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Search: id:A077135
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| A077135 |
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Numbers n whose proper (other than 1 and n) odd divisors are prime and even divisors are 1 less than a prime. |
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+0
2
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| 4, 6, 8, 9, 10, 12, 14, 15, 20, 21, 22, 25, 26, 33, 34, 35, 38, 39, 44, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 92, 93, 94, 95, 106, 111, 115, 116, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 164, 166, 169
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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k is a member if (1) k = p*q p, q are primes; (2) k = 4*p and p, 2p+1 are primes. Are there any other prime signatures k could take?
The odd members (A046315) outnumber the even members. - Robert G. Wilson v (rgwv(AT)rgwv.com), Mar 31 2005
This sequence consists of precisely the semiprimes and numbers of the form 4p where 2p+1 is also prime. n cannot have pq as a proper divisor, with p and q odd primes (not necessarily distinct). Likewise 8 cannot be a proper factor. This eliminates all but the specified cases. Note that this sequence is intended to exclude 1 and primes, although by the natural rules for vacuous quantifiers they should be included. - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jul 28 2007
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MATHEMATICA
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fQ[n_] := Block[{d = Take[ Divisors[n], {2, -2}]}, Union[ Flatten[ PrimeQ[{Select[d, OddQ[ # ] &], Select[d, EvenQ[ # ] &] + 1}]]] == {True}]; Select[ Range[ 176], fQ[ # ] &] (from Robert G. Wilson v Mar 31 2005)
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CROSSREFS
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Cf. A001358, A005384.
Sequence in context: A063806 A063989 A117097 this_sequence A110615 A060679 A051234
Adjacent sequences: A077132 A077133 A077134 this_sequence A077136 A077137 A077138
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KEYWORD
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nonn
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AUTHOR
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Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 29 2002
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EXTENSIONS
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Corrected and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Mar 31 2005
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