A011551 -id:A011551 - cp's OEIS frontend

A175834 Number of real zeros of the polynomial whose coefficients are the decimal expansion of the golden ratio truncated to n places (A011551).

0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 3, 2, 3, 4, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 3, 2

Offset: 0

Michel Lagneau, Dec 05 2010

a(n) = number of real zeros of the polynomial P(n,x) = sum_{k=0..n} p(k) x^k where p(k) are the digits of the decimal expansion of floor(GoldenRatio *10^n) and GoldenRatio = 1.6180339 ....

			a(4) = 2 because 16180 => P(4,x) = 8x+x^2+6x^3+x^4 has 2 real roots :
x0= - 6.053134348… and x1 = 0.

Cf. A173667 A011551.

with(numtheory):Digits:=50: T:=array(1..45):for zz from 0 to 43 do:n:=floor(((1+sqrt(5))/2)*10^zz):   for i from 1 to 43 do: T[i]:=0:od: l:=length(n) : n0:=n:for m from 1 to l do:q:=n0:u:=irem(q,10):v:=iquo(q, 10):n0:=v :u: T[m]:=u:od: x:=fsolve(T[1]+ T[2]*z + T[3]*z^2+ T[4]*z^3+ T[5]*z^4 + T[6]*z^5 + T[7]*z^6 + T[8]*z^7 + T[9]*z^8 + T[10]*z^9+
  T[11]*z^10+ T[12]*z^11 + T[13]*z^12 + T[14]*z^13 + T[15]*z^14+ T[16]*z^15+ T[17]*z^16 + T[18]*z^17 + T[19]*z^18 + T[20]*z^19 + T[21]*z^20 + T[22]*z^21+ T[23]*z^22+ T[24]*z^23 + T[25]*z^24 + T[26]*z^25+ T[27]*z^26+ T[28]*z^27+ T[29]*z^28 + T[30]*z^29 + T[31]*z^30+ T[32]*z^31 + T[33]*z^32 + T[34]*z^33+ T[35]*z^34+ T[36]*z^35 + T[37]*z^36 + T[38]*z^37+ T[39]*z^38 + T[40]*z^39+ T[41]*z^40+ T[42]*z^41 + T[43]*z^42, z, real):x1:=[x]: x2:=nops(x1): printf ( "%d %d %d\n",zz,n,x2):od:

A276198 Smallest prime >= decimal expansion of phi truncated to n places (A011551), where phi is the golden ratio (A001622).

Original entry on oeis.org

2, 17, 163, 1619, 16183, 161807, 1618033, 16180349, 161803403, 1618033999, 16180339933, 161803398917, 1618033988749, 16180339887557, 161803398874991, 1618033988749901, 16180339887498961, 161803398874989661, 1618033988749894853, 16180339887498948503, 161803398874989484891

Offset: 0

Ilya Gutkovskiy, Aug 24 2016

			a(5) = 161807, since this is the smallest prime >= floor(phi*10^5) = 161803.
phi = 1.61803398874989484820458683436563811772...

Eric Weisstein's World of Mathematics, Next Prime
Eric Weisstein's World of Mathematics, Golden Ratio
Eric Weisstein's World of Mathematics, Phi-Prime
Index entries related to "constant primes"

Cf. A000040, A000720, A001622, A007918, A011551, A064117, A064119.

Table[NextPrime[Floor[GoldenRatio 10^n] - 1], {n, 0, 20}]

a(n) = A007918(A011551(n)).
a(n) = A000040(A000720(A011551(n)-1)+1).
a(A064119(n)-1) = A064117(n).

A011543 Decimal expansion of e truncated to n places.

Original entry on oeis.org

2, 27, 271, 2718, 27182, 271828, 2718281, 27182818, 271828182, 2718281828, 27182818284, 271828182845, 2718281828459, 27182818284590, 271828182845904, 2718281828459045, 27182818284590452, 271828182845904523, 2718281828459045235, 27182818284590452353, 271828182845904523536

Offset: 0

N. J. A. Sloane

a(n) <= A011544(n) <= a(n)+1. - Danny Rorabaugh, Mar 07 2015

Danny Rorabaugh, Table of n, a(n) for n = 0..999

Cf. A001113, A011544.

Cf. A011545, A011547, A011549, A011551.

Module[{nn=30,edgs},edgs=RealDigits[E,10,nn][[1]];Table[ FromDigits[ Take[ edgs, n]],{n,nn}]] (* Harvey P. Dale, Oct 04 2017 *)

a(n) = floor(exp(1)*10^n); \\ Michel Marcus, Mar 08 2015

from sympy import E
def a(n): return int(E*10**n)
print([a(n) for n in range(21)]) # Michael S. Branicky, Feb 27 2021

a(n) = floor(e*10^n).

