A366740 Positive integers whose semiprime divisors do not all have different Heinz weights (sum of prime indices, A056239).

90, 180, 210, 270, 360, 420, 450, 462, 525, 540, 550, 630, 720, 810, 840, 858, 900, 910, 924, 990, 1050, 1080, 1100, 1155, 1170, 1260, 1326, 1350, 1386, 1440, 1470, 1530, 1575, 1620, 1650, 1666, 1680, 1710, 1716, 1800, 1820, 1848, 1870, 1890, 1911, 1938, 1980

Offset: 1

Author

Gus Wiseman, Nov 05 2023

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.

From Robert Israel, Nov 06 2023: (Start)

Positive integers divisible by the product of four primes, prime(i)*prime(j)*prime(k)*prime(l), i < j <= k < l, with i + l = j + k.

All positive multiples of terms are terms. (End)

Examples

			The semiprime divisors of 90 are (6,9,10,15), with prime indices ({1,2},{2,2},{1,3},{2,3}) with sums (3,4,4,5), which are not all different, so 90 is in the sequence.
The terms together with their prime indices begin:
    90: {1,2,2,3}
   180: {1,1,2,2,3}
   210: {1,2,3,4}
   270: {1,2,2,2,3}
   360: {1,1,1,2,2,3}
   420: {1,1,2,3,4}
   450: {1,2,2,3,3}
   462: {1,2,4,5}
   525: {2,3,3,4}
   540: {1,1,2,2,2,3}
   550: {1,3,3,5}
   630: {1,2,2,3,4}
   720: {1,1,1,1,2,2,3}
		

Crossrefs

The complement is too dense.

For all divisors instead of just semiprimes we have A299729, strict A316402.

Distinct semi-sums of prime indices are counted by A366739.

Partitions of this type are counted by A366753, non-binary A366754.

A001222 counts prime factors (or prime indices), distinct A001221.

A001358 lists semiprimes, squarefree A006881, conjugate A065119.

A056239 adds up prime indices, row sums of A112798.

A299701 counts distinct subset-sums of prime indices, positive A304793.

A299702 ranks knapsack partitions, counted by A108917, strict A275972.

Semiprime divisors are listed by A367096 and have:

- square count: A056170

- sum: A076290

- squarefree count: A079275

- count: A086971

- firsts: A220264

Cf. A000720, A001248, A008967, A365541, A365920, A366737, A366738, A366741, A367093, A367095, A367097.

Programs

Maple

N:= 10^4: # for terms <= N
P:= select(isprime, [$1..N]): nP:= nops(P):
R:= {}:
for i from 1 while P[i]*P[i+1]^2*P[i+2] < N do
  for j from i+1 while P[i]*P[j]^2 * P[j+1] < N do
    for k from j do
      l:= j+k-i;
      if l <= k or l > nP then break fi;
      v:= P[i]*P[j]*P[k]*P[l];
      if v <= N then
        R:= R union {seq(t,t=v..N,v)};
      fi
od od od:
sort(convert(R,list)); # Robert Israel, Nov 06 2023

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],!UnsameQ@@Total/@Union[Subsets[prix[#],{2}]]&]

Formula

These are numbers k such that A086971(k) > A366739(k).