A357854 Squarefree numbers with a divisor having the same sum of prime indices as their quotient.

1, 30, 70, 154, 165, 210, 273, 286, 390, 442, 462, 561, 595, 646, 714, 741, 858, 874, 910, 1045, 1155, 1173, 1254, 1326, 1330, 1334, 1495, 1653, 1771, 1794, 1798, 1870, 1938, 2139, 2145, 2294, 2415, 2465, 2470, 2530, 2622, 2639, 2730, 2926, 2945, 2958, 3034

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..47.

Example

The terms together with their prime indices begin:
     1: {}
    30: {1,2,3}
    70: {1,3,4}
   154: {1,4,5}
   165: {2,3,5}
   210: {1,2,3,4}
   273: {2,4,6}
   286: {1,5,6}
   390: {1,2,3,6}
For example, 210 has factorization 14*15, and both factors have the same sum of prime indices 5, so 210 is in the sequence.

Mathematica

sumprix[n_]:=Total[Cases[FactorInteger[n], {p_, k_}:>k*PrimePi[p]]];
Select[Range[1000], SquareFreeQ[#]&&MemberQ[sumprix/@Divisors[#], sumprix[#]/2]&]

Crossrefs

The partitions with these Heinz numbers are counted by A237258.

A subset of A319241, squarefree case of A300061.

Squarefree positions of nonzero terms in A357879.

This is the squarefree case of A357976, counted by A002219.

A001222 counts prime factors, distinct A001221.

A056239 adds up prime indices, row sums of A112798.

Cf. A033879, A033880, A064914, A181819, A235130, A237194, A276107, A300273, A321144, A357975, A357976.

Sequence in context: A295102 A131647 A301900 * A071141 A071312 A071142

Adjacent sequences: A357851 A357852 A357853 * A357855 A357856 A357857

Keyword

nonn

Author

Gus Wiseman, Oct 27 2022

Status

approved