A344415 Numbers whose greatest prime index is half their sum of prime indices.

4, 9, 12, 25, 30, 40, 49, 63, 70, 84, 112, 121, 154, 165, 169, 198, 220, 264, 273, 286, 289, 325, 351, 352, 361, 364, 390, 442, 468, 520, 529, 561, 595, 624, 646, 714, 741, 748, 765, 832, 841, 850, 874, 918, 931, 952, 961, 988, 1020, 1045, 1173, 1197, 1224

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.

Links

Table of n, a(n) for n=1..53.

Formula

A061395(a(n)) = A056239(a(n))/2.

Example

The sequence of terms together with their prime indices begins:
       4: {1,1}           198: {1,2,2,5}
       9: {2,2}           220: {1,1,3,5}
      12: {1,1,2}         264: {1,1,1,2,5}
      25: {3,3}           273: {2,4,6}
      30: {1,2,3}         286: {1,5,6}
      40: {1,1,1,3}       289: {7,7}
      49: {4,4}           325: {3,3,6}
      63: {2,2,4}         351: {2,2,2,6}
      70: {1,3,4}         352: {1,1,1,1,1,5}
      84: {1,1,2,4}       361: {8,8}
     112: {1,1,1,1,4}     364: {1,1,4,6}
     121: {5,5}           390: {1,2,3,6}
     154: {1,4,5}         442: {1,6,7}
     165: {2,3,5}         468: {1,1,2,2,6}
     169: {6,6}           520: {1,1,1,3,6}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Max[primeMS[#]]==Total[primeMS[#]]/2&]

Crossrefs

The partitions with these Heinz numbers are counted by A035363.

The conjugate version is A340387.

This sequence is the case of equality in A344414 and A344416.

A001222 counts prime factors with multiplicity.

A025065 counts palindromic partitions, ranked by A265640.

A027187 counts partitions of even length, ranked by A028260.

A056239 adds up prime indices, row sums of A112798.

A058696 counts partitions of even numbers, ranked by A300061.

A301987 lists numbers whose sum of prime indices equals their product.

A322109 ranks partitions of n with no part > n/2, counted by A110618.

A334201 adds up all prime indices except the greatest.

A344291 lists numbers m with A001222(m) <= A056239(m)/2, counted by A110618.

A344296 lists numbers m with A001222(m) >= A056239(m)/2, counted by A025065.

Cf. A000070, A001414, A209816, A301988, A316413, A316428, A320924, A325037, A325038, A325044, A330950, A344293, A344294, A344297.

Sequence in context: A320925 A212875 A069082 * A297414 A076794 A254520

Adjacent sequences: A344412 A344413 A344414 * A344416 A344417 A344418

Keyword

nonn

Author

Gus Wiseman, May 19 2021

Status

approved