A007512 -id:A007512 - cp's OEIS frontend

A064118 Numbers k such that the first k digits of e form a prime.

1, 3, 7, 85, 1781, 2780, 112280, 155025

Offset: 1

Shyam Sunder Gupta, Sep 09 2001

The primes are given in A007512. Sequences A065815, A119344, A136583, A210706,... are analogs for gamma, sqrt(3), sqrt(10), 3^(1/3), .... The MathWorld page about "Constant Primes" lists further examples. - M. F. Hasler, Aug 31 2013

			a(2)=3 because the 3-digit number 271 is prime.

C. A. Pickover, The Mathematics of Oz, "2, 271, 2718281", Chapter 95, Camb.Univ.Press, UK 2002.

Eric Weisstein's World of Mathematics, Constant Primes
Eric Weisstein's World of Mathematics, e Digits
Eric Weisstein's World of Mathematics, e-Prime
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Index entries related to "constant primes".

Cf. A001113.

Cf. A047658.

Do[If[PrimeQ[FromDigits[RealDigits[N[E, n + 10], 10, n][[1]]]], Print[n]], {n, 1, 2300}]

One more term from Robert G. Wilson v, Sep 28 2001
a(6) from Eric W. Weisstein, Jan 17 2005
a(7) from Eric W. Weisstein, Jul 03 2009
a(8) from Eric W. Weisstein, Oct 11 2010

A095935 First prime of length n encountered in the decimal representation of e.

Original entry on oeis.org

2, 71, 271, 4523, 74713, 904523, 2718281, 72407663, 360287471, 7427466391, 75724709369, 749669676277, 8284590452353, 99959574966967, 724709369995957, 2470936999595749, 28459045235360287, 571382178525166427

Offset: 1

Mark Cooke (rm_cooke(AT)yahoo.com), Jul 12 2004

Cf. A007512.

f[n_] := Select[FromDigits /@ Partition[ First[ RealDigits[E, 10, 1000]], n, 1], PrimeQ, 1][[1]]; Array[f, 20] (* Robert G. Wilson v, Nov 14 2011 *)

A210704 Primes formed by initial digits of 3^(1/3) = A002581, i.e., of the form floor[3^(1/3)*10^k].

Original entry on oeis.org

144224957030740838232163, 144224957030740838232163831078010958839186925349935057754641619454168759682999733

Offset: 1

M. F. Hasler, Aug 31 2013

Inspired by prime curios about 4957 (cf. LINKS), one of which honors the late Bruce Murray, 30.11.1931 - 29.8.2013.

See A210706 for the k-values. The keyword "less" for this records means that the next term (2488 digits) cannot be added / displayed here, and instead of listing further primes here, the k-values should be recorded in A210706.

C. Caldwell, G.L. Honaker (Eds.), 4957 (another Prime Pages' Curiosity).

Cf. A005042 (analog for Pi), A007512 (analog for e), A115453 (analog for sqrt(2)), A119343 (analog for sqrt(3)), A072952 (analog for gamma).

With[{cr3=RealDigits[CubeRoot[3],10,100][[1]]},Select[Table[FromDigits[Take[cr3,n]],{n,100}],PrimeQ]] (* Harvey P. Dale, Mar 30 2024 *)

(c=sqrtn(3,3),v=1/*set to 0 for indices instead of values*/)->for(k=0,precision(c),ispseudoprime(p=c\.1^k)&&print1([k,p][1+v]","))

a(n) = floor[A002581 * 10^A210706(n)], where A002581 is taken as a constant.

A040016 Largest prime < e^n.

Original entry on oeis.org

2, 7, 19, 53, 139, 401, 1093, 2971, 8101, 22013, 59863, 162751, 442399, 1202603, 3269011, 8886109, 24154939, 65659969, 178482289, 485165141, 1318815713, 3584912833, 9744803443, 26489122081, 72004899319, 195729609407, 532048240573, 1446257064289, 3931334297131

Offset: 1

Jud McCranie

nonn

A050809 is a subset. Lim_{n --> infinity} a(n+1)/a(n) = e. - Jonathan Vos Post, May 02 2006

			a(20) = floor(e^20) - 54 = 485165195 - 54 = 485165141 as there are no primes p such that 485165141 < p < 485165195.

Giovanni Resta, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, e-Prime.

Cf. A000040, A000149, A007512, A014210, A050808, A050809, A059303, A064118, A095935, A115019, A074496, A118840.

Table[NextPrime[E^n,-1],{n,30}] (* Harvey P. Dale, Jul 24 2016 *)

a(n)=precprime(exp(n)) \\ Charles R Greathouse IV, Mar 26 2013

Edited by N. J. A. Sloane, Dec 22 2006

a(27)-a(29) from Giovanni Resta, Apr 29 2017

A072952 Primes obtained as initial segments of the decimal expansion of the Euler-Mascheroni constant gamma = 0.5772... .

