A064118 -id:A064118 - cp's OEIS frontend

A060421 Numbers k such that the first k digits of the decimal expansion of Pi form a prime.

1, 2, 6, 38, 16208, 47577, 78073, 613373

Offset: 1

Michel ten Voorde, Apr 05 2001

The Brown link states that in 2001 Ed T. Prothro reported discovering that 16208 gives a probable prime and that Prothro verified that all values for 500 through 16207 digits of Pi are composites. - Rick L. Shepherd, Sep 10 2002

The corresponding primes are in A005042. - Alexander R. Povolotsky, Dec 17 2007

			3 is prime, so a(1) = 1; 31 is prime, so a(2) = 2; 314159 is prime, so a(3) = 6; ...

K. S. Brown, Primes in the Decimal Expansion of Pi [Broken link?]
K. S. Brown, Primes in the Decimal Expansion of Pi [Cached copy]
Prime Curios, 314159
Prime Curios, 31415...36307 (16208-digits)
Shyam Sunder Gupta, Mystery of pi, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 19, 473-497.
Eric Weisstein's World of Mathematics, Constant Primes
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Eric Weisstein's World of Mathematics, Pi Digits
Eric Weisstein's World of Mathematics, Pi-Prime

Cf. A000796 (Pi), A005042, A007523, A047658.
Primes in other constants: A064118 (e), A065815 (gamma), A064119 (phi), A118328 (Catalan's constant), A115377 (sqrt(2)), A119344 (sqrt(3)), A228226 (log 2), A228240 (log 10), A119334 (zeta(3)), A122422 (Soldner's constant), A118420 (Glaisher-Kinkelin constant), A174974 (Golomb-Dickman constant), A118327 (Khinchin's constant).
In other bases: A065987 (binary), A065989 (ternary), A065991 (quaternary), A065990 (quinary), A065993 (senary).

Do[If[PrimeQ[FromDigits[RealDigits[N[Pi, n + 10], 10, n][[1]]]], Print[n]], {n, 1, 9016} ]

a(6) = 47577 from Eric W. Weisstein, Apr 01 2006
a(7) = 78073 from Eric W. Weisstein, Jul 13 2006
a(8) = 613373 from Adrian Bondrescu, May 29 2016

A007512 Primes of the form floor(e*10^k), i.e., formed by concatenation of an initial segment of the decimal expansion of e.

Original entry on oeis.org

2, 271, 2718281, 2718281828459045235360287471352662497757247093699959574966967627724076630353547594571

Offset: 1

N. J. A. Sloane, Robert G. Wilson v, Simon Plouffe

The number of digits in a(n) is given in A064118. This allows us to get larger terms that cannot be displayed here, via the given FORMULA. Sequences A005042, A072952, A115453, A119343, A210704, ... are the analogs for Pi, gamma, sqrt(2), sqrt(3), 3^(1/3), ... - M. F. Hasler, Aug 31 2013

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Eric Weisstein's World of Mathematics, e-Prime
Index entries related to "constant primes".

Digits := 110; n0 := evalf(E); for i from 1 to 100 do t1 := trunc(10^i*n0); if isprime(t1) then print(t1); fi; od:

c=exp(1);for(k=0,precision(c),ispseudoprime(c\.1^k) & print1(c\.1^k,",")) \\ M. F. Hasler, Sep 01 2013

a(n) = floor(e*10^(A064118(n)-1)). - M. F. Hasler, Aug 31 2013

Next term is a 1781-digit BPSW-probable prime 2718281828459045235...211151368350627526023. - Randall L Rathbun, Feb 02 2002
Edited by T. D. Noe, Oct 30 2008
Edited by M. F. Hasler, Aug 31 2013

A047658 Numbers k such that the initial k digits in decimal portion of Pi form a prime number.

Original entry on oeis.org

5, 12, 281, 547, 6205, 16350

Offset: 1

Carlos Rivera

Conjecture: this sequence is finite. - Carlos Rivera
Rivera's conjecture that this sequence is finite conflicts with heuristics; the next entry is almost certainly 6205, since floor((Pi-3)*10^6205) is (very) probably prime, though its proof may take decades. - David Broadhurst, Nov 08 2000
Floor((Pi-3)*10^6205) is a strong pseudoprime to all (1229) prime bases a < 10000 (the test took 45 minutes). - Joerg Arndt, Jan 16 2011
Terms for n>=5 are only probable primes. - Dmitry Kamenetsky, Aug 03 2015
Floor((Pi-3)*10^16350) is a probable prime, checked with 25 iterations of the Miller-Rabin test. - Dmitry Kamenetsky, Aug 05 2015
The next term is greater than 65400. - Dmitry Kamenetsky, Aug 09 2015
The next term is greater than 100000. - Michael S. Branicky, Sep 29 2024

			5 gives 14159 (prime); 12 gives 141592653589 (prime) and so on.

