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

1,2

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.

Links

Table of n, a(n) for n=1..66.

Formula

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

Example

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.

Mathematica

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

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.

Sequence in context: A281005 A080259 A067715 * A231564 A325798 A365830

Adjacent sequences: A344288 A344289 A344290 * A344292 A344293 A344294

Keyword

nonn

Author

Gus Wiseman, May 15 2021

Status

approved