Revision History for A362619
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| A362619
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One and all numbers whose greatest prime factor is a mode, meaning it appears at least as many times as each of the others.
(history;
published version)
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#7 by Michael De Vlieger at Thu May 11 08:47:12 EDT 2023
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#6 by Gus Wiseman at Thu May 11 08:12:35 EDT 2023
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#5 by Gus Wiseman at Thu May 11 08:12:03 EDT 2023
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| EXAMPLE
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The prime indicesfactorization of 300 is are {1,1,2*2,3,*3}, *5*5, with modes {12,35} and maximum 35, so 300 is in the sequence.
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| MATHEMATICA
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prixprifacs[n_]:=If[n==1, {}, Flatten[Cases[ConstantArray@@@FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]; ]]];
Select[Range[100], #==1||], MemberQ[Commonest[prixprifacs[#]], Max[prix[#]]]&]prifacs[#]]&]
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| CROSSREFS
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The case of a unique mode is A362616, counted by A362612.
A112798A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
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#4 by Gus Wiseman at Thu May 11 03:28:40 EDT 2023
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| NAME
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NumbersOne and all numbers whose greatest prime factor is a mode, meaning it appears at least as many times as each of the others.
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| LINKS
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Wikipedia, <a href="https://en.wikipedia.org/wiki/Mode_(statistics)">Mode (statistics)</a>.
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| CROSSREFS
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The case of unique complementmode is A362620A362616, counted by A240302A362612.
The case where the maximum is the only modecomplement is A362616A362620, counted by A362612A240302.
A359178, ranks partitions with a unique co-mode, counted by A362610.
A362611 rankscounts modes in prime factorization, countedtriangle byversion A362614.
A362615A362613 counts partitions by number of co-modes in prime factorization, rankedtriangle byversion A362613A362615.
Cf. A000040 pri, A000720 pri_pi, A002865 ptns_multimax, A098859 ptns_dstnct_mults, A215366 h_tri, A237824 ptns_twi_min_geq_max, A237984 ptns_mean_is_part, A275870 collapsible, A327473 h_mean_is_part, A327476 h_ptns_wo_mean, A359908 prix_int_medn.
Cf. A000040, A002865, A237824, A237984, A327473, A327476, A359908.
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#3 by Gus Wiseman at Thu May 11 03:00:01 EDT 2023
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| EXAMPLE
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The prime factorsindices of 300 are {1,1,2,3,3}, with modes {1,3} and maximum 3, so 300 is in the sequence.
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#2 by Gus Wiseman at Wed May 10 00:04:43 EDT 2023
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| NAME
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allocatedNumbers whose greatest prime factor is a mode, meaning it appears at least as many times as each forof Gusthe Wisemanothers.
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| DATA
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83
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| OFFSET
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1,2
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| COMMENTS
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First differs from A304678 in having 300.
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| EXAMPLE
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The prime factors of 300 are {1,1,2,3,3}, with modes {1,3} and maximum 3, so 300 is in the sequence.
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| MATHEMATICA
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prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], #==1||MemberQ[Commonest[prix[#]], Max[prix[#]]]&]
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| CROSSREFS
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Partitions of this type are counted by A171979.
The complement is A362620, counted by A240302
The case where the maximum is the only mode is A362616, counted by A362612.
A112798 lists prime indices, length A001222, sum A056239.
A356862 ranks partitions with a unique mode, counted by A362608.
A359178, ranks partitions with a unique co-mode, counted by A362610.
A362605 ranks partitions with a more than one mode, counted by A362607.
A362606 ranks partitions with a more than one co-mode, counted by A362609.
A362611 ranks modes in prime factorization, counted by A362614.
A362615 counts partitions by number of co-modes, ranked by A362613.
A362621 ranks partitions with median equal to maximum, counted by A053263.
Cf. A000040 pri, A000720 pri_pi, A002865 ptns_multimax, A098859 ptns_dstnct_mults, A215366 h_tri, A237824 ptns_twi_min_geq_max, A237984 ptns_mean_is_part, A275870 collapsible, A327473 h_mean_is_part, A327476 h_ptns_wo_mean, A359908 prix_int_medn.
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| KEYWORD
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allocated
nonn
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| AUTHOR
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Gus Wiseman, May 09 2023
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| STATUS
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approved
editing
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#1 by Gus Wiseman at Fri Apr 28 03:41:10 EDT 2023
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| NAME
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allocated for Gus Wiseman
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| KEYWORD
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allocated
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| STATUS
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approved
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