A000149 -id:A000149 - cp's OEIS frontend

A117879 First semiprime after e^n.

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}.

A254528 Number of decimal digits in the integer part of e^n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 14, 15, 15, 16, 16, 17, 17, 17, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 24, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29, 29, 30, 30, 30, 31, 31, 32, 32, 33, 33

Offset: 0

Robert G. Wilson v, Feb 01 2015

			e^10 = 22026.46579480671..., so a(10) = 5.

Benedict W. J. Irwin, Table of n, a(n) for n = 0..10000

Cf. A000149, A055642.

Cf. A001113, A072334, A091933, A092426, A092511, A092512, A092513 (see their offsets).

f[n_] := 1 + Floor@ Log10@ Exp@ n; Array[f, 75, 0]
Table[Sum[DigitCount[Floor[Exp[1]^k]][[n]], {n, 1, 10}], {k, 0, 150}] (* Benedict W. J. Irwin, Apr 13 2016 *)
IntegerLength[Floor[E^Range[0,80]]] (* Harvey P. Dale, Aug 28 2017 *)

a(n) = localprec(n+1); #Str(floor(exp(n))); \\ Michel Marcus, Dec 05 2020

a(n) = A055642(A000149(n)). - Amiram Eldar, May 25 2024

A045650 Numbers that cannot be expressed as k + floor(log(k)) where k is an integer.

Original entry on oeis.org

3, 9, 23, 58, 153, 409, 1103, 2988, 8112, 22036, 59885, 162766, 442426, 1202618, 3269032, 8886126, 24154969, 65659987, 178482319, 485165215, 1318815755, 3584912868, 9744803469, 26489122153, 72004899362, 195729609454

Offset: 1

David W. Wilson

Complement of A212445. - Michel Marcus, Jun 30 2015

Cf. A212445, A000149.

a(n)=floor(exp(n))+n \\ Charles R Greathouse IV, Jul 02 2015

Conjecture: lim n->infinity a(n) = e^n. - Ron R. King, Nov 10 2006

a(n) = floor(e^n) + n. - Nurdin N. Takenov (greanvert(AT)gmail.com), Mar 10 2007

a(18) onward corrected by Sean A. Irvine, Mar 17 2021

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

A119380 Remainder when the integer part of e^n is divided by the n-th prime number.

Original entry on oeis.org

0, 1, 0, 5, 5, 0, 8, 16, 7, 15, 13, 28, 23, 23, 26, 24, 57, 57, 62, 43, 70, 49, 36, 64, 84, 3, 4, 64, 83, 103, 45, 53, 49, 37, 26, 19, 75, 20, 147, 20, 134, 73, 56, 17, 31, 89, 143, 200, 103, 170, 25, 37, 159, 181, 90, 242, 16, 93, 222, 163, 57, 132, 214, 71, 164, 57, 62, 14

Offset: 1

Axel Harvey, Jul 24 2006

			The sixth term is 0 because e^6 is 403.42879... and 403 is a multiple of 13, the sixth prime.

Iain Fox, Table of n, a(n) for n = 1..10000

Cf. A000149.

A119380:=n->floor(exp(1)^n) mod ithprime(n): seq(A119380(n), n=1..100); # Wesley Ivan Hurt, Nov 30 2017

Table[Mod[Floor[E^n], Prime[n]], {n, 1, 100}] (* Stefan Steinerberger, Jul 26 2006 *)

a(n) = floor(exp(n))%prime(n) \\ Iain Fox, Nov 30 2017

a(n) = floor(e^n) mod prime(n).

More terms from Stefan Steinerberger, Jul 26 2006

A347417 Decimal expansion of Product_{k>=1} (1 + exp(-k)).

Original entry on oeis.org

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

Offset: 1

Christoph B. Kassir, Aug 31 2021

			1.67792868498935419743907236983684684682083...

Cf. A001113, A000149.

evalf(limit(product(1+1/exp(k), k=1..t), t=infinity), 120);

Mathematica
```
Product[1+Exp[-k], {k, 1, Infinity}]
```
PARI
```
prodinf(k=1,1+exp(-k))
```

A096181 Floor (e^(n / log(n))).

Original entry on oeis.org

17, 15, 17, 22, 28, 36, 46, 60, 76, 98, 125, 158, 201, 254, 320, 403, 506, 634, 793, 989, 1233, 1533, 1904, 2360, 2922, 3612, 4459, 5498, 6771, 8328, 10231, 12556, 15393, 18851, 23063, 28189, 34423, 41998, 51195, 62353, 75883, 92274, 112119, 136131

Offset: 2

Robert G. Wilson v, Jun 19 2004

nonn

Cf. A000149, A073226.

Table[ Floor[ E^(n/Log[n])], {n, 2, 45}]

Definition corrected and cross-refs added by Franklin T. Adams-Watters, Jan 25 2010

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}.

A120728 a(n) = floor(e^n) modulo 3.

Original entry on oeis.org

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

Offset: 1

Roger L. Bagula, Aug 19 2006

This sequence is not unique to e; there are infinitely many numbers x such that mod(floor(x^n), 3) will produce the same sequence. - Franklin T. Adams-Watters, Sep 29 2011

Cf. A000149.

Table[Mod[Floor[Exp[n]], 3], {n, 1, 50}]
Table[Floor[Exp[n]] - 3*Floor[Exp[n]/3], {n, 1, 50}]

a(n) = A000149(n) modulo 3. - Jason Yuen, May 25 2025

A140471 Floored n-th power of Viswanath's constant.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 5, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 22, 25, 28, 32, 36, 41, 46, 52, 59, 67, 76, 86, 98, 111, 125, 142, 161, 182, 206, 233, 264, 299, 339, 384, 434, 492, 557, 630, 713, 808, 914, 1035, 1172, 1326, 1502, 1700, 1924

Offset: 0

Alonso del Arte, Jun 28 2008

nonn

For sufficiently large terms of a random Fibonacci sequence, the powers of Viswanath's constant approximate the absolute value of the terms in such a sequence (with a few notable exceptions).

			a(7) = 2 because V^7 is approximately 2.381734947432 and floored that is 2.

Eric Weisstein, Random Fibonacci sequence at MathWorld

Cf. A014217, floored n-th power of the golden ratio; A000149, floored n-th power of e; A001672, floored n-th power of Pi.

V = 1.1319882487943; Table[Floor[V^n], {n, 0, 49}]

a(n) = floor(v^n), where v = 1.1319882487943 as given by A078416.

More terms from Alois P. Heinz, Mar 08 2020

cp's OEIS Frontend

A117879 First semiprime after e^n.

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A254528 Number of decimal digits in the integer part of e^n.

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A045650 Numbers that cannot be expressed as k + floor(log(k)) where k is an integer.

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

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A119380 Remainder when the integer part of e^n is divided by the n-th prime number.

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A347417 Decimal expansion of Product_{k>=1} (1 + exp(-k)).

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A096181 Floor (e^(n / log(n))).

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A117881 First semiprime after Pi^n.

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