login
Powers of squarefree numbers whose prime indices are relatively prime. Heinz numbers of uniform partitions with relatively prime parts.
1

%I #4 Apr 16 2018 18:56:52

%S 2,4,6,8,10,14,15,16,22,26,30,32,33,34,35,36,38,42,46,51,55,58,62,64,

%T 66,69,70,74,77,78,82,85,86,93,94,95,100,102,105,106,110,114,118,119,

%U 122,123,128,130,134,138,141,142,143,145,146,154,155,158,161,165

%N Powers of squarefree numbers whose prime indices are relatively prime. Heinz numbers of uniform partitions with relatively prime parts.

%C A prime index of n is a number m such that prime(m) divides n. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

%C The number of uniform partitions of n with relatively prime parts is A078374(n).

%H A. David Christopher and M. Davamani Christober, <a href="http://emis.impa.br/EMIS/journals/GMN/yahoo_site_admin/assets/docs/1_GMN-2492-V13N2.77213831.pdf">Relatively Prime Uniform Partitions</a>, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp.1-12.

%e Sequence of all uniform relatively prime integer partitions begins (1), (11), (21), (111), (31), (41), (32), (1111), (51), (61), (321), (11111), (52), (71), (43), (2211).

%t Select[Range[200],And[GCD@@PrimePi/@FactorInteger[#][[All,1]]===1,SameQ@@FactorInteger[#][[All,2]]]&]

%Y Cf. A000009, A000837, A007916, A047966, A052409, A052410, A072774, A078374, A289023, A289509, A300486, A302491, A302796.

%K nonn

%O 1,1

%A _Gus Wiseman_, Apr 16 2018