A361855 - OEIS

%I #6 Apr 01 2023 22:03:27

%S 28,40,78,84,171,190,198,220,240,252,280,351,364,390,406,435,714,748,

%T 756,765,777,784,814,840,850,925,988,1118,1197,1254,1330,1352,1419,

%U 1425,1440,1505,1564,1600,1638,1716,1755,1794,1802,1820,1950,2067,2204,2254

%N Numbers > 1 whose prime indices satisfy (maximum) * (length) = 2*(sum).

%C 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.

%C Also positive integers whose prime indices satisfy (maximum) = 2*(mean).

%C Also Heinz numbers of partitions of the same size as their complement (see example).

%F A061395(a(n)) * A001222(a(n)) = 2*A056239(a(n)).

%e The terms together with their prime indices begin:

%e    28: {1,1,4}

%e    40: {1,1,1,3}

%e    78: {1,2,6}

%e    84: {1,1,2,4}

%e   171: {2,2,8}

%e   190: {1,3,8}

%e   198: {1,2,2,5}

%e   220: {1,1,3,5}

%e   240: {1,1,1,1,2,3}

%e   252: {1,1,2,2,4}

%e   280: {1,1,1,3,4}

%e The prime indices of 84 are {1,1,2,4}, with maximum 4, length 4, and sum 8, and 4*4 = 2*8, so 84 is in the sequence.

%e The prime indices of 120 are {1,1,1,2,3}, with maximum 3, length 5, and sum 8, and 3*5 != 2*8, so 120 is not in the sequence.

%e The prime indices of 252 are {1,1,2,2,4}, with maximum 4, length 5, and sum 10, and 4*5 = 2*10, so 252 is in the sequence.

%e The partition (5,2,2,1) with Heinz number 198 has diagram:

%e   o o o o o

%e   o o . . .

%e   o o . . .

%e   o . . . .

%e Since the partition and its complement (shown in dots) both have size 10, 198 is in the sequence.

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[2,100],Max@@prix[#]*PrimeOmega[#]==2*Total[prix[#]]&]

%Y These partitions are counted by A361853, strict A361854.

%Y For median instead of mean we have A361856, counted by A361849.

%Y For minimum instead of mean we have A361908, counted by A118096.

%Y For length instead of mean we have A361909, counted by A237753.

%Y A001222 (bigomega) counts prime factors, distinct A001221 (omega).

%Y A061395 gives greatest prime index.

%Y A112798 lists prime indices, sum A056239.

%Y A326567/A326568 gives mean of prime indices.

%Y Cf. A067801, A237824, A316413, A324521, A326844, A361205, A361851, A361906.

%K nonn

%O 1,1

%A _Gus Wiseman_, Mar 29 2023