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Search: a001672 -id:a001672
Displaying 1-10 of 24 results found.
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A115240
a(n) = A001672(n) - A115239(n).
+20
1
0, 0, 3, 10, 33, 104, 328, 1031, 3241, 10183, 31991, 100503, 315740, 991928, 3116234, 9789941, 30756009, 96622854, 303549648, 953629346, 2995914948, 9411944394, 29568495367, 92892167824, 291829352014, 916808948392, 2880240257014
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
EXAMPLE
a(5) = A001672(5) - A115239(5) = 306 - 273 = 33.
MATHEMATICA
a[1] = Floor[Pi]; a[n_] := a[n] = Floor[a[n - 1]*Pi]; Array[ Floor[Pi^# ] - a[ # ] &, 27] (* Robert G. Wilson v *)
CROSSREFS
Cf. A001672, A115239.
KEYWORD
nonn
AUTHOR
Hieronymus Fischer, Jan 17 2006
EXTENSIONS
More terms from Robert G. Wilson v, Jan 18 2006
STATUS
approved
A119604
Merged values of A014217 = {floor(((1+sqrt(5))/2)^n)}, A000149 = {floor(e^n)}, and A001672 = {floor(Pi^n)}, with multiplicity.
+20
0
1, 1, 1, 1, 2, 2, 3, 4, 6, 7, 9, 11, 17, 20, 29, 31, 46, 54, 76, 97, 122, 148, 199, 306, 321, 403, 521, 842, 961, 1096, 1364, 2206, 2980, 3020, 3571, 5777, 8103, 9349, 9488, 15126, 22026, 24476
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
Table of n, a(n) for n=0..41.
OEIS index entries related to powers of irrational constants.
EXAMPLE
a(8)=6 because floor(((1+sqrt(5))/2)^4)=6, a(9)=7 because floor(e^2)=7 and a(10)=9 because floor(Pi^2)=9.
CROSSREFS
Cf. A001672, A014217, A000149.
KEYWORD
easy,nonn
AUTHOR
Bryan David Levenson, Jun 03 2006
EXTENSIONS
Edited by M. F. Hasler, May 29 2018
STATUS
approved
A137994
a(n) is the smallest integer > a(n-1) such that {Pi^a(n)} < {Pi^a(n-1)}, where {x} = x - floor(x), a(1)=1.
+10
12
1, 3, 81, 264, 281, 472, 1147, 2081, 3207, 3592, 10479, 12128, 65875, 114791, 118885
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The sequence was suggested by Leroy Quet on Pi day 2008, cf. A138324.
The next such number must be greater than 100000. - Hieronymus Fischer, Jan 06 2009
a(16) > 300,000. - Robert Price, Mar 25 2019
LINKS
Table of n, a(n) for n=1..15.
EXAMPLE
a(3)=81, since {Pi^81}=0.0037011283.., but {Pi^k}>=0.0062766802... for 1<=k<=80; thus {Pi^81}<{Pi^k} for 1<=k<81. - Hieronymus Fischer, Jan 06 2009
MATHEMATICA
$MaxExtraPrecision = 10000;
p = .999;
Select[Range[1, 5000],
If[FractionalPart[Pi^#] < p, p = FractionalPart[Pi^#]; True] &] (* Robert Price, Mar 12 2019 *)
PROG
(PARI) default(realprecision, 10^4); print1(a=1); for(i=1, 100, f=frac(Pi^a); until( frac(Pi^a++)<f, ); print1(", "a))
CROSSREFS
Cf. A001203, A138324, A001672.
Cf. A081464, A153669, A153677, A153685, A153693, A153705, A153713, A154130, A153717. - Hieronymus Fischer, Jan 06 2009
KEYWORD
nonn,more,hard,changed
AUTHOR
Leroy Quet and M. F. Hasler, Mar 14 2008
EXTENSIONS
a(11)-a(13) from Hieronymus Fischer, Jan 06 2009
Edited by R. J. Mathar, May 21 2010
a(14)-a(15) from Robert Price, Mar 12 2019
STATUS
approved
A153710
Numbers k such that the fractional part of Pi^k is less than 1/k.
+10
9
1, 3, 5, 9, 10, 11, 59, 81, 264, 281, 472, 3592, 10479, 12128, 65875, 118885
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Numbers k such that fract(Pi^k) < 1/k, where fract(x) = x-floor(x).
The next such number must be greater than 100000.
a(17) > 300000. - Robert Price, Mar 25 2019
LINKS
Table of n, a(n) for n=1..16.
