A344291 Numbers whose sum of prime indices is at least twice their number of prime indices (counted with multiplicity).

1, 3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85

Offset: 1

Author

Gus Wiseman, May 15 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.

Examples

			The sequence of terms together with their prime indices begins:
     1: {}       25: {3,3}      43: {14}       62: {1,11}
     3: {2}      26: {1,6}      44: {1,1,5}    63: {2,2,4}
     5: {3}      27: {2,2,2}    45: {2,2,3}    65: {3,6}
     7: {4}      28: {1,1,4}    46: {1,9}      66: {1,2,5}
     9: {2,2}    29: {10}       47: {15}       67: {19}
    10: {1,3}    30: {1,2,3}    49: {4,4}      68: {1,1,7}
    11: {5}      31: {11}       50: {1,3,3}    69: {2,9}
    13: {6}      33: {2,5}      51: {2,7}      70: {1,3,4}
    14: {1,4}    34: {1,7}      52: {1,1,6}    71: {20}
    15: {2,3}    35: {3,4}      53: {16}       73: {21}
    17: {7}      37: {12}       55: {3,5}      74: {1,12}
    19: {8}      38: {1,8}      57: {2,8}      75: {2,3,3}
    21: {2,4}    39: {2,6}      58: {1,10}     76: {1,1,8}
    22: {1,5}    41: {13}       59: {17}       77: {4,5}
    23: {9}      42: {1,2,4}    61: {18}       78: {1,2,6}
For example, the prime indices of 45 are {2,2,3} with sum 7 >= 2*3, so 45 is in the sequence.
		

Crossrefs

The partitions with these Heinz numbers are counted by A110618.

The conjugate version is A322109.

The case of equality is A340387, counted by A035363.

The 5-smooth case is A344293, with non-3-smooth case A344294.

The opposite version is A344296.

The conjugate opposite version is A344414.

The conjugate case of equality is A344415.

A001221 counts distinct prime indices.

A001222 counts prime indices with multiplicity.

A056239 adds up prime indices, row sums of A112798.

Cf. A001358, A025065, A084127, A244990, A344292.

Programs

Mathematica

Select[Range[100],PrimeOmega[#]<=Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/2&]

Formula

A056239(a(n)) >= 2*A001222(a(n)).