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A376521 Sorted positions of first appearances in the run-compression (A037201) of the first differences (A001223) of the prime numbers (A000040).

%I A376521 #5 Sep 28 2024 07:38:27
%S A376521 1,2,3,8,22,28,32,42,91,141,172,198,242,259,341,400,556,692,1119,1737,
%T A376521 1779,2072,2101,2913,3126,3204,3246,3457,3598,4294,4383,7596,7651,
%U A376521 8284,11986,13729,14220,15101,16273,18217,22303,29523,30243,32236,32808,32820
%N A376521 Sorted positions of first appearances in the run-compression (A037201) of the first differences (A001223) of the prime numbers (A000040).
%C A376521 We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, we can remove all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).
%e A376521 The sequence of prime numbers (A000040) is:
%e A376521   2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, ...
%e A376521 with first differences (A001223):
%e A376521   1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, ...
%e A376521 with run-compression (A037201):
%e A376521   1, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, ...
%e A376521 with first appearances at (A376521):
%e A376521   1, 2, 3, 8, 22, 28, 32, 42, 91, 141, 172, 198, 242, 259, 341, 400, 556, 692, 1119, ...
%t A376521 q=First/@Split[Differences[Select[Range[1000],PrimeQ]]];
%t A376521 Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]
%Y A376521 These are the sorted positions of first appearances in A037201.
%Y A376521 For positions of twos instead of first appearances we have A376343.
%Y A376521 The unsorted version is A376520.
%Y A376521 A000040 lists the prime numbers, differences A001223.
%Y A376521 A003242 counts compressed compositions, ranks A333489.
%Y A376521 A333254 lists run-lengths of differences between consecutive primes.
%Y A376521 A373948 encodes compression using compositions in standard order.
%Y A376521 Cf. A007921, A030173, A061398, A064113, A373817, A373822, A373947, A376305, A376308, A376312.
%K A376521 nonn
%O A376521 1,2
%A A376521 _Gus Wiseman_, Sep 26 2024