A011543 -id:A011543 - cp's OEIS frontend

A175830 Number of real-valued zeros of the polynomial whose coefficients are the leading n+1 decimal digits of Euler's constant, A011543(n).

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

Offset: 0

Michel Lagneau, Dec 05 2010

a(n) is the number of real-valued zeros of the polynomial P(n,x) = sum_{k=0..n} e(k) x^k where e = 2.7182818... and A011543(n) = sum_{k>=0} e(k)*10^k.

			a(0) = 0 because 2 => P(0,x)=2 is a constant and has no real root.
a(2) = 2  because 271 => P(2,x) = 1+7x + 2x^2 has 2 real roots.
a(13) = 3 because 27182818284590  => P(13,x) = 9x +5x^2 +4x^3 +8x^4 +2x^5 +8x^6 +x^7 +8x^8 +2x^9 +8x^10 +x^11 +7x^12 +2x^13 has 3 real roots, -3.664218401…, -0.7829315178… and 0.

Cf. A173667, A175799, A001113, A011543.

with(numtheory):Digits:=50: T:=array(1..45):for zz from 0 to 43 do:n:=floor(exp(1)*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: ~

A276196 Smallest prime >= decimal expansion of e truncated to n places (A011543).

Original entry on oeis.org

2, 29, 271, 2719, 27191, 271829, 2718281, 27182839, 271828199, 2718281831, 27182818309, 271828182863, 2718281828489, 27182818284617, 271828182845909, 2718281828459051, 27182818284590483, 271828182845904533, 2718281828459045269, 27182818284590452387, 271828182845904523609

Offset: 0

Ilya Gutkovskiy, Aug 24 2016

			a(5) = 271829, since this is the smallest prime >= floor(e*10^5) = 271828.
e = 2.71828182845904523536028747135266249775724...

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

Cf. A000040, A000720, A001113, A007512, A007918, A011543, A064118.

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

a(n) = A007918(A011543(n)).
a(n) = A000040(A000720(A011543(n)-1)+1).
a(A064118(n)-1) = A007512(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 *)

A138115 Triangle read by rows: row n lists the first n digits of the decimal expansion of e.

Original entry on oeis.org

2, 2, 7, 2, 7, 1, 2, 7, 1, 8, 2, 7, 1, 8, 2, 2, 7, 1, 8, 2, 8, 2, 7, 1, 8, 2, 8, 1, 2, 7, 1, 8, 2, 8, 1, 8, 2, 7, 1, 8, 2, 8, 1, 8, 2, 2, 7, 1, 8, 2, 8, 1, 8, 2, 8, 2, 7, 1, 8, 2, 8, 1, 8, 2, 8, 4, 2, 7, 1, 8, 2, 8, 1, 8, 2, 8, 4, 5

Offset: 1

Omar E. Pol, Mar 14 2008

Also, successive digits of A011543.

			Triangle begins:
..... 2
.... 2,7
... 2,7,1
.. 2,7,1,8
. 2,7,1,8,2

Cf. A001113, A011543, A138068, A138069.

Module[{nn=20,ed},ed=RealDigits[E,10,nn][[1]];Flatten[Table[Take[ed,n],{n,nn}]]] (* Harvey P. Dale, Dec 24 2014 *)

A179852 Number of terms in the simple continued fraction for (e truncated to n decimal digits).

Original entry on oeis.org

1, 4, 6, 8, 12, 13, 15, 17, 23, 24, 31, 23, 29, 29, 34, 38, 36, 42, 38, 47, 44, 44, 50, 48, 49, 54, 60, 56, 60, 61, 60, 60, 70, 68, 77, 72, 71, 82, 68, 81, 78, 83, 82, 82, 89, 91, 102, 90, 98, 99, 103, 97, 90, 107, 107, 112, 111, 120, 109, 115, 118, 112, 111, 127, 128, 133, 117, 117, 117, 139, 136

Offset: 1

Umut Uludag, Jan 11 2011

			a(1): simple continued fraction (SCF) for 2 = 2 (1 term);
a(2): SCF for 2.7 = 2+1/(1+1/(2+1/3)) (4 terms);
a(3): SCF for 2.71 = 2+1/(1+1/(2+1/(2+1/(4+1/3)))) (6 terms).

Cf. A080235, A080229, A011543.

Digits := 120 ;
A011543 := proc(n) floor(exp(1)*10^n) ; end proc:A179852 := proc(n) A011543(n-1)/10^(n-1) ; numtheory[cfrac](%) ; op(%) ; nops(%) ; end proc: # R. J. Mathar, Jan 14 2011

A129897 a(n) is the integer that results from truncating both e and Pi to n digits, adding the results, and discarding the decimal point.

