A383534 Irregular triangle read by rows where row n lists the positive first differences of the 0-prepended prime indices of n.

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, 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, 4, 2, 1, 1, 8, 15, 1, 1, 4, 1, 2, 2, 5, 1, 5, 16, 1, 1, 3, 2

Offset: 1

Author

Gus Wiseman, May 20 2025

Keywords

• nonn
• tabf

Comments

Also differences of distinct 0-prepended prime indices of n.

Examples

			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).
Rows begin:
    1: ()        16: (1)        31: (11)
    2: (1)       17: (7)        32: (1)
    3: (2)       18: (1,1)      33: (2,3)
    4: (1)       19: (8)        34: (1,6)
    5: (3)       20: (1,2)      35: (3,1)
    6: (1,1)     21: (2,2)      36: (1,1)
    7: (4)       22: (1,4)      37: (12)
    8: (1)       23: (9)        38: (1,7)
    9: (2)       24: (1,1)      39: (2,4)
   10: (1,2)     25: (3)        40: (1,2)
   11: (5)       26: (1,5)      41: (13)
   12: (1,1)     27: (2)        42: (1,1,2)
   13: (6)       28: (1,3)      43: (14)
   14: (1,3)     29: (10)       44: (1,4)
   15: (2,1)     30: (1,1,1)    45: (2,1)
		

Crossrefs

Row-lengths are A001221, sums A061395.

Rows start with A055396, end with A241919.

For multiplicities instead of differences we have A124010 (prime signature).

Including difference 0 gives A287352, without prepending A355536.

Positions of first appearances of rows are A358137.

Positions of strict rows are A383512, counted by A098859.

Positions of non-strict rows are A383513, counted by A336866.

Heinz numbers of rows are A383535.

Restricting to rows of squarefree index gives A384008.

Without prepending we get A384009.

A000040 lists the primes, differences A001223.

A056239 adds up prime indices, row sums of A112798, counted by A001222.

A320348 counts strict partitions with distinct 0-appended differences, ranks A325388.

A325324 counts partitions with distinct 0-appended differences, ranks A325367.

Cf. A005117, A122111, A130091, A325325, A325349, A325366, A325368, A381431, A383506.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[DeleteCases[Differences[Prepend[prix[n],0]],0],{n,100}]

Formula

a(A005117(n)) = A384008(n).