A035484 -id:A035484 - cp's OEIS frontend

A127029 Let f(n) = exp(Pi*sqrt(n)); sequence gives numbers n such that f(n) - floor(f(n)) < 1/10^4.

652, 2608, 5868, 14370, 22905, 23325, 28201, 34218, 45374, 54295, 55003, 59266, 66499, 76750, 87228, 91311, 95041, 160874, 169259, 184867, 192157, 196681, 197869, 206495, 206656, 210532, 210676, 230491, 251572, 273117, 289732, 301417, 311669, 317780, 321484

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

Cf. A035484, A127022, A127023, A127024, A127025, A127026, A127027, A127028, A127030.

$MaxExtraPrecision = 1000; a = {}; Do[If[((Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]]) > 0) && ((Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]]) < 10^(-4)), AppendTo[a, x]], {x, 1, 20000}]; a

a(5)-a(35) from Jon E. Schoenfield, Sep 04 2017

A127022 Let f(k) = exp(Pi*sqrt(k)); sequence gives numbers k such that ceiling(f(k)) - f(k) < 1/10^3.

Original entry on oeis.org

25, 37, 43, 58, 67, 74, 163, 232, 522, 719, 1169, 1245, 1467, 1850, 1872, 2086, 3368, 4075, 5773, 7685, 7802, 7942, 8325, 9728, 10032, 11682, 12158, 13574, 17908, 18505, 19183, 19396, 20039, 20244, 20584, 22241, 23773, 23778, 23834, 25004, 27573, 28071, 32497

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

JungHwan Min, Table of n, a(n) for n = 1..5000

Cf. A035484, A127023, A127024, A127025.

SetDefaultRealField(RealField(500)); R:= RealField(); [n: n in [1..50000] | Ceiling(Exp(Pi(R)*Sqrt(n))) - Exp(Pi(R)*Sqrt(n)) lt 1/1000]; // G. C. Greubel, Jun 02 2019

a = {}; Do[If[(1 - (Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]]) > 0) && (1 - ( Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]])< 10^(-3)), AppendTo[a, x]], {x, 1, 1000}]; a
Reap[Block[{$MaxExtraPrecision = Infinity}, Do[If[N[FractionalPart[Exp[Pi Sqrt[n]]], 8] > .999, Sow[n]], {n, 2000}]]][[-1, 1]] (* JungHwan Min, Mar 20 2016 *)

default(realprecision, 500); c(n) = exp(Pi*sqrt(n));
for(n=1, 50000, if( ceil(c(n)) - c(n) <1/1000, print1(n", "))) \\ G. C. Greubel, Jun 02 2019

a(16)-a(43) added (from JungHwan Min's b-file) by Jon E. Schoenfield, Sep 04 2017

A127024 Let f(n) = exp(Pi*sqrt(n)); sequence gives numbers n such that ceiling(f(n)) - f(n) < 1/10^5.

Original entry on oeis.org

58, 67, 163, 232, 1467, 4075, 343732, 357711, 478233, 486396, 881967, 1003957, 1033466, 1045512, 1053883, 1091706, 1208198, 1240173, 1341615, 1844122, 1878006, 1964724, 2177184, 2259143, 2276046, 2279335, 2488542, 2691364, 2850458, 3157407, 3262163, 3310971

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

J.-P. Serre, "Lectures on the Mordell-Weil theorem".

Anthony Canu, Table of n, a(n) for n = 1..50

Cf. A035484, A127022, A127023, A127025.

a = {}; Do[If[(1 - (Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]]) > 0) && (1 - ( Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]])< 10^(-5)), AppendTo[a, x]], {x, 1, 1000}]; a

a(6)-a(32) from Jon E. Schoenfield, Sep 04 2017

A127025 Let f(n) = exp(Pi*sqrt(n)); sequence gives numbers n such that ceiling(f(n)) - f(n) < 1/10^6.

Original entry on oeis.org

58, 163, 1467, 478233, 881967, 1053883, 1341615, 1844122, 3498092, 6069493, 6396611, 8707530, 10414308, 13340780, 16039620, 17013933, 17226343, 18577932, 19390220, 21991290, 24529596, 26202225, 26634713, 26651262, 26848308, 27497372, 32149837, 35437319, 35892748

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

Anthony Canu, Table of n, a(n) for n = 1..182

Cf. A035484, A127022, A127023, A127024.

a = {}; Do[If[(1 - (Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]]) > 0) && (1 - ( Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]])< 10^(-6)), AppendTo[a, x]], {x, 1, 1000}]; a

is(n)=my(t);default(realprecision, 40);default(realprecision, Pi*sqrt(n)\log(10)+40); t=exp(Pi*sqrt(n));ceil(t)-t<1e-6 \\ Charles R Greathouse IV, Feb 20 2012

a(4)-a(20) from Charles R Greathouse IV, Feb 20 2012

a(21)-a(36) from Charles R Greathouse IV, Feb 23 2012

A127023 Let f(n) = exp(Pi*sqrt(n)); sequence gives numbers n such that ceiling(f(n)) - f(n) < 1/10^4.

