login
Heinz numbers of integer partitions of n into distinct perfect squares (A033461).
8

%I #9 Apr 13 2019 09:02:14

%S 1,2,7,14,23,46,53,97,106,151,161,194,227,302,311,322,371,419,454,541,

%T 622,661,679,742,827,838,1009,1057,1082,1193,1219,1322,1358,1427,1589,

%U 1619,1654,1879,2018,2114,2143,2177,2231,2386,2437,2438,2741,2854,2933

%N Heinz numbers of integer partitions of n into distinct perfect squares (A033461).

%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

%C Also products of distinct elements of A011757.

%e The sequence of terms together with their prime indices begins:

%e 1: {}

%e 2: {1}

%e 7: {4}

%e 14: {1,4}

%e 23: {9}

%e 46: {1,9}

%e 53: {16}

%e 97: {25}

%e 106: {1,16}

%e 151: {36}

%e 161: {4,9}

%e 194: {1,25}

%e 227: {49}

%e 302: {1,36}

%e 311: {64}

%e 322: {1,4,9}

%e 371: {4,16}

%e 419: {81}

%e 454: {1,49}

%e 541: {100}

%t Select[Range[1000],And@@Cases[FactorInteger[#],{p_,k_}:>k==1&&IntegerQ[Sqrt[PrimePi[p]]]]&]

%Y Cf. A001156, A005117, A011757, A033461, A052335, A056239, A062457, A078135, A112798, A117144, A276078.

%Y Cf. A109298, A324524, A324525, A324571, A324572, A324588.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 08 2019