A376596.
%Y A376653 The exclusive version is a(n) - 1 = A376654(n), except first term.
%Y A376653 For squarefree instead of prime-power we have A376655.
%Y A376653 A000961 lists prime-powers inclusive, exclusive A246655.
%Y A376653 A001597 lists perfect-powers, complement A007916.
%Y A376653 A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
%Y A376653 A064113 lists positions of adjacent equal prime gaps.
%Y A376653 For prime-powers inclusive: A057820 (first differences), A376597 (inflections and undulations), A376598 (nonzero curvature).
%Y A376653 For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376599 (non-prime-power).
%Y A376653 Cf. A053707, A069623, A093555, A174965, A251092, A333254, A361102.
%K A376653 nonn
%O A376653 1,2
%A A376653 _Gus Wiseman_, Oct 06 2024