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A376520 Position of first appearance of 2n in the run-compression (A037201) of the first differences (A001223) of the prime numbers (A000040).

%I A376520 #5 Sep 28 2024 07:38:22
%S A376520 2,3,8,22,32,42,28,259,91,141,172,242,341,400,556,692,198,1119,3126,
%T A376520 2072,1779,1737,7596,2913,3246,2101,3598,7651,4383,4294,3457,8284,
%U A376520 14220,11986,15101,3204,32808,18217,16273,42990,22303,37037,13729,43117,32820,70501
%N A376520 Position of first appearance of 2n in the run-compression (A037201) of the first differences (A001223) of the prime numbers (A000040).
%C A376520 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 A376520 The sequence of prime numbers (A000040) is:
%e A376520   2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, ...
%e A376520 with first differences (A001223):
%e A376520   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 A376520 with run-compression (A037201):
%e A376520   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 A376520 with first appearance of 2n at (A376520):
%e A376520   2, 3, 8, 22, 32, 42, 28, 259, 91, 141, 172, 242, 341, 400, 556, 692, 198, 1119, ...
%t A376520 mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
%t A376520 q=First/@Split[Differences[Select[Range[10000],PrimeQ]]];
%t A376520 Table[Position[q,2k][[1,1]],{k,mnrm[Rest[q]/2]}]
%Y A376520 This is the position of first appearance of 2n in A037201.
%Y A376520 For positions of twos instead of first appearances we have A376343.
%Y A376520 The sorted version is A376521.
%Y A376520 A000040 lists the prime numbers, differences A001223.
%Y A376520 A000961 and A246655 list prime-powers, differences A057820.
%Y A376520 A003242 counts compressed compositions, ranks A333489.
%Y A376520 A005117 lists squarefree numbers, differences A076259 (ones A375927).
%Y A376520 A013929 lists nonsquarefree numbers, differences A078147.
%Y A376520 A116861 counts partitions by compressed sum, compositions A373949.
%Y A376520 A116608 counts partitions by compressed length, compositions A333755.
%Y A376520 A274174 counts contiguous compositions, ranks A374249.
%Y A376520 A333254 lists run-lengths of differences between consecutive primes.
%Y A376520 A373948 encodes compression using compositions in standard order.
%Y A376520 Cf. A007921, A030173, A064113, A373817, A373822, A373947, A376305, A376308, A376312.
%K A376520 nonn
%O A376520 1,1
%A A376520 _Gus Wiseman_, Sep 26 2024