A046974 -id:A046974 - cp's OEIS frontend

A011545 a(n) is the integer whose decimal digits are the first n+1 decimal digits of Pi.

3, 31, 314, 3141, 31415, 314159, 3141592, 31415926, 314159265, 3141592653, 31415926535, 314159265358, 3141592653589, 31415926535897, 314159265358979, 3141592653589793, 31415926535897932, 314159265358979323, 3141592653589793238, 31415926535897932384

Offset: 0

N. J. A. Sloane

Number of collisions occurring in a system consisting of an infinitely massive, rigid wall at the origin, a ball with mass m stationary at position x1 > 0, and a ball with mass (10^2n)m at position x2 > x1 and rolling toward the origin, assuming perfectly elastic collisions and no friction. - Richard Holmes, Jun 17 2021

Wolfgang Haken (1977) conjectured that no term of this sequence is a perfect square, and estimated the probability that this conjecture is false to be smaller than 10^-9. - Paolo Xausa, Jul 15 2023

Martin Gardner, Fractal Music, Hypercards and More: Mathematical Recreations from Scientific American Magazine, W. H. Freemand and Company, New York, NY, 1992, pp. 274-275.

Paolo Xausa, Table of n, a(n) for n = 0..100
G. Galperin, Playing pool with π (the number π from a billiard point of view), Regular and Chaotic Dynamics, 8 (2003), 375-394.
Wolfgang Haken, An attempt to understand the four color problem, in Journal of Graph Theory, Vol. 1, Issue 3, 1977, pp. 193-206.
G. Sanderson, Why do colliding blocks compute pi?, a 3Blue1Brown YouTube video, Jan 20 2019.

Cf. A000796 (decimal expansion of Pi), A089281, A078604, A089282, A089283, A089284, A089285, A089286, A089287, A089288, A089289, A046974, A089290.

s=RealDigits[Pi, 10, 30][[1]]; Table[FromDigits[Take[s, n]], {n, Length[s]}]
(* Or: *)
a[n_] := IntegerPart[Pi*10^n]; Table[a[n], {n, 0, 9}] (* Peter Luschny, Mar 15 2024 *)

A011545(n)={localprec(n+3); Pi\10^-n} \\ M. F. Hasler, Mar 15 2024

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

Definition corrected by M. F. Hasler, Mar 15 2024

A019676 Decimal expansion of Pi/9.

Original entry on oeis.org

3, 4, 9, 0, 6, 5, 8, 5, 0, 3, 9, 8, 8, 6, 5, 9, 1, 5, 3, 8, 4, 7, 3, 8, 1, 5, 3, 6, 9, 7, 7, 2, 2, 5, 4, 2, 6, 8, 8, 5, 7, 4, 3, 7, 7, 7, 0, 8, 3, 4, 5, 0, 9, 1, 2, 1, 9, 4, 3, 8, 2, 8, 8, 0, 3, 4, 2, 0, 1, 8, 2, 2, 9, 2, 0, 6, 8, 9, 8, 8, 8, 7, 3, 6, 4, 4, 8, 3, 1, 3, 9, 2, 6, 9, 0, 1, 8, 9, 6

Offset: 0

N. J. A. Sloane

Overlapping interpretation of A046974, as in where numbers are put in a decimal digit place and overlapping digits are added. - Eliora Ben-Gurion, Dec 22 2019

			0.349065850398865915384738153697722542688574377708345091219438288034201... - _Vladimir Joseph Stephan Orlovsky_, Dec 02 2009

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for transcendental numbers

Cf. A000796 (Pi).

Pi(RealField(120))/9; // G. C. Greubel, Jan 01 2020

evalf(Pi/9, 120); # G. C. Greubel, Jan 01 2020

RealDigits[N[Pi/9, 120]]//First (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

default(realprecision, 120); Pi/9 \\ G. C. Greubel, Jan 01 2020

numerical_approx(pi/9, digits=120) # G. C. Greubel, Jan 01 2020

Equals Sum_{k>=0} A046974(k)/10^(k+1) = 3/10 + 4/100 + 8/10^3 + 9/10^4 + 14/10^5 + 23/10^6 + ... - Eliora Ben-Gurion, Dec 22 2019

Equals Integral_{x=0..oo} x^(7/2)/(1 + x^9) dx. - Amiram Eldar, Aug 12 2020

A099534 a(n)=Sum of the first n decimal places of e.

Original entry on oeis.org

7, 8, 16, 18, 26, 27, 35, 37, 45, 49, 54, 63, 63, 67, 72, 74, 77, 82, 85, 91, 91, 93, 101, 108, 112, 119, 120, 123, 128, 130, 136, 142, 144, 148, 157, 164, 171, 176, 183, 185, 189, 196, 196, 205, 208, 214, 223, 232, 241, 246, 255, 260, 267, 271, 280, 286, 292

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Oct 22 2004

			Decimal places of e are: 718281828459045... so the sums are: 7, 7+1, 7+1+8,
7+1+8+2,... = 7,8,16,18,...

