A366740 - OEIS

%I #9 Nov 06 2023 22:58:49

%S 90,180,210,270,360,420,450,462,525,540,550,630,720,810,840,858,900,

%T 910,924,990,1050,1080,1100,1155,1170,1260,1326,1350,1386,1440,1470,

%U 1530,1575,1620,1650,1666,1680,1710,1716,1800,1820,1848,1870,1890,1911,1938,1980

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

%C 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.

%C From _Robert Israel_, Nov 06 2023: (Start)

%C 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.

%C All positive multiples of terms are terms. (End)

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

%e 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.

%e The terms together with their prime indices begin:

%e     90: {1,2,2,3}

%e    180: {1,1,2,2,3}

%e    210: {1,2,3,4}

%e    270: {1,2,2,2,3}

%e    360: {1,1,1,2,2,3}

%e    420: {1,1,2,3,4}

%e    450: {1,2,2,3,3}

%e    462: {1,2,4,5}

%e    525: {2,3,3,4}

%e    540: {1,1,2,2,2,3}

%e    550: {1,3,3,5}

%e    630: {1,2,2,3,4}

%e    720: {1,1,1,1,2,2,3}

%p N:= 10^4: # for terms <= N

%p P:= select(isprime, [$1..N]): nP:= nops(P):

%p R:= {}:

%p for i from 1 while P[i]*P[i+1]^2*P[i+2] < N do

%p   for j from i+1 while P[i]*P[j]^2 * P[j+1] < N do

%p     for k from j do

%p       l:= j+k-i;

%p       if l <= k or l > nP then break fi;

%p       v:= P[i]*P[j]*P[k]*P[l];

%p       if v <= N then

%p         R:= R union {seq(t,t=v..N,v)};

%p       fi

%p od od od:

%p sort(convert(R,list)); # _Robert Israel_, Nov 06 2023

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[1000],!UnsameQ@@Total/@Union[Subsets[prix[#],{2}]]&]

%Y The complement is too dense.

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

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

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

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

%Y A001358 lists semiprimes, squarefree A006881, conjugate A065119.

%Y A056239 adds up prime indices, row sums of A112798.

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

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

%Y Semiprime divisors are listed by A367096 and have:

%Y - square count: A056170

%Y - sum: A076290

%Y - squarefree count: A079275

%Y - count: A086971

%Y - firsts: A220264

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

%K nonn

%O 1,1

%A _Gus Wiseman_, Nov 05 2023