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 A383534 #7 May 21 2025 16:41:53
%S A383534 1,2,1,3,1,1,4,1,2,1,2,5,1,1,6,1,3,2,1,1,7,1,1,8,1,2,2,2,1,4,9,1,1,3,
%T A383534 1,5,2,1,3,10,1,1,1,11,1,2,3,1,6,3,1,1,1,12,1,7,2,4,1,2,13,1,1,2,14,1,
%U A383534 4,2,1,1,8,15,1,1,4,1,2,2,5,1,5,16,1,1,3,2
%N A383534 Irregular triangle read by rows where row n lists the positive first differences of the 0-prepended prime indices of n.
%C A383534 Also differences of distinct 0-prepended prime indices of n.
%F A383534 a(A005117(n)) = A384008(n).
%e A383534 The prime indices of 140 are {1,1,3,4}, zero prepended {0,1,1,3,4}, differences (1,0,2,1), positive (1,2,1).
%e A383534 Rows begin:
%e A383534 1: () 16: (1) 31: (11)
%e A383534 2: (1) 17: (7) 32: (1)
%e A383534 3: (2) 18: (1,1) 33: (2,3)
%e A383534 4: (1) 19: (8) 34: (1,6)
%e A383534 5: (3) 20: (1,2) 35: (3,1)
%e A383534 6: (1,1) 21: (2,2) 36: (1,1)
%e A383534 7: (4) 22: (1,4) 37: (12)
%e A383534 8: (1) 23: (9) 38: (1,7)
%e A383534 9: (2) 24: (1,1) 39: (2,4)
%e A383534 10: (1,2) 25: (3) 40: (1,2)
%e A383534 11: (5) 26: (1,5) 41: (13)
%e A383534 12: (1,1) 27: (2) 42: (1,1,2)
%e A383534 13: (6) 28: (1,3) 43: (14)
%e A383534 14: (1,3) 29: (10) 44: (1,4)
%e A383534 15: (2,1) 30: (1,1,1) 45: (2,1)
%t A383534 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A383534 Table[DeleteCases[Differences[Prepend[prix[n],0]],0],{n,100}]
%Y A383534 Row-lengths are A001221, sums A061395.
%Y A383534 Rows start with A055396, end with A241919.
%Y A383534 For multiplicities instead of differences we have A124010 (prime signature).
%Y A383534 Including difference 0 gives A287352, without prepending A355536.
%Y A383534 Positions of first appearances of rows are A358137.
%Y A383534 Positions of strict rows are A383512, counted by A098859.
%Y A383534 Positions of non-strict rows are A383513, counted by A336866.
%Y A383534 Heinz numbers of rows are A383535.
%Y A383534 Restricting to rows of squarefree index gives A384008.
%Y A383534 Without prepending we get A384009.
%Y A383534 A000040 lists the primes, differences A001223.
%Y A383534 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A383534 A320348 counts strict partitions with distinct 0-appended differences, ranks A325388.
%Y A383534 A325324 counts partitions with distinct 0-appended differences, ranks A325367.
%Y A383534 Cf. A005117, A122111, A130091, A325325, A325349, A325366, A325368, A381431, A383506.
%K A383534 nonn,tabf
%O A383534 1,2
%A A383534 _Gus Wiseman_, May 20 2025