A146340
Numbers k such that continued fraction of (1 + sqrt(k))/2 has period 17.
Original entry on oeis.org
521, 617, 709, 1433, 1597, 2549, 2909, 2965, 3161, 3581, 3821, 4013, 4285, 4649, 5501, 5585, 5693, 5813, 6197, 6409, 7825, 7853, 8093, 8125, 8573, 8917, 9281, 9665, 9677, 9925, 10265, 10597, 10973, 11273, 12085, 12805, 13061, 13109, 13613, 13957, 14677
Offset: 1
a(1) = 521 because continued fraction of (1+sqrt(521))/2 = 11, 1, 10, 2, 5, 4, 2, 1, 1, 1, 1, 2, 4, 5, 2, 10, 1, 21, 1, 10, 2, 5, 4, 2, 1, 1, 1, 1, 2, 4, 5, 2, 10, 1, 21, 1, 10, 2, 5, ... has period (1, 10, 2, 5, 4, 2, 1, 1, 1, 1, 2, 4, 5, 2, 10, 1, 21) length 17.
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A146326 := proc(n) if not issqr(n) then numtheory[cfrac]( (1+sqrt(n))/2, 'periodic','quotients') ; nops(%[2]) ; else 0 ; fi; end: isA146340 := proc(n) RETURN(A146326(n) = 17) ; end: for n from 2 do if isA146340(n) then printf("%d,\n",n) ; fi; od: # R. J. Mathar, Sep 06 2009
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cf17Q[n_]:=Module[{s=(1+Sqrt[n])/2},If[IntegerQ[s], 1,Length[ ContinuedFraction[ s][[2]]]]==17]; Select[Range[5000],cf17Q] (* Harvey P. Dale, Dec 20 2017 *)
998 and 1006 removed, sequence extended by
R. J. Mathar, Sep 06 2009
A146343
a(n) = smallest number k such that the continued fraction of (1 + sqrt(k))/2 has period n.
Original entry on oeis.org
5, 2, 17, 6, 41, 18, 89, 31, 73, 43, 265, 94, 421, 118, 193, 172, 521, 106, 241, 151, 337, 489, 433, 268, 929, 211, 409, 334, 673, 379, 937, 463, 601, 331, 769, 721, 2297, 619, 1033, 718, 1777, 394, 1753, 604, 1993, 634, 1249, 526, 3649, 694
Offset: 1
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A := proc(n) local c; try c := numtheory[cfrac](1/2+sqrt(n)/2,'periodic','quotients') ; RETURN(nops(c[2]) ); catch: RETURN(-1) end try ; end: A146343 := proc(n) for k from 1 do if A(k) = n then RETURN(k); fi; od: end: for n from 1 to 30 do printf("%d,",A146343(n)) ; od: # R. J. Mathar, Nov 08 2008
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nn = 50; t = Table[0, {nn}]; cnt = 0; k = 1; While[cnt < nn, k++; cf = ContinuedFraction[(1 + Sqrt[k])/2]; If[Head[cf[[-1]]] === List, len = Length[cf[[-1]]]; If[len <= nn && t[[len]] == 0, t[[len]] = k; cnt++]]]; t
a(6) changed to 18, a(25) to 929, a(28) to 334 by
R. J. Mathar, Nov 08 2008
A146350
Primes p such that continued fraction of (1+sqrt(p))/2 has period 5 : primes in A146330.
Original entry on oeis.org
41, 149, 157, 181, 269, 397, 761, 941, 1013, 2081, 2153, 2477, 2693, 3181, 3221, 3533, 4253, 4409, 5273, 5297, 5741, 6949, 8069, 8501, 8597, 9293, 10301, 10357, 10957, 11321, 12281, 12589, 13313, 17477, 19477, 19949, 20369, 21433, 22397, 23957, 26309
Offset: 1
A146351
Primes p such that continued fraction of (1 + sqrt(p))/2 has period 6: primes in A146331.
Original entry on oeis.org
19, 59, 107, 131, 499, 659, 1627, 1907, 2251, 2467, 3803, 4139, 4283, 5827, 6779, 9539, 10067, 11491, 12619, 13763, 16987, 18587, 18803, 19507, 22003, 23003, 23819, 24859, 28643, 30859, 37507, 40939, 42083, 42299, 43403, 43867, 44563, 52747, 53507, 55339
Offset: 1
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A146326 := proc(n) if not issqr(n) then numtheory[cfrac]( (1+sqrt(n))/2, 'periodic','quotients') ; nops(%[2]) ; else 0 ; fi; end: isA146351 := proc(n) RETURN(isprime(n) and A146326(n) = 6) ; end: for n from 2 to 13000 do if isA146351(n) then printf("%d,\n",n) ; fi; od: # R. J. Mathar, Sep 06 2009
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Select[Prime[Range[10000]],Length[ContinuedFraction[(1+Sqrt[#])/2][[2]]] == 6&] (* Harvey P. Dale, Dec 22 2013 *)
A146352
Primes p such that continued fraction of (1 + sqrt(p))/2 has period 7: primes in A146332.
