A006988 -id:A006988 - cp's OEIS frontend

A055739 [e^n]-th prime.

2, 3, 17, 71, 251, 857, 2767, 8803, 27211, 82939, 249779, 744949, 2201273, 6463081, 18858529, 54764947, 158330573, 456016933, 1309050653, 3746543923, 10694444393, 30453898201, 86534078387, 245401348403, 694683409429, 1963275871663, 5540095680547, 15611517864749

Offset: 0

Robert G. Wilson v, Jun 09 2000

Amiram Eldar, Table of n, a(n) for n = 0..39

Cf. A000149, A033844, A006988.

Table[Prime[Floor[N[E^n]]],{n,0,25}] (* Typographical error corrected by Harvey P. Dale, Dec 27 2019 *)

a(n) = prime(A000149(n)). - Amiram Eldar, Jul 22 2025

a(25)-a(27) from Amiram Eldar, Jul 22 2025

A065857 The (10^n)-th composite number.

Original entry on oeis.org

4, 18, 133, 1197, 11374, 110487, 1084605, 10708555, 106091745, 1053422339, 10475688327, 104287176419, 1039019056246, 10358018863853, 103307491450820, 1030734020030318, 10287026204717358, 102692313540015924, 1025351434864118026, 10239531292310798956, 102270102190290407386

Offset: 0

Labos Elemer, Nov 26 2001

			The 100th composite number is C(100)=133, while the 100th prime is 541. In general: A000720(m) < A062298(m) < m < A002808(m) < A000040(m), for example pi(100)=25 < 75 < 100 < C(100)=133 < prime(100)=541.

A. E. Bojarincev, Asymptotic expressions for the n-th composite number. Univ. Mat. Zap. 6:21-43(1967). [in Russian]
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 133, p. 45, Ellipses, Paris 2008.

Chai Wah Wu, Table of n, a(n) for n = 0..22
Laurentiu Panaitopol, Some properties of the series of composed [sic] numbers, Journal of Inequalities in Pure and Applied Mathematics 2:3 (2001).
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), pp. 64-94.

Cf. A033844, A002808, A062298, A000720, A006988, A065855, A065856.

Composite[n_Integer] := Block[ {k = n + PrimePi[n] + 1 }, While[ k != n + PrimePi[k] + 1, k = n + PrimePi[k] + 1]; Return[k]];
Table[Composite[10^n], {n, 0, 9}]

a(n)=my(k=10^n);forcomposite(n=4,2*k+2,if(k--==0,return(n))) \\ Charles R Greathouse IV, May 30 2013

a(n) = A002808(A011557(n)).

a(n) = 10^(n + n/log n + 2n/log^2 + 4n/log^3 n + O(n/log^4 n)). See Bojarincev for an asymptotic expansion. - Charles R Greathouse IV, May 30 2013

More terms from Robert G. Wilson v, Nov 26 2001
a(14) from Lekraj Beedassy, Jul 14 2008
a(15)-a(19) from Chai Wah Wu, Apr 16 2018
a(20) from Chai Wah Wu, Aug 23 2018

A300397 a(n) = number of primes that end in 1 among the first 10^n primes.

Original entry on oeis.org

1, 24, 245, 2484, 24967, 249934, 2499755, 24999435, 249997579, 2499995131, 24999983109, 249999976990

Offset: 1

G. L. Honaker, Jr., Mar 05 2018

			a(1) = 1 because the first 10 primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, among which there is only one prime ending in 1.

Cf. A006988, A030430, A185705, A300398, A300399, A300400.

a(n) = #select(x->((x % 10) == 1), primes(10^n)); \\ Michel Marcus, Mar 07 2018

a(n) = A185705(A006988(n)). - Michel Marcus, Mar 20 2018

a(3)-a(8) from Chuck Gaydos
a(9) from Jon E. Schoenfield, Mar 20 2018
a(10)-a(12) from Giovanni Resta, Mar 27 2018

A300398 a(n) = number of primes that end in 3 among the first 10^n primes.