A139260 Decimal expansion of Euler's constant (or Euler-Mascheroni constant) gamma truncated to n places.

Original entry on oeis.org

5, 57, 577, 5772, 57721, 577215, 5772156, 57721566, 577215664, 5772156649, 57721566490, 577215664901, 5772156649015, 57721566490153, 577215664901532, 5772156649015328, 57721566490153286, 577215664901532860

Offset: 1

Omar E. Pol, May 01 2008

			n ... Successive digits of a(n)
1 ............ ( 5 )
2 ......... . ( 5 7 )
3 .......... ( 5 7 7 )
4 ......... ( 5 7 7 2 )
5 ........ ( 5 7 7 2 1 )
6 ....... ( 5 7 7 2 1 5 )
7 ...... ( 5 7 7 2 1 5 6 )
8 ..... ( 5 7 7 2 1 5 6 6 )
9 .... ( 5 7 7 2 1 5 6 6 4 )
10 .. ( 5 7 7 2 1 5 6 6 4 9 )

G. C. Greubel, Table of n, a(n) for n = 1..995

Decimal expansion of gamma: A001620. Cf. A011543, A011545 A011551, A139258, A139259, A139261.

With[{em=RealDigits[EulerGamma,10,20][[1]]},Table[FromDigits[Take[ em,n]], {n,20}]] (* Harvey P. Dale, Nov 22 2011 *)

A138116 Triangle read by rows: row n lists the first n digits of the decimal expansion of golden ratio phi.

Original entry on oeis.org

1, 1, 6, 1, 6, 1, 1, 6, 1, 8, 1, 6, 1, 8, 0, 1, 6, 1, 8, 0, 3, 1, 6, 1, 8, 0, 3, 3, 1, 6, 1, 8, 0, 3, 3, 9, 1, 6, 1, 8, 0, 3, 3, 9, 8, 1, 6, 1, 8, 0, 3, 3, 9, 8, 8, 1, 6, 1, 8, 0, 3, 3, 9, 8, 8, 7, 1, 6, 1, 8, 0, 3, 3, 9, 8, 8, 7, 4

Offset: 1

Omar E. Pol, Mar 14 2008

Also, successive digits of A011551.

			Triangle begins:
      1
     1,6
    1,6,1
   1,6,1,8
  1,6,1,8,0

Cf. A001622 (phi), A011551, A138072, A138073, A175834 (polynomial roots).

s={};Do[AppendTo[s,RealDigits[GoldenRatio,10,n][[1]]],{n,12}];s//Flatten (* James C. McMahon, Jun 05 2025 *)

A195607 Numerator of floor(Phi*10^n)/10^n, where phi = (sqrt(5) + 1)/2 = A001622 is the Golden Ratio.

Original entry on oeis.org

1, 8, 161, 809, 809, 161803, 1618033, 16180339, 80901699, 404508497, 16180339887, 80901699437, 1618033988749, 8090169943749, 161803398874989, 809016994374947, 4045084971874737, 40450849718747371, 25281781074217107, 8090169943749474241

Offset: 0

M. F. Hasler, following a suggestion by Eric Angelini, Sep 21 2011

Numerator of the decimal fraction of phi = 1.61803... truncated to a given number of decimal places.

The corresponding sequence for 1/phi = 0.61803... = phi-1 (also called the Golden Ratio) has a very similar behavior, because for both, the truncated decimal expansion can be simplified by the same factors 2^k*5^m.

			a(3) = 161 is the numerator of 1.61 = 161/100.
a(4) = 809 is the numerator of 1.618 = 1618/1000 = 809/500.

Cf. A001622, A011551.

Cf. A195603 (analog for Pi), A195604 (for e), A195606 (for gamma).

Floor[GoldenRatio #]/#&/@(10^Range[0,20])//Numerator (* Harvey P. Dale, Apr 14 2023 *)

a(n,c=sqrt(5)/2+.5)=numerator(c\.1^n/10^n)  \\ M. F. Hasler, Sep 21 2011

cp's OEIS Frontend

A175834 Number of real zeros of the polynomial whose coefficients are the decimal expansion of the golden ratio truncated to n places (A011551).

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A276198 Smallest prime >= decimal expansion of phi truncated to n places (A011551), where phi is the golden ratio (A001622).

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A011543 Decimal expansion of e truncated to n places.

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A139260 Decimal expansion of Euler's constant (or Euler-Mascheroni constant) gamma truncated to n places.

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Mathematica

A138116 Triangle read by rows: row n lists the first n digits of the decimal expansion of golden ratio phi.

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Mathematica

A195607 Numerator of floor(Phi*10^n)/10^n, where phi = (sqrt(5) + 1)/2 = A001622 is the Golden Ratio.

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