Chris K. Caldwell, Prime Curios: 14159...07021 (547-digits)
Shyam Sunder Gupta, Mystery of pi, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 19, 473-497.

Cf. A000796 (Pi), A060421, A064118.

nn=1000; d=RealDigits[Pi-3, 10, nn][[1]]; Select[Range[nn], PrimeQ[FromDigits[Take[d, #]]] &]

is(n)=isprime((Pi-3)*10^n\1) \\ Charles R Greathouse IV, Aug 28 2015

from sympy import S, isprime
pi_digits = str(S.Pi.n(10**5))[2:-1]
def afind():
    kint = 0
    for k in range(len(pi_digits)):
        kint = 10*kint + int(pi_digits[k])
        if isprime(kint):
            print(k+1, end=", ")
afind() # Michael S. Branicky, Jan 29 2023

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

A109892 a(n) = least integer of the form (n!+1)(n!+2)...(n!+k)/n!.

Original entry on oeis.org

2, 6, 84, 20475, 234531275, 199200973555045, 16481425431663122588749, 173392935733620216899469862542865, 300717095810709134168380432250652303057474577

Offset: 1

Amarnath Murthy, Jul 13 2005

nonn

Equivalently, binomial(n!+n,n). Proof: (n!+1)(n!+2)...(n!+k) == k! mod n! == 0 mod n! if and only if k >= n (for n >= 2). - Paul D Hanna and Robert Israel, Aug 31 2010.
Note that k <= n. Subsidiary sequence to be investigated: n such that k < n.
This is just a coincidence, but k=2,6,84 are also such that floor(exp(1)*10^k) is a prime, cf. A064118. - M. F. Hasler, Aug 31 2013

			a(4)=25*26*27*28/24=20475.

Cf. A105291.

A109892 := proc(n) local k,fn; k := 1; fn := n! ; while mul(fn+i,i=1..k) mod fn <> 0 do k := k+1; od ; RETURN(mul(fn+i,i=1..k)/fn) ; end: seq(A109892(n),n=1..10) ; # R. J. Mathar, Aug 15 2007

Table[(n+n!)!/(n!*(n!)!), {n, 1, 9}] (* Jean-François Alcover, Mar 04 2014, after first comment *)

Corrected and extended by R. J. Mathar, Aug 15 2007

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}.

A210706 Numbers k such that floor[ 3^(1/3)*10^k ] is prime.

Original entry on oeis.org

23, 80, 2487

Offset: 1

M. F. Hasler, Aug 31 2013

Inspired by prime curios about 4957 (cf. LINKS), one of which honors the late Bruce Murray (Nov 30 1931 - Aug 29 2013).
Meant to be a "condensed" version of A210704, see there for more.
Alternate definition: Numbers k such that concatenation of the first (k+1) digits of A002581 yields a prime.

			t = 3^(1/3) = 1.44224957030740838232163831... multiplied by 10^23 yields
t*10^23 = 144224957030740838232163.831..., the integer part of which is the prime A210704(1), therefore a(1)=23.

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

Cf. A002581 = decimal expansion of 3^(1/3).

Cf. A065815 (analog for gamma), A060421 (1+ analog for Pi), A064118 (1+ analog for exp(1)), A119344 (1 + analog for sqrt(3)), A136583 (1+ analog for sqrt(10)).

\p2999
t=sqrtn(3,3);for(i=1,2999,ispseudoprime(t\.1^i)&print1(i","))

a(n) = (length of A210704(n)) - 1, where "length" means number of decimal digits.

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

cp's OEIS Frontend

A060421 Numbers k such that the first k digits of the decimal expansion of Pi form a prime.

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A007512 Primes of the form floor(e*10^k), i.e., formed by concatenation of an initial segment of the decimal expansion of e.

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A047658 Numbers k such that the initial k digits in decimal portion of Pi form a prime number.

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A040016 Largest prime < e^n.

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A109892 a(n) = least integer of the form (n!+1)(n!+2)...(n!+k)/n!.

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A117879 First semiprime after e^n.

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A210706 Numbers k such that floor[ 3^(1/3)*10^k ] is prime.

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