Original entry on oeis.org

3, 26, 253, 2515, 25007, 250110, 2500209, 25000135, 250002220, 2500000243, 25000027724, 250000009260

Offset: 1

G. L. Honaker, Jr., Mar 05 2018

			a(1) = 3 because the first 10 primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, among which there are three primes ending in 3.

Caldwell and Honaker, Prime Curios!

Cf. A006988, A030431, A185712, A300397, A300399, A300400.

a(n) = #select(x->((x % 10) == 3), primes(10^n)); \\ Michel Marcus, Mar 07 2018

a(n) = A185712(A006988(n)). - Michel Marcus, Mar 20 2018

a(3)-a(8) from Chuck Gaydos
a(9) from Jon E. Schoenfield, Mar 20 2018
a(10)-a(12) from Giovanni Resta, Mar 27 2018

A300399 a(n) = number of primes that end in 7 among the first 10^n primes.

Original entry on oeis.org

2, 24, 254, 2508, 25015, 250014, 2500283, 25000402, 250002505, 2500005319, 25000014226, 250000051446

Offset: 1

G. L. Honaker, Jr., Mar 05 2018

			a(1) = 2 because the first 10 primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, among which there are two primes ending in 7.

Cf. A006988, A030432, A185714, A300397, A300398, A300400.

a(n) = #select(x->((x % 10) == 7), primes(10^n)); \\ Michel Marcus, Mar 07 2018

a(n) = A185714(A006988(n)). - Michel Marcus, Mar 20 2018

a(3)-a(8) from Chuck Gaydos
a(9) from Jon E. Schoenfield, Mar 20 2018
a(10)-a(12) from Giovanni Resta, Mar 27 2018

A300400 a(n) = number of primes that end in 9 among the first 10^n primes.

Original entry on oeis.org

2, 24, 246, 2491, 25009, 249940, 2499751, 25000026, 249997694, 2499999305, 24999974939, 249999962302

Offset: 1

G. L. Honaker, Jr., Mar 05 2018

			a(1) = 2 because the first 10 primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, among which there are two primes ending in 9.

Caldwell and Honaker, Prime Curios!

Cf. A006988, A030433, A185715, A300397, A300398, A300399.

a(n) = #select(x->((x % 10) == 9), primes(10^n)); \\ Michel Marcus, Mar 07 2018

a(n) = A185715(A006988(n)). - Michel Marcus, Mar 20 2018

a(3)-a(8) from Chuck Gaydos
a(9) from Jon E. Schoenfield, Mar 20 2018
a(10)-a(12) from Giovanni Resta, Mar 27 2018

A335331 a(n) = prime(k) where k is the n-th 7-smooth number.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 43, 47, 53, 61, 71, 73, 89, 97, 103, 107, 113, 131, 149, 151, 173, 181, 197, 223, 227, 229, 251, 263, 281, 307, 311, 349, 359, 379, 409, 419, 433, 463, 503, 521, 541, 571, 593, 613, 659, 691, 701, 719, 761, 809, 827, 853, 863

Offset: 1

David A. Corneth, Jun 01 2020

nonn

At A110069 we look for numbers of the form n = (d_1 + d_2 + ... + d_k)*prime(d_1*d_2*...*d_k) where d_1 d_2 ... d_k is the decimal expansion of n. As the largest prime that can be among the digits of a base-10 number is 7, the product of digits is 7-smooth. Hence the factor prime(d_1*d_2*...*d_k) is a term from this sequence. As lots of numbers have a product of digits of, say, 210^4, it would help to know prime(210^4) in advance. That's a(5817) of this sequence as 210^4 is the 5817th 7-smooth number. Precomputing such numbers is a computational benefit.

David A. Corneth, Table of n, a(n) for n = 1..8862

Cf. A002473, A006988, A033844, A110069.

A055680 (5^n)-th prime.

