cp's OEIS Frontend

A325180 Heinz number 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 2.

%I A325180 #6 Apr 10 2019 22:02:51
%S A325180 5,8,10,12,20,21,35,36,42,49,54,60,63,70,81,84,90,98,100,105,126,135,
%T A325180 140,147,150,189,196,210,225,275,294,315,385,441,500,539,550,605,700,
%U A325180 750,770,825,847,980,1050,1078,1100,1125,1155,1210,1250,1331,1372,1375
%N A325180 Heinz number 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 2.
%C A325180 The enumeration of these partitions by sum is given by A325182.
%C A325180 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H A325180 Gus Wiseman, <a href="/A325180/a325180.png">Young diagrams corresponding to the first 96 terms</a>.
%e A325180 The sequence of terms together with their prime indices begins:
%e A325180     5: {3}
%e A325180     8: {1,1,1}
%e A325180    10: {1,3}
%e A325180    12: {1,1,2}
%e A325180    20: {1,1,3}
%e A325180    21: {2,4}
%e A325180    35: {3,4}
%e A325180    36: {1,1,2,2}
%e A325180    42: {1,2,4}
%e A325180    49: {4,4}
%e A325180    54: {1,2,2,2}
%e A325180    60: {1,1,2,3}
%e A325180    63: {2,2,4}
%e A325180    70: {1,3,4}
%e A325180    81: {2,2,2,2}
%e A325180    84: {1,1,2,4}
%e A325180    90: {1,2,2,3}
%e A325180    98: {1,4,4}
%e A325180   100: {1,1,3,3}
%e A325180   105: {2,3,4}
%t A325180 durf[n_]:=Length[Select[Range[PrimeOmega[n]],Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]][[#]]>=#&]];
%t A325180 codurf[n_]:=If[n==1,0,Max[PrimeOmega[n],PrimePi[FactorInteger[n][[-1,1]]]]];
%t A325180 Select[Range[1000],codurf[#]-durf[#]==2&]
%Y A325180 Numbers k such that A263297(k) - A257990(k) = 2.
%Y A325180 Positions of 2's in A325178.
%Y A325180 Cf. A006918, A056239, A093641, A112798, A325164, A325170, A325179, A325182, A325192, A325197.
%K A325180 nonn
%O A325180 1,1
%A A325180 _Gus Wiseman_, Apr 08 2019