Original entry on oeis.org

5, 577, 5772156649015328606065120900824024310421

Offset: 1

Shyam Sunder Gupta, Aug 12 2002

The next term (a(4)) has 185 digits and is too large to include. - Harvey P. Dale, May 14 2013
Sequence A065815 gives the number of digits of a(n), resp. numbers k such that a(n) = floor(gamma*10^k). Sequences A005042, A007512, A115453, A119343, A210704, ... are the analog of the present sequence for Pi, e, sqrt(2), sqrt(3), 3^(1/3), ... - M. F. Hasler, Aug 31 2013
The original wording of the definition (and example) was "primes found in the decimal expansion..." which could as well refer to the sequence (5,7,7,215664901,5,3,2, ...) or (5,7,72156649, ...) or (5,7,7215664901, ...) (analogs to A047777 or A195834), or to the sequence (5,7,57, ...), analog to A198018. - M. F. Hasler, Sep 01 2013

			a(2) = 577, since 577 is the second prime obtained as initial segment of the decimal expansion of Euler-Mascheroni constant gamma = 0.577215664... .

Amiram Eldar, Table of n, a(n) for n = 1..4
Eric Weisstein's World of Mathematics, Constant Primes.
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant Digits.
Index entries related to "constant primes".

Cf. A001620, A047777, A065815, A195834, A198018.

Analogous sequences: A005042 (Pi), A007512 (e), A115453 (sqrt(2)), A119343 (sqrt(3)), A210704 (3^(1/3)).

nn=200;With[{emc=RealDigits[EulerGamma,10,nn][[1]]},Select[Table[ FromDigits[ Take[emc,n]],{n,nn}],PrimeQ]] (* Harvey P. Dale, May 14 2013 *)

default(realprecision, 777); /* use that many digits */
A072952={(c=Euler, v=1/*set to 0 for indices (i.e., A065815) instead of values*/)->for(k=0, precision(c), ispseudoprime(p=c\.1^k)&&print1([k, p][1+v]", "))} \\ M. F. Hasler, Aug 31 2013

A117879 First semiprime after e^n.

Original entry on oeis.org

4, 4, 9, 21, 55, 155, 407, 1099, 2981, 8105, 22033, 59881, 162757, 442417, 1202611, 3269021, 8886117, 24154953, 65659981, 178482301, 485165203, 1318815739, 3584912849, 9744803447, 26489122131, 72004899341, 195729609431

Offset: 0

Jonathan Vos Post, May 02 2006

Semiprime analog of A074496 = first prime after e^n. Lim_{n->infinity} a(n+1)/a(n) = e. There are numbers where floor(e^n) is itself a semiprime, as with floor(e^6) = 403 = 13 * 31, floor(e^15) = 3269017 = 773 * 4229, floor(e^20) = 485165195 = 5 * 97033039, floor(e^22) = 3584912846 = 2 * 1792456423, floor(e^24) = 26489122129 = 103 * 257175943.

Harvey P. Dale, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, e-Prime.

Cf. A000040, A000149, A001358, A007512, A014210, A050808, A050809, A059303, A064118, A095935, A115019, A074496, A118840.

fsa[n_]:=Module[{i=1,c=Floor[E^n]},While[PrimeOmega[c+i]!=2,i++];c+i]; Array[fsa,30,0] (* Harvey P. Dale, Oct 18 2013 *)

a(n) = Smallest semiprime > e^n. Smallest semiprime > floor(e^n). a(n) = min{s > A000149(n) and s in A001358}.

A229155 Number of digits of the n-th term of the decimal expansion of e = exp(1) cut into chunks of primes.

Original entry on oeis.org

1, 1, 649, 1, 1, 2, 29, 1, 1, 2, 1, 1, 1, 53, 1872, 3, 5

Offset: 1

M. F. Hasler, Sep 15 2013

Trying to cut the decimal expansion A001113 of e=2.718281828... into "prime chunks", one gets (2, 7, p, 5, 3, 11, q, 7, 3, 61, 3, 3, 2, r, ...) where p, q, r are 649-, 29-, 53-digit primes, respectively. The size of p makes it impossible to register this more fundamental sequence in the OEIS as it is done in A047777 for Pi. This led us to store just the length of the terms in this sequence.

Sequence A121267 is a (not exact) analog for Pi; note that A047777 requires all primes to be distinct, while we allow repetition of 7, 3, 2, ... as seen in the above example. If we did not, the terms following 29 would be 2, 2, 6, 3, 7, 8, 3, 441, 9, 17, ... instead of 1, 1, 2, 1, 1, 1, 53, ...

