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%I #13 May 15 2022 11:40:05
%S 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,
%T 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,
%U 1,2,1,2,1,1,1,1,1,2,1,2,2,1,1,2,1,1,2
%N 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.
%C If the prime signature of n (nonincreasing version) is viewed as a partition, row n gives the conjugate partition.
%F Row n is identical to row A124859(n) of table A212171.
%e 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}.
%e From _Gus Wiseman_, Mar 31 2022: (Start)
%e Rows begin:
%e 1: () 16: (1,1,1,1) 31: (1)
%e 2: (1) 17: (1) 32: (1,1,1,1,1)
%e 3: (1) 18: (2,1) 33: (2)
%e 4: (1,1) 19: (1) 34: (2)
%e 5: (1) 20: (2,1) 35: (2)
%e 6: (2) 21: (2) 36: (2,2)
%e 7: (1) 22: (2) 37: (1)
%e 8: (1,1,1) 23: (1) 38: (2)
%e 9: (1,1) 24: (2,1,1) 39: (2)
%e 10: (2) 25: (1,1) 40: (2,1,1)
%e 11: (1) 26: (2) 41: (1)
%e 12: (2,1) 27: (1,1,1) 42: (3)
%e 13: (1) 28: (2,1) 43: (1)
%e 14: (2) 29: (1) 44: (2,1)
%e 15: (2) 30: (3) 45: (2,1)
%e (End)
%t Table[Length/@Table[Select[Last/@FactorInteger[n],#>=k&],{k,Max@@Last/@FactorInteger[n]}],{n,2,100}] (* _Gus Wiseman_, Mar 31 2022 *)
%Y Row lengths are A051903(n); row sums are A001222(n).
%Y Cf. A217171.
%Y These partitions are ranked by A238745.
%Y For prime indices A296150 instead of exponents we get A321649, rev A321650.
%Y A000700 counts self-conjugate partitions, ranked by A088902.
%Y A003963 gives product of prime indices, conjugate A329382.
%Y A008480 gives number of permutations of prime indices, conjugate A321648.
%Y A056239 adds up prime indices, row sums of A112798.
%Y A124010 gives prime signature, sorted A118914, length A001221.
%Y A352486-A352490 are sets related to the fixed points of A122111.
%Y Cf. A000701, A000720, A046682, A238747, A258116, A319005, A330644, A352491.
%K nonn,tabf
%O 2,6
%A _Matthew Vandermast_, Apr 28 2014