cp's OEIS Frontend

A376654 Sorted positions of first appearances in the second differences of consecutive prime-powers exclusive (A246655).

%I A376654 #5 Oct 07 2024 14:06:21
%S A376654 3,4,9,11,17,24,44,46,47,59,67,68,70,79,117,120,177,178,198,205,206,
%T A376654 215,243,244,303,324,326,401,465,483,604,800,879,938,1032,1054,1076,
%U A376654 1233,1280,1720,1889,1890,1905,1939,1959,1961,2256,2289,2409,2879,3149
%N A376654 Sorted positions of first appearances in the second differences of consecutive prime-powers exclusive (A246655).
%e A376654 The prime-powers exclusive (A246655) are:
%e A376654   2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, ...
%e A376654 with first differences (A057820 except first term) :
%e A376654   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, 3, ...
%e A376654 with first differences (A376596 except first term):
%e A376654   0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, -4, ...
%e A376654 with first appearances (A376654):
%e A376654   1, 3, 4, 9, 11, 17, 24, 44, 46, 47, 59, 67, 68, 70, 79, 117, 120, 177, 178, 198, ...
%t A376654 q=Differences[Select[Range[1000],PrimePowerQ[#]&],2];
%t A376654 Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]
%Y A376654 For first differences we have A376340.
%Y A376654 These are the sorted positions of first appearances in A376596 except first term.
%Y A376654 The inclusive version is a(n) + 1 = A376653(n), except first term.
%Y A376654 For squarefree instead of prime-power we have A376655.
%Y A376654 A000961 lists prime-powers inclusive, exclusive A246655.
%Y A376654 A001597 lists perfect-powers, complement A007916.
%Y A376654 A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
%Y A376654 For prime-powers inclusive: A057820 (first differences), A376597 (inflections and undulations), A376598 (nonzero curvature).
%Y A376654 For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376599 (non-prime-power).
%Y A376654 Cf. A025475, A053707, A057820, A064113, A069623, A093555, A174965, A333254, A336416, A361102.
%K A376654 nonn
%O A376654 1,1
%A A376654 _Gus Wiseman_, Oct 06 2024