A074496 -id:A074496 - cp's OEIS frontend

A040016 Largest prime < e^n.

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

A074497 a(n) = the smallest prime > Pi^n.

Original entry on oeis.org

2, 5, 11, 37, 101, 307, 967, 3023, 9491, 29819, 93683, 294211, 924281, 2903689, 9122207, 28658153, 90032231, 282844571, 888582407, 2791563953, 8769956803, 27551631847, 86556004201, 271923706939, 854273519929, 2683779414353, 8431341691901, 26487841119121

Offset: 0

Joseph L. Pe, Sep 26 2002

The corresponding indices in A000040 are the sequence {1, 3, 5, 12, 26, 63, 163, 434, 1176, 3230, 9043, 25550, 73051, ...}. - L. Edson Jeffery, Jan 17 2014

			The first prime > Pi^3 = 31.006.... is 37, so a(3) = 37.

Cf. A000040, A014210, A014211, A074496.

a[n_] := NextPrime[Pi^n];  a /@ Range[0,10] (* Giovanni Resta, Jan 17 2014 *)

with \p1000: for(n=0,30,print1(nextprime(Pi^n)","))

More terms from Ralf Stephan, Mar 19 2003

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}.

A117836 Semiprimes of the form floor(e^k).

Original entry on oeis.org

403, 3269017, 485165195, 3584912846, 26489122129, 29048849665247, 639843493530054949, 114200738981568428366295718, 66631762164108958342448140502408732626873, 492345828601205839975486205911330449483779

Offset: 1

Jonathan Vos Post, Apr 30 2006

nonn

			a(1) = 403 = floor(e^6) = 13 * 31.
a(2) = 3269017 = floor(e^15) = 773 * 4229.
a(3) = 485165195 = floor(e^20) = 5 * 97033039.
a(4) = 3584912846 = floor(e^22) = 2 * 1792456423.
a(5) = 26489122129 = floor(e^24) = 103 * 257175943.

Giovanni Resta, Table of n, a(n) for n = 1..17

Cf. A000040, A000149, A001358, A074496.

Select[Table[Floor[Exp[k]],{k,100}],PrimeOmega[#]==2&] (* James C. McMahon, Sep 14 2024 *)

A000149 Intersection A001358.

a(6)-a(10) from Giovanni Resta, Jun 15 2016

A117839 Primes of the form floor(Pi^k + e^k).

Original entry on oeis.org

2, 5, 17, 9255121991, 28870447577

Offset: 1

Jonathan Vos Post, Apr 30 2006

Intersection of A000040 and A061675.

The next term has 1535 digits. - Harvey P. Dale, Apr 26 2011

See also A059792 (Numbers k such that floor(Pi^k) is prime) and their corresponding primes A077547.
See also A059303 (Numbers k such that floor(e^k) + 1 is prime) and their corresponding primes A118840.
Cf. A000040, A059303, A059792, A061675, A077547, A074496.

Select[Table[Floor[\[Pi]^n+E^n],{n,0,5000}],PrimeQ]  (* Harvey P. Dale, Apr 26 2011 *)

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

A040016 Largest prime < e^n.

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A074497 a(n) = the smallest prime > Pi^n.

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A117879 First semiprime after e^n.

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A117836 Semiprimes of the form floor(e^k).

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A117839 Primes of the form floor(Pi^k + e^k).

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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