Original entry on oeis.org

5, 58, 585, 5859, 58597, 585987, 5859873, 58598744, 585987447, 5859874481, 58598744819, 585987448203, 5859874482048, 58598744820487, 585987448204883, 5859874482048838, 58598744820488384, 585987448204883846, 5859874482048838473, 58598744820488384737

Offset: 0

Kevin J Davis (kevin_j_davis(AT)btinternet.com), Jun 04 2007

			From _Jon E. Schoenfield_, Feb 24 2021: (Start)
n=1: 2      + 3      = 5      --> a(1) = 5;
n=2: 2.7    + 3.1    = 5.8    --> a(2) = 58;
n=3: 2.71   + 3.14   = 5.85   --> a(3) = 585;
n=4: 2.718  + 3.141  = 5.859  --> a(4) = 5859;
n=5: 2.7182 + 3.1415 = 5.8597 --> a(5) = 58597; etc.
(End)

Cf. A011543, A011545.

Module[{nn=20,pi,e},pi=RealDigits[Pi,10,nn][[1]];e=RealDigits[E,10,nn] [[1]]; Table[FromDigits[Take[pi,n]]+FromDigits[Take[e,n]],{n,nn}]] (* Harvey P. Dale, May 28 2016 *)

a(n) = floor(floor(Pi*10^n) + floor(exp(1)*10^n)), Pi=3.14159..., e=2.71828....

Corrected and extended by Harvey P. Dale, May 28 2016

A249126 Product of the first n digits of Pi (treated as a number) and the first n digits of e (treated as a number).

Original entry on oeis.org

6, 837, 85094, 8537238, 853922530, 85397212652, 8539729843352, 853973398759468, 85397341863406230, 8539734219628209684, 853973422224398765940, 85397342226185298383510, 8539734222671268708689351, 853973422267317150596927230, 85397342226735418150399772016

Offset: 1

Harvey P. Dale, Oct 21 2014

			The first digit of Pi is 3, the first digit of e is 2, and 2 times 3 = 6, so a(1) = 6.  The first two digits of Pi are 31, the first two digits of e are 27, and 31 times 27 = 837, so a(2) = 837.

Cf. A011543 (e), A011545 (Pi).

Module[{nn=20,p,e},p=RealDigits[Pi,10,nn][[1]];e=RealDigits[E,10,nn][[1]];Table[FromDigits[Take[p,n]]FromDigits[Take[e,n]],{n,nn}]]

a(n)=n--; floor(10^n * Pi) * floor(10^n * exp(1)) \\ Charles R Greathouse IV, Oct 21 2014

a(n+1) = floor(10^n * Pi) * floor(10^n * e). - Charles R Greathouse IV, Oct 21 2014
a(n) ~ Pi*e/100 * 100^n + O(10^n). - Charles R Greathouse IV, Oct 21 2014
a(n) = A011543(n-1)*A011545(n-1). - Michel Marcus, Oct 26 2014

A283158 Numbers k such that A011544(k-1) is a prime.

Original entry on oeis.org

1, 85, 555, 1508, 1781, 4224, 7037, 43740

Offset: 1

XU Pingya, Mar 01 2017

For k <= 16000, there are seven primes in sequence A011544.

Round(e*10^112279) = floor(e*10^112279), and floor(e*10^112279) = A011544(112279) (=A007512(7)) is a prime. Thus 112280 = A064118(7) is also a term.

Eric Weisstein's World of Mathematics, e-Prime

Cf. A007512, A011543, A011544, A064118.

Do[If[PrimeQ[Round[E*10^(n-1)]],Print[n]],{n,16000}]

a(2) = 85 added by Jason Yuen, Jun 16 2025

a(8) from Michael S. Branicky, Jun 24 2025

cp's OEIS Frontend

A175830 Number of real-valued zeros of the polynomial whose coefficients are the leading n+1 decimal digits of Euler's constant, A011543(n).

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A276196 Smallest prime >= decimal expansion of e truncated to n places (A011543).

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

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A138115 Triangle read by rows: row n lists the first n digits of the decimal expansion of e.

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A179852 Number of terms in the simple continued fraction for (e truncated to n decimal digits).

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A129897 a(n) is the integer that results from truncating both e and Pi to n digits, adding the results, and discarding the decimal point.

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A249126 Product of the first n digits of Pi (treated as a number) and the first n digits of e (treated as a number).

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A283158 Numbers k such that A011544(k-1) is a prime.

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