Original entry on oeis.org
89, 109, 113, 137, 373, 389, 509, 653, 797, 853, 997, 1009, 1493, 1997, 2309, 2621, 2677, 3797, 4973, 7817, 7873, 9829, 9833, 12197, 12269, 12821, 14009, 15773, 16661, 16673, 18253, 18269, 20389, 21557, 24197, 24533, 25037, 25741, 30677, 31973, 33941, 34253, 35977
Offset: 1
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A146326 := proc(n) if not issqr(n) then numtheory[cfrac]( (1+sqrt(n))/2, 'periodic','quotients') ; nops(%[2]) ; else 0 ; fi; end: isA146352 := proc(n) RETURN(isprime(n) and A146326(n) = 7) ; end: for n from 2 to 13000 do if isA146352(n) then printf("%d,\n",n) ; fi; od: # R. J. Mathar, Sep 06 2009
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Select[Range[2*10^4], PrimeQ[#] && Length[ContinuedFraction[(1+Sqrt[#])/2][[2]]] == 7 &] (* Amiram Eldar, Mar 30 2020 *)
A146353
Primes p such that continued fraction of (1 + sqrt(p))/2 has period 8; primes in A146333.
Original entry on oeis.org
31, 71, 383, 503, 743, 983, 1327, 2543, 4271, 5711, 6151, 8543, 9871, 14503, 17783, 21191, 22031, 25463, 35023, 35759, 36263, 36559, 40543, 46471, 47711, 60727, 66343, 72551, 73751, 75767, 81551, 83639, 91463, 98327, 142183, 159407, 160343, 193031, 195743, 218623
Offset: 1
A146354
Primes p such that continued fraction of (1 + sqrt(p))/2 has period 9: primes in A143577.
Original entry on oeis.org
73, 97, 233, 277, 349, 353, 613, 821, 877, 1181, 1277, 1613, 1637, 1693, 2357, 2777, 3557, 3989, 4157, 4517, 4889, 4933, 5261, 6113, 7213, 9133, 9181, 9749, 10313, 10909, 11057, 11213, 11257, 12161, 12301, 13033, 16217, 16741, 17989, 19469
Offset: 1
A-number in definition corrected. 1613 and 4933 inserted, 9421 deleted, extended beyond 9749 by
R. J. Mathar, Nov 09 2008
A146355
Primes p such that continued fraction of (1 + sqrt(p))/2 has period 10 : primes in A146335.
Original entry on oeis.org
43, 67, 563, 827, 1787, 1811, 2099, 2459, 5107, 7643, 8363, 9323, 9371, 9467, 12251, 13499, 23539, 24251, 28411, 35059, 41843, 47563, 49531, 51419, 57731, 66851, 82787, 94547, 109267, 123499, 123923, 126443, 127643, 134363, 135467, 138587, 162251, 180419, 181019
Offset: 1
A146356
Primes p such that continued fraction of (1 + sqrt(p))/2 has period 11: primes in A146335.
Original entry on oeis.org
541, 593, 661, 701, 857, 1061, 1109, 1217, 1237, 1709, 1733, 1949, 2333, 2557, 2957, 3229, 3677, 3701, 4373, 5081, 5237, 5309, 6133, 7013, 8693, 9533, 10333, 10853, 12437, 14197, 19213, 20693, 21101, 23173, 29753, 30949, 33797, 36677, 37781, 37993, 41813
Offset: 1
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A146326 := proc(n) if not issqr(n) then numtheory[cfrac]( (1+sqrt(n))/2, 'periodic','quotients') ; nops(%[2]) ; else 0 ; fi; end: isA146356 := proc(n) RETURN(isprime(n) and A146326(n) = 11) ; end: for n from 2 to 30000 do if isA146356(n) then printf("%d,\n",n) ; fi; od: # R. J. Mathar, Sep 06 2009
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Select[Prime[Range[5000]],Length[ContinuedFraction[(1+Sqrt[#])/2][[2]]] == 11&] (* Harvey P. Dale, Apr 27 2016 *)
A146357
Primes p such that continued fraction of (1 + sqrt(p))/2 has period 12 : primes in A146336.
Original entry on oeis.org
103, 127, 239, 263, 479, 887, 1567, 2711, 5743, 5903, 8311, 8447, 10567, 10847, 12391, 14783, 14831, 15887, 18191, 22343, 23447, 28151, 31391, 32359, 40087, 40343, 42703, 53407, 60103, 60623, 64231, 75431, 79943, 81559, 83663, 93503, 114167, 130199, 135119, 141863
Offset: 1
Period length in definition corrected, 103 added, 607 and 2063 removed. -
R. J. Mathar, Nov 08 2008
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