EXAMPLE
a(4) = 9 since fract(Pi^9) = 0.0993... < 1/9, but fract(Pi^k) = 0.3891..., 0.2932..., 0.5310... for 6 <= k <= 8, which all are greater than 1/k.
MATHEMATICA
Select[Range[1000], N[FractionalPart[Pi^#], 100] < (1/#) &] (* G. C. Greubel, Aug 25 2016 *)
PROG
(PARI) isok(k) = frac(Pi^k) < 1/k; \\ Michel Marcus, Feb 11 2014
CROSSREFS
Cf. A153662, A153670, A153678, A153686, A153694, A153702, A153714, A154130, A153718.
Cf. A001672.
KEYWORD
nonn,more
AUTHOR
Hieronymus Fischer, Jan 08 2009
EXTENSIONS
a(16) from Robert Price, Mar 25 2019
STATUS
approved
A153712
Numbers k such that the fractional part of Pi^k is greater than 1-(1/k).
+10
9
1, 2, 15, 22, 58, 109, 157, 1030, 1071, 1274, 2008, 2322, 5269, 151710
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Numbers k such that fract(Pi^k) > 1-(1/k), where fract(x) = x-floor(x).
The next such number must be greater than 100000.
a(15) > 300000. - Robert Price, Mar 25 2019
LINKS
Table of n, a(n) for n=1..14.
EXAMPLE
a(3) = 15, since fract(Pi^15) = 0.969... > 0.933... = 1 - (1/15), but fract(Pi^k) <= 1 - (1/k) for 3 <= k <= 14.
MATHEMATICA
Select[Range[1000], N[FractionalPart[Pi^#], 100] > 1 - (1/#) &] (* G. C. Greubel, Aug 25 2016 *)
CROSSREFS
Cf. A153664, A153672, A153680, A153688, A153696, A153704, A153716, A154130, A153720.
Cf. A001672.
KEYWORD
nonn,more
AUTHOR
Hieronymus Fischer, Jan 06 2009
EXTENSIONS
a(14) from Robert Price, Mar 25 2019
STATUS
approved
A002160
Nearest integer to Pi^n.
(Formerly M2841 N1142)
+10
8
1, 3, 10, 31, 97, 306, 961, 3020, 9489, 29809, 93648, 294204, 924269, 2903677, 9122171, 28658146, 90032221, 282844564, 888582403, 2791563950, 8769956796, 27551631843, 86556004192, 271923706894, 854273519914, 2683779414318, 8431341691876, 26487841119104, 83214007069230
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
REFERENCES
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 122.
J. T. Peters, Ten-Place Logarithm Table. Vols. 1 and 2, rev. ed. Ungar, NY, 1957, Vol. 1 (Appendix), p. 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Table of n, a(n) for n=0..28.
EXAMPLE
a(0) = 1 because Pi^0 = 1;
a(2) = 10 because Pi^2 = 9.8696...;
a(10) = 93648 because Pi^10 = 93648.047476...
MAPLE
a := []: Digits := 1000: for n from 0 to 50 do: a := [op(a), round(Pi^n)]: od: seq(a[i+1], i=0..50);
MATHEMATICA
Round[Pi^Range[0, 40]] (* Harvey P. Dale, Jun 10 2024 *)
PROG
(PARI) apply( A002160(n)=Pi^n\/1, [0..50]) \\ An error message will say so if default(realprecision) must be increased. - M. F. Hasler, May 27 2018
CROSSREFS
Cf. A000227 (e^n), A001672 (floor(Pi^n)), A001673 (ceiling(Pi^n)).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 29 2003
Edited by M. F. Hasler, May 27 2018
STATUS
approved
A153711
Minimal exponents m such that the fractional part of Pi^m obtains a maximum (when starting with m=1).
+10
8
1, 2, 15, 22, 58, 157, 1030, 5269, 145048, 151710
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Recursive definition: a(1)=1, a(n) = least number m>a(n-1) such that the fractional part of Pi^m is greater than the fractional part of Pi^k for all k, 1<=k<m.
The next such number must be greater than 100000.
a(11) > 300000. - Robert Price, Mar 25 2019
LINKS
Table of n, a(n) for n=1..10.
FORMULA
Recursion: a(1):=1, a(k):=min{ m>1 | fract(Pi^m) > fract(Pi^a(k-1))}, where fract(x) = x-floor(x).