Cf. A046975 for version of this sequence including the initial 2 of e. A039918 and A046974 for analogous sequences for Pi.

a(n)=A046975(n+1)-2

A074850 Partial products of successive digits in the decimal expansion of Pi.

Original entry on oeis.org

3, 3, 12, 12, 60, 540, 1080, 6480, 32400, 97200, 486000, 3888000, 34992000, 244944000, 2204496000, 6613488000, 13226976000, 39680928000, 317447424000, 1269789696000, 7618738176000, 15237476352000, 91424858112000

Offset: 1

Zak Seidov, Sep 10 2002

Because 33rd digit in the decimal expansion of Pi, pi(33) = 0, all a(n>32) = 0.

Partial sums of digits of the decimal expansion of Pi are in A046974.

			a(3)=12 because pi(1)=3, pi(1)=1, pi(3)=4 and a(3)=3*1*4=12.

Cf. A000796, A046974, A073055.

ppi={3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8, 3, 2, 7, 9, 5, 0, 2, 8}; Table[Product[ppi[[i]], {i, n}], {n, 1, 33}]
Rest[FoldList[Times,1,RealDigits[Pi,10,30][[1]]]] (* Harvey P. Dale, Jan 23 2015 *)

a(n) = pi(1)*...*pi(n); pi(n)=A000796(n).

a(n) = A073055(n), n>0. - R. J. Mathar, Dec 15 2020

A089290 Digital root of floor(Pi*10^n), Pi=3.14....

Original entry on oeis.org

3, 4, 8, 9, 5, 5, 7, 4, 9, 3, 8, 7, 7, 5, 5, 8, 1, 4, 3, 7, 4, 6, 3, 7, 1, 4, 3, 6, 8, 6, 6, 2, 2, 4, 3, 2, 6, 7, 7, 5, 6, 3, 3, 6, 6, 6, 9, 7, 3, 4, 4, 9, 8, 1, 1, 1, 8, 3, 3, 7, 2, 7, 7, 9, 3, 3, 1, 9, 1, 7, 2, 2, 8, 1, 9, 6, 8, 8, 7, 7, 7, 6, 3, 5, 4, 4, 7

Offset: 0

Reinhard Zumkeller, Oct 30 2003

			n=10: floor(Pi*10^n) = 31415926535 -> 3 + 1 + 4 + 1 + 5 + 9 + 2 + 6 + 5 + 3 + 5 = 44 -> 4 + 4 = 8, so a(10)=8.

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Cf. A046974, A000796.

A089290 := proc(n) Digits:=n+20: return ((floor(Pi*10^n)-1) mod 9) + 1: end: seq(A089290(n),n=0..100); # Nathaniel Johnston, May 04 2011

Table[ Mod[ Floor[ Pi*10^n], 9], {n, 0, 104}] /. 0 -> 9 (* Robert G. Wilson v, Oct 31 2003 *)

a(n) = A010888(A011545(n)).

More terms from Ray Chandler and Robert G. Wilson v, Oct 31 2003

A099536 Sum of the first n digits of Zeta(3) (Apery's constant), including the initial 1.

Original entry on oeis.org

1, 3, 3, 5, 5, 10, 16, 25, 25, 28, 29, 34, 43, 48, 57, 61, 63, 71, 76, 79, 88, 97, 104, 107, 115, 116, 122, 123, 128, 129, 130, 134, 138, 147, 156, 165, 165, 172, 178, 182, 191, 199, 205, 207, 216, 218, 221, 225, 225, 229, 238, 246, 254, 262, 263, 270, 279, 281

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Oct 22 2004

			Zeta(3)=1.20205690... so sequence begins 1, 1+2, 1+2+0, 1+2+0+2, 1+2+0+2+0,
1+2+0+2+0+5,... which gives 1, 3, 3, 5, 5, 10, ...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Analogous sequences for other constants: A096535 (log 2), A099534 and A046975 (e), A039918 and A046974 (Pi).

Apéry's number or Apéry's constant zeta(3) is A002117. - N. J. A. Sloane, Jul 11 2023

Accumulate[RealDigits[Zeta[3],10,120][[1]]] (* Harvey P. Dale, Jan 18 2012 *)

A076787 Pisumprimes: prime(k), where k is the sum of the first n digits of Pi.