Original entry on oeis.org

37, 58, 67, 163, 232, 719, 1169, 1467, 4075, 5773, 19183, 33563, 65477, 67893, 68996, 70273, 81194, 90857, 106194, 112070, 117434, 120332, 122456, 133074, 167196, 210500, 226081, 254883, 261084, 263987, 270154, 281128, 298455, 301487, 313447, 321349, 325779

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

Cf. A035484, A127022, A127024, A127025.

a = {}; Do[If[(1 - (Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]]) > 0) && (1 - ( Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]])< 10^(-4)), AppendTo[a, x]], {x, 1, 1000}]; a

a(9)-a(37) from Jon E. Schoenfield, Sep 04 2017

A127031 Let f(n) = exp(Pi*sqrt(n)); sequence gives numbers n such that f(n) - floor(f(n)) < 1/10^6.

Original entry on oeis.org

652, 2608, 880111, 2720885, 4089051, 4619054, 5046630, 5409046, 5433402, 5603556, 5645558, 7278138, 7466589, 10037029, 10730786, 10823358, 11540978, 11860073, 12898258, 14554227, 15107659, 15602035, 15896143, 17070573, 18204473, 19252185, 19425342, 19556500

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

a(212)=195246501 is the smallest integer such that the fractional part of e^(Pi*sqrt(n)) begins with exactly 8 zeros. - Anthony Canu, Oct 11 2017

			5 is not in the sequence since exp(Pi*sqrt(5)) = 1124.186... has fractional part 0.186... which is greater than 1/10^6. But exp(Pi*sqrt(652)) has fractional part 0.0000000001637... which is less than 1/10^6, so 652 is in the sequence. - _Michael B. Porter_, Aug 24 2016

Anthony Canu, Table of n, a(n) for n = 1..217

Cf. A035484, A127022, A127023, A127024, A127025, A127026, A127027, A127028, A127029.

$MaxExtraPrecision = 1000; a = {}; Do[If[((Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]]) > 0) && ((Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]]) < 10^(-6)), AppendTo[a, x]], {x, 1, 100000}]; a

search(a,b)=my(t,prec=default(realprecision), nprec=round(Pi*sqrt(b)/log(10)+20)); default(realprecision,nprec); for(n=floor(a),b,t=exp(Pi*sqrt(n));if(t-floor(t)<.000001, print(n))); default(realprecision,prec) \\ Charles R Greathouse IV, Jul 28 2009

a(3)-a(14) from Charles R Greathouse IV, Jul 28 2009
a(15)-a(18) from Anthony Canu, Aug 24 2016
a(19) from Anthony Canu, Aug 31 2016
a(20)-a(28) from Anthony Canu, Mar 03 2017

A056581 Nearest integer to 1/(A056580(n) - exp(sqrt(n)*Pi)).

Original entry on oeis.org

-7, -51, 4, -2, -5, 110, 15, -3, 3, 5, -7, -3, 19, 4, 5, -3, 430, 141, 4, 4, -2, 574, 3, 7, 1518, -3, 62, 84, -2, -10, 11, -7, -13, -4, 4, -3, 45551, -5, 3, 3, 2, -33, 4494, -8, -5, -6, 3, -2, 7, 2, 9, -3, -4, -4, 3, -17, -2, 5624716, 147, -5, 4, 3, 3, 2, 6, -2, 747638

Offset: 1

Henry Bottomley, Jun 30 2000

sign

A measure of how close e^(Pi*sqrt(n)) is to an integer (higher absolute value of a(n) means closer, negative value means the closest integer is smaller than it).
The sign convention is chosen so that most terms and in particular record values such as those occurring for the Heegner numbers A003173, are positive, so that A069014 lists record indices of this sequence (except for A069014(2)=2 instead of 3 for signed values). The sequence is not defined for n=0,-1 where e^(sqrt(n)*Pi) is an integer. - M. F. Hasler, Apr 15 2008
Negative resp. positive values of a(n) correspond to 2nd resp. 3rd term of the continued fraction expansion of exp(sqrt(n)*Pi), up to a difference of -1 or -2 depending on the direction of rounding. - M. F. Hasler, Apr 15 2008

			a(6)=110, since e^(Pi*sqrt(6)) = 2197.9908695... and 1/(2198-2197.9908695...) = 109.52... which rounds to 110.
e^(Pi*sqrt(163)) = 262537412640768743.9999999999992500725971981... (the Ramanujan number) and so a(163)=1333462407513.

For links, references and more information see A019296 and other cross-referenced sequences.

