A050808 -id:A050808 - cp's OEIS frontend

A050809 Primes of the form floor( exp(k) ).

2, 7, 65659969, 5184705528587072464087, 14302079958348104463583671072905261080748384225250684971, 17199742630376622641833783925547830057256484050709158699244513

Offset: 1

Patrick De Geest, Oct 15 1999

			a(3) = floor(e^18) = 65659969, which is prime.

Amiram Eldar, Table of n, a(n) for n = 1..9
Eric Weisstein's World of Mathematics, e-Prime.

Cf. A050808 (values of k), A000149, A040016, A037028, A000227, A004791.

Select[Table[Floor[Exp[n]], {n, 150}], PrimeQ] (* Jayanta Basu, Jun 01 2013 *)

Corrected by Naohiro Nomoto, Feb 22 2001

A059303 Numbers k such that floor(e^k) + 1 is prime.

Original entry on oeis.org

0, 1, 5, 7, 10, 105, 22959, 34888

Offset: 1

Felice Russo, Jan 25 2001

0 followed by all k such that ceiling(e^k) is prime. - Jeppe Stig Nielsen, Feb 12 2024

Eric Weisstein's World of Mathematics, e-Prime

Cf. A001671, A050808.

Select[Range[40000], PrimeQ[Floor[E^#] + 1] &] (* G. C. Greubel, Jan 06 2017 *)

isok(k) = isprime(floor(exp(k))+1) \\ Michel Marcus, Jun 08 2013

Corrected by Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

a(7)-a(8) from Donovan Johnson

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

A113887 Numbers n such that floor(exp(sqrt(n))) is a prime number.

Original entry on oeis.org

1, 3, 4, 6, 10, 12, 17, 21, 26, 30, 53, 54, 58, 83, 95, 109, 111, 128, 131, 137, 145, 157, 165, 166, 181, 195, 202, 244, 261, 265, 290, 306, 324, 343, 353, 369, 386, 415, 417, 418, 438, 468, 473, 503, 633, 704, 735, 758, 859, 903, 919, 955, 979, 987, 1008, 1016

Offset: 1

Stefan Steinerberger, Jan 28 2006

nonn

Cf. A050808.

Select[ Range@1044, PrimeQ@ Floor@ Exp@ Sqrt@ # &] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Jan 30 2006

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}.

A074221 Numbers k such that floor(k^e) is prime.

Original entry on oeis.org

3, 4, 5, 11, 22, 26, 32, 42, 44, 45, 84, 95, 97, 165, 183, 198, 200, 201, 204, 214, 235, 247, 258, 260, 267, 294, 310, 365, 381, 387, 389, 393, 442, 456, 463, 507, 543, 544, 551, 588, 609, 628, 676, 693, 718, 729, 736, 755, 767, 795, 810, 842, 865, 877, 885

Offset: 0

Jon Perry, Sep 17 2002

nonn

			Floor(3^e) = 19.

Cf. A001113, A050808.

Select[Range[1000],PrimeQ[Floor[#^E]]&] (* Harvey P. Dale, Jul 21 2013 *)

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}.

A370305 Numbers k such that the distance from exp(k) to the closest average of two consecutive primes is less than 1.

Original entry on oeis.org

1, 3, 16, 61, 74, 91, 113, 1441, 1566, 2170, 2499

Offset: 1

Jeppe Stig Nielsen, Feb 14 2024

Explicitly, abs( e^k - (prevprime(e^k)+nextprime(e^k))/2 ) < 1.
For k>1, the formula (prevprime(e^k)+nextprime(e^k))/2 either gives floor(e^k), for k = 61, 74, 2170, ..., or gives ceiling(e^k), for k = 3, 16, 91, 113, 1441, 1566, 2499, ... This partitions {a(n)}\{1} into two subsequences each of which can be conjectured to have relative density 1/2.
In cases k = 16, 61, 113, 2499, ... the distance is actually less than 0.5. Then the formula (prevprime(e^k)+nextprime(e^k))/2 yields round(e^k), the nearest integer to e^k.

			For k=16, e^16 is about 8886110.52. The next prime is 8886113, and the previous prime is 8886109, and their average 8886111 is at a distance of about 0.48 away from e^16.

Cf. A037028, A040016, A050808, A059303, A074496.

default(realprecision,2000);for(k=1,+oo,r=exp(k);abs(r-(precprime(r)+nextprime(r))/2)<1&&print1(k,", "))

cp's OEIS Frontend

A050809 Primes of the form floor( exp(k) ).

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A059303 Numbers k such that floor(e^k) + 1 is prime.

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A040016 Largest prime < e^n.

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A113887 Numbers n such that floor(exp(sqrt(n))) is a prime number.

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A117879 First semiprime after e^n.

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A074221 Numbers k such that floor(k^e) is prime.

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A117881 First semiprime after Pi^n.

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A370305 Numbers k such that the distance from exp(k) to the closest average of two consecutive primes is less than 1.

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PARI