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A258882
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Primitive weird numbers of the form 2^k*p*q with k > 0 and where p < q are odd primes.
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25
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70, 836, 7192, 7912, 9272, 10792, 17272, 73616, 83312, 113072, 519712, 539744, 555616, 682592, 786208, 1188256, 1229152, 1901728, 2081824, 2189024, 3963968, 4128448, 4145216, 4486208, 4559552, 4632896, 4960448, 5440192, 5568448, 6460864, 6621632, 7354304, 7470272, 8000704, 8134208
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OFFSET
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1,1
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COMMENTS
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The number of terms < 10^n: 0, 1, 2, 5, 9, 15, 35, 61, 114, 204, 380, 696, 1703, 3548, 6726, 13137, ....
If 2^k*p*q is a weird number, it is necessarily primitive, and 2^(k+1) < p < 2^(k+2)-2 < q < 2^(2k+1).
No odd weird numbers are known and any even weird number must have at least 3 distinct prime factors, since all numbers of the form 2^k*p^m are deficient or pseudoperfect or perfect (iff m = 1 and p = 2^(k+1)-1 is a Mersenne prime). Sequence A258333 lists the number of terms in this sequence for given k. - M. F. Hasler, Jul 11 2016
Kravitz has shown that 2^k*p*q is a primitive weird number when the primes p and q satisfy p = (2^(k+1)*q-q-1)/(q+1-2^(k+1)). Many terms in this sequence are of this form, e.g., a(n) with n = 1, 2, 3, 4, 6, 7, 9, 10, 15, 23, 26, 38, 45, 75, 94, 144, 157, 187, 287, 327, 368, 370, 459, 607, 657, 658, .... Sequences A242025, A242998, ... are related to the special case where q is a Mersenne prime (A000668). - M. F. Hasler, Jul 12 2016
Weird numbers of the form 2^k*p*q are always primitive, so this condition could be omitted in the definition of this sequence. - M. F. Hasler, Jul 13 2016
About 35 years after Kravitz's work, the topic of weird numbers has regained interest after a CWU press release about students who used Kravitz's formula to find a large PWN of this form. See A242025 and A320875. - M. F. Hasler, Nov 20 2018
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REFERENCES
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S. Kravitz, A search for large weird numbers. J. Recreational Math. 9 (1976), 82-85 (1977). Zbl 0365.10003
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LINKS
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R. Bagula et al., A very big weird number, Number Theory group on LinkedIn (web.archive.org snapshot; page no longer available). Dec. 2013
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FORMULA
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EXAMPLE
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a(2) = A002975(2) = 836 = 2^2*11*19.
A002975(3) = 4030 = 2*5*13*31 is not in this sequence since it is not of the required form.
The same is true for A002975(4) = 5830.
a(3) = A002975(5) = 7192 = 2^3*29*31, etc.
A002975(179) = 2319548096 = 2^6 * 137^2 * 1931 is the first term of A002975 with only two odd prime divisors, but not of the required form. - M. F. Hasler, Nov 20 2018
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MATHEMATICA
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(* copy the terms from A002975, assign them equal to 'lst' and then *) fQ[n_] := Block[{m = n}, While[ Mod[m, 2] == 0, m /= 2]; PrimeOmega@ m == 2]; Select[lst, fQ]
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PROG
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(PARI) select(t->factor(t)[, 2][^1]=[1, 1]~, A002975) \\ Assuming that A002975 is defined as set or vector. - M. F. Hasler, Jul 11 2016
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CROSSREFS
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Cf. A320875 (more general case of Karavitz' formula).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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