Cf. A003173, A019296-A019297, A035484, A056580, A058292, A060456, A069014, A138851.

default(realprecision,100); dZ(x)=round(x)-x
A056581(n)=round(1/dZ(exp(sqrt(n)*Pi)))

a(n) = 1/(A056580(n) - e^(sqrt(n)*Pi)).

A019296 ={-1, 0} U { n | abs(A056581(n)) > 100} U { some n for which abs(A056581(n)) = 100 }. - M. F. Hasler, Apr 15 2008

Definition, formulas and values corrected and extended by M. F. Hasler, Apr 15 2008

A127026 Let f(n) = exp(Pi*sqrt(n)); sequence gives numbers n such that f(n)-floor(f(n)) < 1/10.

Original entry on oeis.org

2, 30, 33, 42, 56, 96, 100, 103, 140, 148, 164, 171, 172, 182, 187, 188, 205, 211, 223, 226, 239, 241, 243, 253, 268, 300, 318, 328, 359, 361, 364, 379, 387, 394, 408, 410, 421, 423, 425, 426, 436, 455, 465, 467, 490, 492, 509, 529, 536, 546, 579, 583, 587

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

Cf. A035484, A127022, A127023, A127024, A127025, A127027, A127028, A127029, A127030.

a = {}; Do[If[((Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]]) > 0) && (( Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]])< 10^(-1)), AppendTo[a, x]], {x, 1, 1000}]; a

A127027 Let f(n) = exp(Pi*sqrt(n)); sequence gives numbers n such that f(n) - floor(f(n)) < 1/10^2.

Original entry on oeis.org

103, 148, 164, 205, 223, 226, 268, 359, 630, 652, 940, 1005, 1194, 1213, 1332, 1353, 1441, 1481, 1519, 1750, 1823, 1825, 1835, 1930, 1951, 1961, 2309, 2339, 2347, 2357, 2498, 2511, 2527, 2554, 2608, 2683, 3086, 3108, 3157, 3377, 3646, 3653, 3656, 3738, 3762

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

Cf. A035484, A127022, A127023, A127024, A127025, A127026, A127028, A127029, A127030.

a = {}; Do[If[((Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]]) > 0) && (( Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]])< 10^(-2)), AppendTo[a, x]], {x, 1, 1000}]; a
epsQ[n_]:=Module[{c=Exp[Pi*Sqrt[n]],min=1/10^2},0Harvey P. Dale, Apr 06 2012 *)

Corrected by Harvey P. Dale, Apr 06 2012

a(27)-a(45) from Jon E. Schoenfield, Sep 04 2017

A127028 Let f(n) = exp(Pi*sqrt(n)); sequence gives numbers n such that f(n) - floor(f(n)) < 1/10^3.

Original entry on oeis.org

148, 268, 652, 1194, 1332, 1519, 2608, 5650, 5774, 5868, 6632, 7260, 9058, 10153, 10210, 11076, 12314, 13049, 13097, 14370, 15696, 16170, 18736, 19571, 21655, 21814, 22905, 23325, 24496, 27435, 28201, 29239, 31214, 34218, 36166, 36287, 38805, 38990, 42761

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

Cf. A035484, A127022, A127023, A127024, A127025, A127026, A127027, A127029, A127030.

a = {}; Do[If[((Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]]) > 0) && (( Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]])< 10^(-3)), AppendTo[a, x]], {x, 1, 1000}]; a

a(7)-a(39) from Jon E. Schoenfield, Sep 04 2017

cp's OEIS Frontend

A127029 Let f(n) = exp(Pi*sqrt(n)); sequence gives numbers n such that f(n) - floor(f(n)) < 1/10^4.

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A127022 Let f(k) = exp(Pi*sqrt(k)); sequence gives numbers k such that ceiling(f(k)) - f(k) < 1/10^3.

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A127024 Let f(n) = exp(Pi*sqrt(n)); sequence gives numbers n such that ceiling(f(n)) - f(n) < 1/10^5.

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A127025 Let f(n) = exp(Pi*sqrt(n)); sequence gives numbers n such that ceiling(f(n)) - f(n) < 1/10^6.

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A127023 Let f(n) = exp(Pi*sqrt(n)); sequence gives numbers n such that ceiling(f(n)) - f(n) < 1/10^4.

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A127031 Let f(n) = exp(Pi*sqrt(n)); sequence gives numbers n such that f(n) - floor(f(n)) < 1/10^6.

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A056581 Nearest integer to 1/(A056580(n) - exp(sqrt(n)*Pi)).

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A127026 Let f(n) = exp(Pi*sqrt(n)); sequence gives numbers n such that f(n)-floor(f(n)) < 1/10.

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A127027 Let f(n) = exp(Pi*sqrt(n)); sequence gives numbers n such that f(n) - floor(f(n)) < 1/10^2.