This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A376343 #5 Sep 28 2024 07:36:19 %S A376343 2,4,6,9,12,15,18,24,26,31,33,37,39,41,44,47,50,53,57,62,73,75,81,90, %T A376343 95,99,102,105,108,127,129,131,135,139,156,158,161,163,167,173,182, %U A376343 187,190,193,196,205,210,214,216,232,235,241,244,247,254,263,265,270 %N A376343 Positions of twos in the run-compressed (A037201) first differences (A001223) of the primes (A000040). %C A376343 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). %F A376343 For just the odd primes we have a(n) - 1. %e A376343 The sequence of prime numbers (A000040) is: %e A376343 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, ... %e A376343 with first differences (A001223): %e A376343 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 A376343 with run-compression (A037201): %e A376343 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 A376343 with twos at (A376343): %e A376343 2, 4, 6, 9, 12, 15, 18, 24, 26, 31, 33, 37, 39, 41, 44, 47, 50, 53, 57, 62, 73, ... %t A376343 Join@@Position[First/@Split[Differences[Select[Range[100],PrimeQ]]],2] %Y A376343 Positions of 2's in A037201. %Y A376343 The repeats were at positions A064113 before being omitted. %Y A376343 A variation for squarefree numbers is A376342. %Y A376343 A000040 lists the prime numbers, differences A001223. %Y A376343 A000961 and A246655 list prime-powers, differences A057820. %Y A376343 A005117 lists squarefree numbers, differences A076259 (ones A375927). %Y A376343 A013929 lists nonsquarefree numbers, differences A078147. %Y A376343 A333254 lists run-lengths of differences between consecutive primes. %Y A376343 Cf. A007921, A030173, A053289, A061398, A373817, A373822, A373947, A376305, A376308. %K A376343 nonn %O A376343 1,1 %A A376343 _Gus Wiseman_, Sep 26 2024