A307516 Numbers whose maximum prime index and minimum prime index differ by more than 1.

10, 14, 20, 21, 22, 26, 28, 30, 33, 34, 38, 39, 40, 42, 44, 46, 50, 51, 52, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 74, 76, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 98, 99, 100, 102, 104, 105, 106, 110, 111, 112, 114, 115, 116, 117, 118

Offset: 1

Author

Gus Wiseman, Apr 12 2019

Keywords

• nonn

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose maximum and minimum parts differ by more than 1. The enumeration of these partitions by sum is given by A000094.

Differs from A069900 first at n = 43.

Examples

			The sequence of terms together with their prime indices begins:
   10: {1,3}
   14: {1,4}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   33: {2,5}
   34: {1,7}
   38: {1,8}
   39: {2,6}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   46: {1,9}
   50: {1,3,3}
   51: {2,7}
   52: {1,1,6}
   55: {3,5}
		

Crossrefs

Positions of numbers > 1 in A243055. Complement of A000961 and A256617.

Cf. A000094, A000245, A001222, A052126, A056239, A061395, A064989, A069900, A105441, A112798, A307517, A325196.

Programs

Maple

with(numtheory):
q:= n-> (l-> pi(l[-1])-pi(l[1])>1)(sort([factorset(n)[]])):
select(q, [$2..200])[];  # Alois P. Heinz, Apr 12 2019

Mathematica

Select[Range[100],PrimePi[FactorInteger[#][[-1,1]]]-PrimePi[FactorInteger[#][[1,1]]]>1&]