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Position of first appearance of n in A376264 (run-sums of first differences of nonsquarefree numbers), or 0 if there are none.
3

%I #6 Sep 27 2024 08:32:57

%S 2,8,3,1,6222,14,308540,18

%N Position of first appearance of n in A376264 (run-sums of first differences of nonsquarefree numbers), or 0 if there are none.

%F A376264(a(n)) = n.

%e The sequence of nonsquarefree numbers (A013929) is:

%e 4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, ...

%e with first differences (A078147):

%e 4, 1, 3, 4, 2, 2, 4, 1, 2, 1, 4, 4, 4, 4, 1, 3, 1, 1, 2, 2, 2, 4, 3, 1, ...

%e with runs:

%e (4),(1),(3),(4),(2,2),(4),(1),(2),(1),(4,4,4,4),(1),(3),(1,1),(2,2,2), ...

%e with sums (A376264):

%e 4, 1, 3, 4, 4, 4, 1, 2, 1, 16, 1, 3, 2, 6, 4, 3, 1, 8, 3, 1, 4, 1, 3, 4, 4, ...

%e with first appearances at (A376265):

%e 2, 8, 3, 1, 6222, 14, 308540, 18, ...

%t mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];

%t q=Total/@Split[Differences[Select[Range[10000],!SquareFreeQ[#]&]]]//Most;

%t Table[Position[q,k][[1,1]],{k,mnrm[q]}]

%Y This is the position of first appearance of n in A376264.

%Y The sorted version is A376266.

%Y For run-lengths instead of firsts of run-sums we have A376267.

%Y For compression instead of firsts of run-sums we have A376312.

%Y A000040 lists the prime numbers, differences A001223.

%Y A000961 and A246655 list prime-powers, differences A057820.

%Y A003242 counts compressed compositions, ranks A333489.

%Y A005117 lists squarefree numbers, differences A076259 (ones A375927).

%Y A013929 lists nonsquarefree numbers, differences A078147.

%Y A333254 lists run-lengths of differences between consecutive primes.

%Y A376305 gives run-compression of first differences of squarefree numbers.

%Y A376307 gives run-sums of first differences of squarefree numbers.

%Y Cf. A037201, A053797, A053806, A120992, A373198, A373413, A373822, A375707, A376306, A376310, A376311.

%K nonn,more

%O 1,1

%A _Gus Wiseman_, Sep 27 2024