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Numbers whose least prime factor has the greatest exponent.
3

%I #13 Sep 18 2024 08:43:08

%S 1,2,3,4,5,7,8,9,11,12,13,16,17,19,20,23,24,25,27,28,29,31,32,37,40,

%T 41,43,44,45,47,48,49,52,53,56,59,60,61,63,64,67,68,71,72,73,76,79,80,

%U 81,83,84,88,89,92,96,97,99,101,103,104,107,109,112,113,116

%N Numbers whose least prime factor has the greatest exponent.

%C First differs from A334298 in having 600 and lacking 180.

%C Also numbers whose minimum part in prime factorization is a unique mode.

%C If k is a term, then so are all powers of k. - _Robert Israel_, Sep 17 2024

%H Robert Israel, <a href="/A364160/b364160.txt">Table of n, a(n) for n = 1..10000</a>

%e The prime factorization of 600 is 2*2*2*3*5*5, and 3 > max(1,2), so 600 is in the sequence.

%e The prime factorization of 180 is 2*2*3*3*5, but 2 <= max(2,1), so 180 is not in the sequence.

%e The terms together with their prime indices begin:

%e 1: {} 29: {10} 67: {19}

%e 2: {1} 31: {11} 68: {1,1,7}

%e 3: {2} 32: {1,1,1,1,1} 71: {20}

%e 4: {1,1} 37: {12} 72: {1,1,1,2,2}

%e 5: {3} 40: {1,1,1,3} 73: {21}

%e 7: {4} 41: {13} 76: {1,1,8}

%e 8: {1,1,1} 43: {14} 79: {22}

%e 9: {2,2} 44: {1,1,5} 80: {1,1,1,1,3}

%e 11: {5} 45: {2,2,3} 81: {2,2,2,2}

%e 12: {1,1,2} 47: {15} 83: {23}

%e 13: {6} 48: {1,1,1,1,2} 84: {1,1,2,4}

%e 16: {1,1,1,1} 49: {4,4} 88: {1,1,1,5}

%e 17: {7} 52: {1,1,6} 89: {24}

%e 19: {8} 53: {16} 92: {1,1,9}

%e 20: {1,1,3} 56: {1,1,1,4} 96: {1,1,1,1,1,2}

%e 23: {9} 59: {17} 97: {25}

%e 24: {1,1,1,2} 60: {1,1,2,3} 99: {2,2,5}

%e 25: {3,3} 61: {18} 101: {26}

%e 27: {2,2,2} 63: {2,2,4} 103: {27}

%e 28: {1,1,4} 64: {1,1,1,1,1,1} 104: {1,1,1,6}

%p filter:= proc(n) local F,i;

%p F:= ifactors(n)[2];

%p if nops(F) = 1 then return true fi;

%p i:= min[index](F[..,1]);

%p andmap(t -> F[t,2] < F[i,2], {$1..nops(F)} minus {i})

%p end proc:

%p filter(1):= true:

%p select(filter, [$1..200]); # _Robert Israel_, Sep 17 2024

%t Select[Range[100],First[Last/@FactorInteger[#]] > Max@@Rest[Last/@FactorInteger[#]]&]

%Y Allowing any unique mode gives A356862, complement A362605.

%Y Allowing any unique co-mode gives A359178, complement A362606.

%Y The even case is A360013, counted by A241131.

%Y For greatest instead of least we have A362616, counted by A362612.

%Y These partitions are counted by A364193.

%Y A027746 lists prime factors (with multiplicity).

%Y A112798 lists prime indices, length A001222, sum A056239.

%Y A362611 counts modes in prime factorization, triangle A362614.

%Y A362613 counts co-modes in prime factorization, triangle A362615.

%Y A363486 gives least mode in prime indices, A363487 greatest.

%Y Cf. A098859, A327473, A327476, A360014, A360015, A362610, A364061, A364062.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jul 14 2023