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A384009 Irregular triangle read by rows where row n lists the positive first differences of the prime indices of n.

%I A384009 #6 May 23 2025 10:15:03
%S A384009 1,2,1,3,1,1,2,2,4,1,5,3,1,1,3,6,1,1,7,4,2,1,2,4,1,8,1,2,5,5,1,2,3,6,
%T A384009 9,1,1,10,2,3,1,3,6,7,2,1,1,11,1,7,1,1,4,2,12,1,2,4,13,8,4,1,1,2,8,9,
%U A384009 14,5,1,3,3,2,1,5,5,1,1,15,1,2,2,10,3,1,6,6
%N A384009 Irregular triangle read by rows where row n lists the positive first differences of the prime indices of n.
%C A384009 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%e A384009 The prime indices of 60 are {1,1,2,3}, differences (0,1,1), positive (1,1).
%e A384009 Rows begin:
%e A384009      1: ()     16: ()       31: ()       46: (8)
%e A384009      2: ()     17: ()       32: ()       47: ()
%e A384009      3: ()     18: (1)      33: (3)      48: (1)
%e A384009      4: ()     19: ()       34: (6)      49: ()
%e A384009      5: ()     20: (2)      35: (1)      50: (2)
%e A384009      6: (1)    21: (2)      36: (1)      51: (5)
%e A384009      7: ()     22: (4)      37: ()       52: (5)
%e A384009      8: ()     23: ()       38: (7)      53: ()
%e A384009      9: ()     24: (1)      39: (4)      54: (1)
%e A384009     10: (2)    25: ()       40: (2)      55: (2)
%e A384009     11: ()     26: (5)      41: ()       56: (3)
%e A384009     12: (1)    27: ()       42: (1,2)    57: (6)
%e A384009     13: ()     28: (3)      43: ()       58: (9)
%e A384009     14: (3)    29: ()       44: (4)      59: ()
%e A384009     15: (1)    30: (1,1)    45: (1)      60: (1,1)
%t A384009 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A384009 Table[DeleteCases[Differences[prix[n]],0],{n,100}]
%Y A384009 Row-lengths are A001221(n) - 1, sums A243055.
%Y A384009 For multiplicities instead of differences we have A124010 (prime signature).
%Y A384009 Positions of non-strict rows are a subset of A325992.
%Y A384009 Including difference 0 gives A355536, 0-prepended A287352.
%Y A384009 The 0-prepended version is A383534.
%Y A384009 A000040 lists the primes, differences A001223.
%Y A384009 A056239 adds up prime indices, row sums of A112798, counted by A001222.
%Y A384009 Cf. A039956, A048767, A055396, A061395, A130091, A320348, A325325, A325349, A325368, A358137, A381431.
%K A384009 nonn,tabf
%O A384009 1,2
%A A384009 _Gus Wiseman_, May 23 2025