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

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, 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, 14, 5, 1, 3, 3, 2, 1, 5, 5, 1, 1, 15, 1, 2, 2, 10, 3, 1, 6, 6

Offset

1,2

Comments

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.

Links

Table of n, a(n) for n=1..86.

Example

The prime indices of 60 are {1,1,2,3}, differences (0,1,1), positive (1,1).
Rows begin:
     1: ()     16: ()       31: ()       46: (8)
     2: ()     17: ()       32: ()       47: ()
     3: ()     18: (1)      33: (3)      48: (1)
     4: ()     19: ()       34: (6)      49: ()
     5: ()     20: (2)      35: (1)      50: (2)
     6: (1)    21: (2)      36: (1)      51: (5)
     7: ()     22: (4)      37: ()       52: (5)
     8: ()     23: ()       38: (7)      53: ()
     9: ()     24: (1)      39: (4)      54: (1)
    10: (2)    25: ()       40: (2)      55: (2)
    11: ()     26: (5)      41: ()       56: (3)
    12: (1)    27: ()       42: (1,2)    57: (6)
    13: ()     28: (3)      43: ()       58: (9)
    14: (3)    29: ()       44: (4)      59: ()
    15: (1)    30: (1,1)    45: (1)      60: (1,1)

Mathematica

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

Crossrefs

Row-lengths are A001221(n) - 1, sums A243055.

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

Positions of non-strict rows are a subset of A325992.

Including difference 0 gives A355536, 0-prepended A287352.

The 0-prepended version is A383534.

A000040 lists the primes, differences A001223.

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

Cf. A039956, A048767, A055396, A061395, A130091, A320348, A325325, A325349, A325368, A358137, A381431.

Sequence in context: A140583 A237266 A123507 * A188804 A122580 A265332

Adjacent sequences: A384006 A384007 A384008 * A384010 A384011 A384012

Keyword

nonn,tabf

Author

Gus Wiseman, May 23 2025

Status

approved