cp's OEIS Frontend

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

%I A366740 #9 Nov 06 2023 22:58:49
%S A366740 90,180,210,270,360,420,450,462,525,540,550,630,720,810,840,858,900,
%T A366740 910,924,990,1050,1080,1100,1155,1170,1260,1326,1350,1386,1440,1470,
%U A366740 1530,1575,1620,1650,1666,1680,1710,1716,1800,1820,1848,1870,1890,1911,1938,1980
%N A366740 Positive integers whose semiprime divisors do not all have different Heinz weights (sum of prime indices, A056239).
%C A366740 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 A366740 From _Robert Israel_, Nov 06 2023: (Start)
%C A366740 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 A366740 All positive multiples of terms are terms. (End)
%F A366740 These are numbers k such that A086971(k) > A366739(k).
%e A366740 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 A366740 The terms together with their prime indices begin:
%e A366740     90: {1,2,2,3}
%e A366740    180: {1,1,2,2,3}
%e A366740    210: {1,2,3,4}
%e A366740    270: {1,2,2,2,3}
%e A366740    360: {1,1,1,2,2,3}
%e A366740    420: {1,1,2,3,4}
%e A366740    450: {1,2,2,3,3}
%e A366740    462: {1,2,4,5}
%e A366740    525: {2,3,3,4}
%e A366740    540: {1,1,2,2,2,3}
%e A366740    550: {1,3,3,5}
%e A366740    630: {1,2,2,3,4}
%e A366740    720: {1,1,1,1,2,2,3}
%p A366740 N:= 10^4: # for terms <= N
%p A366740 P:= select(isprime, [$1..N]): nP:= nops(P):
%p A366740 R:= {}:
%p A366740 for i from 1 while P[i]*P[i+1]^2*P[i+2] < N do
%p A366740   for j from i+1 while P[i]*P[j]^2 * P[j+1] < N do
%p A366740     for k from j do
%p A366740       l:= j+k-i;
%p A366740       if l <= k or l > nP then break fi;
%p A366740       v:= P[i]*P[j]*P[k]*P[l];
%p A366740       if v <= N then
%p A366740         R:= R union {seq(t,t=v..N,v)};
%p A366740       fi
%p A366740 od od od:
%p A366740 sort(convert(R,list)); # _Robert Israel_, Nov 06 2023
%t A366740 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A366740 Select[Range[1000],!UnsameQ@@Total/@Union[Subsets[prix[#],{2}]]&]
%Y A366740 The complement is too dense.
%Y A366740 For all divisors instead of just semiprimes we have A299729, strict A316402.
%Y A366740 Distinct semi-sums of prime indices are counted by A366739.
%Y A366740 Partitions of this type are counted by A366753, non-binary A366754.
%Y A366740 A001222 counts prime factors (or prime indices), distinct A001221.
%Y A366740 A001358 lists semiprimes, squarefree A006881, conjugate A065119.
%Y A366740 A056239 adds up prime indices, row sums of A112798.
%Y A366740 A299701 counts distinct subset-sums of prime indices, positive A304793.
%Y A366740 A299702 ranks knapsack partitions, counted by A108917, strict A275972.
%Y A366740 Semiprime divisors are listed by A367096 and have:
%Y A366740 - square count: A056170
%Y A366740 - sum: A076290
%Y A366740 - squarefree count: A079275
%Y A366740 - count: A086971
%Y A366740 - firsts: A220264
%Y A366740 Cf. A000720, A001248, A008967, A365541, A365920, A366737, A366738, A366741, A367093, A367095, A367097.
%K A366740 nonn
%O A366740 1,1
%A A366740 _Gus Wiseman_, Nov 05 2023