A325179 - OEIS

%I #6 Apr 10 2019 22:03:00

%S 3,4,6,15,18,25,27,30,45,50,75,175,245,250,343,350,375,490,525,625,

%T 686,735,875,1029,1225,1715,3773,4802,5929,7203,7546,9317,11319,11858,

%U 12005,14641,16807,17787,18634,18865,26411,27951,29282,29645,41503,43923,46585

%N Heinz numbers of integer partitions such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 1.

%C The enumeration of these partitions by sum is given by A325181.

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

%H Gus Wiseman, <a href="/A325179/a325179.png">Young diagrams for the first 32 terms</a>.

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

%e     3: {2}

%e     4: {1,1}

%e     6: {1,2}

%e    15: {2,3}

%e    18: {1,2,2}

%e    25: {3,3}

%e    27: {2,2,2}

%e    30: {1,2,3}

%e    45: {2,2,3}

%e    50: {1,3,3}

%e    75: {2,3,3}

%e   175: {3,3,4}

%e   245: {3,4,4}

%e   250: {1,3,3,3}

%e   343: {4,4,4}

%e   350: {1,3,3,4}

%e   375: {2,3,3,3}

%e   490: {1,3,4,4}

%e   525: {2,3,3,4}

%e   625: {3,3,3,3}

%t durf[n_]:=Length[Select[Range[PrimeOmega[n]],Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]][[#]]>=#&]];

%t codurf[n_]:=If[n==1,0,Max[PrimeOmega[n],PrimePi[FactorInteger[n][[-1,1]]]]];

%t Select[Range[1000],codurf[#]-durf[#]==1&]

%Y Numbers k such that A263297(k) - A257990(k) = 1.

%Y Positions of 1's in A325178.

%Y Cf. A056239, A093641, A112798, A325180, A325181, A325192, A325196, A325198.

%K nonn

%O 1,1

%A _Gus Wiseman_, Apr 08 2019