A340933 Numbers whose least prime index is even. Heinz numbers of integer partitions whose last part is even.

3, 7, 9, 13, 15, 19, 21, 27, 29, 33, 37, 39, 43, 45, 49, 51, 53, 57, 61, 63, 69, 71, 75, 77, 79, 81, 87, 89, 91, 93, 99, 101, 105, 107, 111, 113, 117, 119, 123, 129, 131, 133, 135, 139, 141, 147, 151, 153, 159, 161, 163, 165, 169, 171, 173, 177, 181, 183

Offset: 1

Author

Gus Wiseman, Feb 12 2021

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. 1 has no prime indices so is not counted.

Examples

			The sequence of terms together with their prime indices begins:
      3: {2}         51: {2,7}         99: {2,2,5}
      7: {4}         53: {16}         101: {26}
      9: {2,2}       57: {2,8}        105: {2,3,4}
     13: {6}         61: {18}         107: {28}
     15: {2,3}       63: {2,2,4}      111: {2,12}
     19: {8}         69: {2,9}        113: {30}
     21: {2,4}       71: {20}         117: {2,2,6}
     27: {2,2,2}     75: {2,3,3}      119: {4,7}
     29: {10}        77: {4,5}        123: {2,13}
     33: {2,5}       79: {22}         129: {2,14}
     37: {12}        81: {2,2,2,2}    131: {32}
     39: {2,6}       87: {2,10}       133: {4,8}
     43: {14}        89: {24}         135: {2,2,2,3}
     45: {2,2,3}     91: {4,6}        139: {34}
     49: {4,4}       93: {2,11}       141: {2,15}
		

Crossrefs

These partitions are counted by A026805.

Looking at length or at maximum gives A028260/A244990, counted by A027187.

If all prime indices are even we get A066207, counted by A035363.

The complement is {1} \/ A340932, counted by A026804.

A001222 counts prime factors.

A005843 lists even numbers.

A031215 lists even-indexed primes.

A055396 selects least prime index.

A056239 adds up prime indices.

A058695 counts partitions of even numbers, ranked by A300061.

A061395 selects greatest prime index.

A112798 lists the prime indices of each positive integer.

Cf. A006141, A030229, A067661, A101708, A236913, A339846, A340601, A340602, A340605, A340854/A340855.

Programs

Mathematica

Select[Range[2,100],EvenQ[PrimePi[FactorInteger[#][[1,1]]]]&]

Formula

A055396(a(n)) belongs to A005843.

Closed under multiplication.