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A366740
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Positive integers whose semiprime divisors do not all have different Heinz weights (sum of prime indices, A056239).
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14
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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
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OFFSET
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1,1
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COMMENTS
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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.
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)
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LINKS
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FORMULA
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EXAMPLE
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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}
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MAPLE
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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:
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MATHEMATICA
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prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1000], !UnsameQ@@Total/@Union[Subsets[prix[#], {2}]]&]
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CROSSREFS
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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.
A299701 counts distinct subset-sums of prime indices, positive A304793.
Semiprime divisors are listed by A367096 and have:
Cf. A000720, A001248, A008967, A365541, A365920, A366737, A366738, A366741, A367093, A367095, A367097.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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