Original entry on oeis.org

5, 7, 19, 23, 43, 83, 97, 127, 151, 167, 193, 239, 283, 337, 389, 409, 421, 439, 487, 509, 563, 571, 607, 631, 647, 661, 727, 743, 757, 811, 863, 907, 907, 919, 977, 1031, 1051, 1061, 1117, 1181

Offset: 1

Cino Hilliard, Nov 16 2002

The sum of the reciprocals of this sequence diverges; it grows as log log n, just as the sum of the reciprocals of the primes does. - Franklin T. Adams-Watters, Mar 30 2006

Prime[#]&/@Accumulate[RealDigits[Pi,10,40][[1]]] (* Harvey P. Dale, Sep 30 2012 *)

\\ pi digit sum index primes; pisump.gp Primes whose index is the sequential sum of digits of pi
{ pisump(n) = default(realprecision, 100000); p = Pi/10; default(realprecision,28); sr=0; s=0; for(x=1, n, d = p*10; d1=floor(d); s+=d1; p = frac(d); d = p*10; p2=prime(s); sr+=1/p2+0.; print1(p2, ", "); ); print(" "); print(sr); }

The digits of Pi are added d_1+d_2..d_i and the prime whose index is the i-th sum is chosen. E.g. for Pi = 3.14149265358979... the first Pisumprime is prime (3) the second is prime(4), 3rd prime(8) etc. Let d_1, d_2, ..d_i be the expansion of the decimal digits of Pi. Then Pisumprime(n) = prime(d_1), prime (d_1+d_2), ...prime(sum(d_i, i=1..n)). This can be generalized to pisumprime(n, z) where z is the nesting level of prime(x). for z=1 we have prime() for z=2 we have prime (prime(x)), z=3 prime(prime(prime(x))) etc.

a(n)=A000040(A046974(n)) - Franklin T. Adams-Watters, Mar 30 2006

Edited by T. D. Noe, Jun 24 2009

A099535 Sum of the first n decimal places of log(2).

Original entry on oeis.org

6, 15, 18, 19, 23, 30, 31, 39, 39, 44, 49, 58, 67, 71, 76, 79, 79, 88, 92, 93, 100, 102, 105, 107, 108, 110, 111, 115, 120, 128, 129, 136, 142, 147, 153, 161, 161, 168, 173, 178, 178, 178, 179, 182, 186, 189, 195, 195, 197, 202, 207, 209, 214, 218, 219, 221, 221

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Oct 22 2004

			log(2) decimals = 693147180559945... so the sums are 6, 6+9, 6+9+3, 6+9+3+1,... which are 6, 15, 18, 19, ...

Cf. A039918 and A046974 for Pi, A046975 and A099534 for e.

Accumulate[RealDigits[Log[2],10,100][[1]]]  (* Harvey P. Dale, Mar 23 2011 *)

A099538 Sum of the first n digits of sqrt(2), including the initial "1".

Original entry on oeis.org

1, 5, 6, 10, 12, 13, 16, 21, 27, 29, 32, 39, 42, 42, 51, 56, 56, 60, 68, 76, 76, 77, 83, 91, 99, 106, 108, 112, 114, 114, 123, 129, 138, 146, 146, 153, 161, 166, 172, 181, 187, 194, 195, 203, 210, 215, 218, 225, 231, 240, 244, 252, 252, 259, 262, 263, 270, 276, 282

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Oct 22 2004

			sqrt(2)=1.41421356237... so the sums are 1, 1+4, 1+4+1, 1+4+1+4, 1+4+1+4+2,...
which gives 1, 5, 6, 10, 12,...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A002193 for digits of sqrt(2). Other sequences like this one for other constants: A099534-A099537, A039918, A046974, A046975.

Accumulate[RealDigits[Sqrt[2],10,60][[1]]] (* Harvey P. Dale, May 30 2012 *)

A099539 Sum of the first n decimal places of sqrt(2).

Original entry on oeis.org

4, 5, 9, 11, 12, 15, 20, 26, 28, 31, 38, 41, 41, 50, 55, 55, 59, 67, 75, 75, 76, 82, 90, 98, 105, 107, 111, 113, 113, 122, 128, 137, 145, 145, 152, 160, 165, 171, 180, 186, 193, 194, 202, 209, 214, 217, 224, 230, 239, 243, 251, 251, 258, 261, 262, 269, 275, 281

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Oct 22 2004

Cf. A099538 for a version of this sequence resulting from including all digits of sqrt(2) and not just the digits after the decimal point.

			Decimal places of sqrt(2) are 41421356237... so sums are 4, 4+1, 4+1+4, 4+1+4+2,... which gives 4, 5, 9, 11, ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A099538 and A099534, A099535, A099536, A099537, A039918, A046974, A046975 for analogous sequences based on other constants.

Accumulate[Rest[RealDigits[N[Sqrt[2],70]][[1]]]] (* Harvey P. Dale, Dec 12 2010 *)

a(n) = A099538(n+1) - 1.

cp's OEIS Frontend

A011545 a(n) is the integer whose decimal digits are the first n+1 decimal digits of Pi.

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A019676 Decimal expansion of Pi/9.

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A099534 a(n)=Sum of the first n decimal places of e.

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A074850 Partial products of successive digits in the decimal expansion of Pi.

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A089290 Digital root of floor(Pi*10^n), Pi=3.14....

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Maple

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A099536 Sum of the first n digits of Zeta(3) (Apery's constant), including the initial 1.

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A076787 Pisumprimes: prime(k), where k is the sum of the first n digits of Pi.

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A099535 Sum of the first n decimal places of log(2).

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