Joseph L. Pe, Trying to Write e as a Concatenation of Primes (2009)

Cf. A007512, A001113, A047658, A064118.

default(realprecision,2000);c=exp(1)/10;for(k=1,9e9,ispseudoprime(c\.1^k) & !print1(k,",") & k=0*c=frac(c*10^k))

a(15)-a(17) from Jinyuan Wang, Mar 26 2020

A117881 First semiprime after Pi^n.

Original entry on oeis.org

4, 4, 10, 33, 106, 309, 965, 3022, 9489, 29813, 93649, 294209, 924271, 2903678, 9122173, 28658147, 90032221, 282844574, 888582413, 2791563955, 8769956797, 27551631845, 86556004193, 271923706897, 854273519921, 2683779414319

Offset: 0

Jonathan Vos Post, May 02 2006

Pi and semiprime analog of A074496 First prime after e^n. Lim_{n->infinity} a(n+1)/a(n) = Pi. See also A000796 Decimal expansion of Pi. There are numbers where floor(Pi^n) is itself a semiprime, as with floor(Pi^2) = 9, floor(Pi^6) = 961 = 31^2, floor(Pi^9) = 29809 = 13 * 2293, floor(Pi^25) = 2683779414317 = 5749 * 466825433.

			a(3) = 33 because Pi^3 = 31.0062766... floor(Pi^3) = 31 is prime hence 31 + 2 = 33 is a term.

Eric Weisstein's World of Mathematics, e-Prime.

Cf. A000040, A000149, A000796, A001358, A007512, A014210, A050808, A050809, A059303, A064118, A095935, A115019, A074496, A118840.

fsp[n_]:=Module[{k=Ceiling[Pi^n]},While[PrimeOmega[k]!=2,k++];k]; Array[fsp,30,0]

a(n) = min{s in A001358 and s > Pi^n}.

A229178 Primes seen in decimal expansion of e, contiguous, smallest and distinct, with terms larger than 10^100 replaced by 0.

Original entry on oeis.org

2, 7, 0, 5, 3, 11, 80232878250981945581530175671, 73, 61, 332069, 811, 2509961, 81881593, 41, 0, 30592123, 66771943252786753, 0, 0, 6343, 490769, 64237, 2229435236612557250881, 47, 79, 223, 151, 97, 477806056967, 253801, 71, 0, 0, 9467, 83, 48197

Offset: 1

M. F. Hasler, Sep 15 2013

Sequence A047777 is the analog for Pi.

Without the admittedly arbitrary clause "terms larger than ... replaced by zero", the sequence could not be stored beyond its second term, since the 3rd term would have 649 digits (and the 15th term would have 441 digits). Although this restriction is arbitrary, several other (maybe more natural) alternatives (for example, larger than the concatenation of the preceding/following 10 terms...) would yield the same initial terms.

Cf. A001113, A007512, A047777, A064118, A229155.

default(realprecision,5000);c=exp(1)/10;u=[];for(k=0,9e9,ispseudoprime(c\.1^k) & !setsearch(u,c\.1^k) & (u=setunion(u,Set(c\.1^k))) & !print1(c\.1^k,",") & k=0*c=frac(c*10^k))

More terms from Jinyuan Wang, Mar 26 2020

A242835 Primes formed by the initial digits of the decimal expansion of the square root of 5.

Original entry on oeis.org

2, 223, 22360679774997896964091

Offset: 1

Felix Fröhlich, May 23 2014

There is no other term with less than 111 digits.

a(4) has 1506 digits. - Hiroaki Yamanouchi, Sep 11 2014

			Decimal expansion of sqrt(5) begins: 2.23606797749978969640917366... - _Wesley Ivan Hurt_, May 26 2014

Cf. A005042, A007512, A072952, A119343.

Cf. A002163 (Decimal Expansion of sqrt(5)).

Module[{d=25,f},f=RealDigits[Sqrt[5],10,d][[1]];Select[ Table[ FromDigits[ Take[f,n]],{n,d}],PrimeQ]] (* Harvey P. Dale, Oct 29 2020 *)

cp's OEIS Frontend

A064118 Numbers k such that the first k digits of e form a prime.

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A095935 First prime of length n encountered in the decimal representation of e.

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A210704 Primes formed by initial digits of 3^(1/3) = A002581, i.e., of the form floor[3^(1/3)*10^k].

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A040016 Largest prime < e^n.

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A072952 Primes obtained as initial segments of the decimal expansion of the Euler-Mascheroni constant gamma = 0.5772... .

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A117879 First semiprime after e^n.

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A229155 Number of digits of the n-th term of the decimal expansion of e = exp(1) cut into chunks of primes.

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A117881 First semiprime after Pi^n.

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