A325179 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.

3, 4, 6, 15, 18, 25, 27, 30, 45, 50, 75, 175, 245, 250, 343, 350, 375, 490, 525, 625, 686, 735, 875, 1029, 1225, 1715, 3773, 4802, 5929, 7203, 7546, 9317, 11319, 11858, 12005, 14641, 16807, 17787, 18634, 18865, 26411, 27951, 29282, 29645, 41503, 43923, 46585

Offset: 1

Author

Gus Wiseman, Apr 08 2019

Keywords

• nonn

Comments

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

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

Examples

			The sequence of terms together with their prime indices begins:
    3: {2}
    4: {1,1}
    6: {1,2}
   15: {2,3}
   18: {1,2,2}
   25: {3,3}
   27: {2,2,2}
   30: {1,2,3}
   45: {2,2,3}
   50: {1,3,3}
   75: {2,3,3}
  175: {3,3,4}
  245: {3,4,4}
  250: {1,3,3,3}
  343: {4,4,4}
  350: {1,3,3,4}
  375: {2,3,3,3}
  490: {1,3,4,4}
  525: {2,3,3,4}
  625: {3,3,3,3}
		

Links

• Gus Wiseman, Young diagrams for the first 32 terms.

Crossrefs

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

Positions of 1's in A325178.

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

Programs

Mathematica

durf[n_]:=Length[Select[Range[PrimeOmega[n]],Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]][[#]]>=#&]];
codurf[n_]:=If[n==1,0,Max[PrimeOmega[n],PrimePi[FactorInteger[n][[-1,1]]]]];
Select[Range[1000],codurf[#]-durf[#]==1&]