A238744 Irregular table read by rows: T (n, k) gives the number of primes p such that p^k divides n; table omits all zero values.

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2

Offset: 2

Author

Matthew Vandermast, Apr 28 2014

Keywords

• nonn
• tabf

Comments

If the prime signature of n (nonincreasing version) is viewed as a partition, row n gives the conjugate partition.

Examples

			24 = 2^3*3 is divisible by two prime numbers (2 and 3), one square of a prime (4 = 2^2), and one cube of a prime (8 = 2^3); therefore, row 24 of the table is {2,1,1}.
From _Gus Wiseman_, Mar 31 2022: (Start)
Rows begin:
     1: ()        16: (1,1,1,1)    31: (1)
     2: (1)       17: (1)          32: (1,1,1,1,1)
     3: (1)       18: (2,1)        33: (2)
     4: (1,1)     19: (1)          34: (2)
     5: (1)       20: (2,1)        35: (2)
     6: (2)       21: (2)          36: (2,2)
     7: (1)       22: (2)          37: (1)
     8: (1,1,1)   23: (1)          38: (2)
     9: (1,1)     24: (2,1,1)      39: (2)
    10: (2)       25: (1,1)        40: (2,1,1)
    11: (1)       26: (2)          41: (1)
    12: (2,1)     27: (1,1,1)      42: (3)
    13: (1)       28: (2,1)        43: (1)
    14: (2)       29: (1)          44: (2,1)
    15: (2)       30: (3)          45: (2,1)
(End)
		

Crossrefs

Row lengths are A051903(n); row sums are A001222(n).

Cf. A217171.

These partitions are ranked by A238745.

For prime indices A296150 instead of exponents we get A321649, rev A321650.

A000700 counts self-conjugate partitions, ranked by A088902.

A003963 gives product of prime indices, conjugate A329382.

A008480 gives number of permutations of prime indices, conjugate A321648.

A056239 adds up prime indices, row sums of A112798.

A124010 gives prime signature, sorted A118914, length A001221.

A352486-A352490 are sets related to the fixed points of A122111.

Cf. A000701, A000720, A046682, A238747, A258116, A319005, A330644, A352491.

Programs

Mathematica

Table[Length/@Table[Select[Last/@FactorInteger[n],#>=k&],{k,Max@@Last/@FactorInteger[n]}],{n,2,100}] (* Gus Wiseman, Mar 31 2022 *)

Formula

Row n is identical to row A124859(n) of table A212171.