EXAMPLE
a(3)=15, since fract(Pi^15)= 0.9693879984..., but fract(Pi^k)<=0.8696... for 1<=k<=14; thus fract(Pi^15)>fract(Pi^k) for 1<=k<15 and 15 is the minimal exponent > 2 with this property.
MATHEMATICA
$MaxExtraPrecision = 100000;
p = 0; Select[Range[1, 10000],
If[FractionalPart[Pi^#] > p, p = FractionalPart[Pi^#]; True] &] (* Robert Price, Mar 25 2019 *)
CROSSREFS
Cf. A153663, A153671, A153679, A153687, A153695, A153707, A153715, A154130, A153719.
Cf. A001672.
KEYWORD
nonn,more
AUTHOR
Hieronymus Fischer, Jan 06 2009
EXTENSIONS
a(9)-a(10) from Robert Price, Mar 25 2019
STATUS
approved
A153715
Greatest number m such that the fractional part of Pi^A153711(m) >= 1-(1/m).
+10
8
1, 7, 32, 53, 189, 2665, 10810, 26577, 128778, 483367
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..10.
FORMULA
a(n) = floor(1/(1-fract(Pi^A153711(n)))), where fract(x) = x-floor(x).
EXAMPLE
a(3) = 32, since 1-(1/33) = 0.9696... > fract(Pi^A153711(3)) = fract(Pi^15) = 0.96938... >= 0.96875 = 1-(1/32).
MATHEMATICA
$MaxExtraPrecision = 100000;
A153711 = {1, 2, 15, 22, 58, 157, 1030, 5269, 145048, 151710};
Floor[1/(1-FractionalPart[Pi^A153711])] (* Robert Price, Apr 18 2019 *)
CROSSREFS
Cf. A153663, A153671, A153679, A153687, A153695, A091560, A153711, A154130, A153723.
Cf. A001672.
KEYWORD
nonn,more
AUTHOR
Hieronymus Fischer, Jan 06 2009
EXTENSIONS
a(9)-a(10) from Robert Price, Apr 18 2019
STATUS
approved
A153714
Greatest number m such that the fractional part of Pi^A153710(n) <= 1/m.
+10
7
7, 159, 50, 10, 21, 55, 117, 270, 307, 744, 757, 7804, 13876, 62099, 70718, 154755
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..16.
FORMULA
a(n) = floor(1/fract(Pi^A153710(n))), where fract(x) = x-floor(x).
EXAMPLE
a(2)=159 since 1/160<fract(Pi^A153710(2))=fract(Pi^3)=0.0062766...<=1/159.
MATHEMATICA
A153710 = {1, 3, 5, 9, 10, 11, 59, 81, 264, 281, 472, 3592, 10479,
12128, 65875, 118885};
Table[fp = FractionalPart[Pi^A153710[[n]]]; m = Floor[1/fp];
While[fp <= 1/m, m++]; m - 1, {n, 1, Length[A153710]}] (* Robert Price, May 10 2019 *)
CROSSREFS
Cf. A153662, A153670, A153678, A153686, A153694, A153702, A153710, A154130, A153722.
Cf. A001672.
KEYWORD
nonn,more
AUTHOR
Hieronymus Fischer, Jan 06 2009
EXTENSIONS
a(16) from Robert Price, May 10 2019
STATUS
approved
A153716
Greatest number m such that the fractional part of Pi^A153712(n) >= 1-(1/m).
+10
7
1, 7, 32, 53, 189, 131, 2665, 10810, 2693, 1976, 3697, 4289, 26577, 483367
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..14.
FORMULA
a(n) = floor(1/(1-fract(Pi^A153712(n)))), where fract(x) = x-floor(x).
EXAMPLE
a(3) = 32, since 1-(1/33) = 0.9696... > fract(Pi^A153712(3)) = fract(Pi^15) = 0.96938... >= 0.96875 = 1-(1/32).
MATHEMATICA
A153712 = {1, 2, 15, 22, 58, 109, 157, 1030, 1071, 1274, 2008, 2322,
5269, 151710};
Table[Floor[1/(1 - FractionalPart[Pi^A153712[[n]]])], {n, 1,
Length[A153712]}] (* Robert Price, May 10 2019 *)
CROSSREFS
Cf. A153664, A153672, A153680, A153688, A153696, A153704, A153712, A154130, A153724.
Cf. A001672.
KEYWORD
nonn,more
AUTHOR
Hieronymus Fischer, Jan 06 2009
EXTENSIONS
a(14) from Robert Price, May 10 2019
STATUS
approved
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