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A307516
Numbers whose maximum prime index and minimum prime index differ by more than 1.
3
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
1,1
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.
EXAMPLE
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}
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&]
CROSSREFS
Positions of numbers > 1 in A243055. Complement of A000961 and A256617.
Sequence in context: A157138 A131693 A255532 * A069900 A073492 A073493
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 12 2019
STATUS
approved