Original entry on oeis.org

2, 11, 97, 691, 4637, 28687, 171529, 994837, 5653807, 31643347, 174978379, 958200857, 5205329029, 28089376111, 150722620067, 804829625447, 4279524344539, 22671428650501, 119712643014673, 630280777956749, 3309723954037103, 17339016680833993, 90641632321932691, 472916224088644129, 2463008836130170681

Offset: 0

Robert G. Wilson v, Jun 09 2000

Cf. A033844, A006988.

Mathematica
```
Table[Prime[5^n],{n,0,16}]
```

a(n)=prime(5^n) \\ Charles R Greathouse IV, Nov 02 2014

a(17)-a(24) from Charles R Greathouse IV, Nov 02 2014

A065139 Numbers n such that the sum of prime(n) and pi(n) is divisible by n.

Original entry on oeis.org

1, 2, 7, 9, 23, 57, 149, 368, 921, 5863, 14531, 36087, 36255, 36257, 233084, 1505984, 1151321194, 1151321361, 7826138122, 967823489175, 967823489458, 967823489477, 967823489490, 967823489491, 2545928465925, 123116092093107, 123116092093185, 123116092094024

Offset: 1

Labos Elemer, Oct 16 2001

nonn

			p(233084) = 3242497, Pi(233084) = 20679; sum = 3263176 = 14*233084; order of quotient is log(n).
(prime(1505984) + pi(1505984))/1505984 = (23981141 + 114603)/1505984 = 16.

Giovanni Resta, Table of n, a(n) for n = 1..33

Cf. A000040, A000720, A006988.

Do[ If[ IntegerQ[ (Prime[n] + PrimePi[n]) /n ], Print[n]], {n, 1, 5*10^7} ]
Select[ Range[10^8], IntegerQ[(Prime[ # ] + PrimePi[ # ])/ # ] & ]

k=0;n=0;forprime(p=2,4e9,if(isprime(n++),k++);if((k+p)%n==0,print1(n", "))) \\ Charles R Greathouse IV, Aug 21 2011

Solutions to pi(x) + prime(x) = A000720(x) + A000040(x) = 0 (mod x).

a(17)-a(19) from Donovan Johnson, Aug 21 2011

a(20)-a(28) from Giovanni Resta, Oct 15 2019

A099260 Number of decimal digits in (10^n)-th prime number.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 0

Rick L. Shepherd, Oct 10 2004

As lim {n->oo} p_n/(n log n) = 1 is equivalent to the prime number theorem, a good first approximation (without having done any detailed analysis) should be a(n)=floor(log_10((10^n)*log(10^n))), which correctly generates all the first 22 terms and predicts that the sequence will continue 24,25,...,43,44,46,47,...,435,436,438,439,...,4344,4345,4347,4348,...,4503,4504 through the first 4500 terms (with only 5,45,437,4346 not appearing - compare with the digits of log_10(e) in A002285).

Many terms of this sequence can be determined exactly using Dusart's bounds. The first missing terms are 5, 44, 435, 4344, 43430, 434295, 4342946, 43429449, 434294483, 4342944820, ....

			a(4) = 6 because A006988(4) = prime(10^4) = 104729 has six decimal digits.

Pierre Dusart, Estimates of Some Functions Over Primes without R.H.

Cf. A006988 ((10^n)-th prime), A006880 (pi(10^n)), A099261 (bit lengths).

Table[IntegerLength[Prime[10^n]],{n,0,75}] (* Harvey P. Dale, Dec 11 2020 *)

a(n)=if(n<3,return(n+1));my(l=n*log(10),ll=log(l),lb=ceil(log(l+ll-1+(ll-2.2)/l)/log(10)),ub=ceil(log(l+ll-1+(ll-2)/l)/log(10)));if(lb==ub,n+lb,error("Cannot determine a("n")"))

Extension, comment, link, and Pari program from Charles R Greathouse IV, Aug 03 2010

cp's OEIS Frontend

A055739 [e^n]-th prime.

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A065857 The (10^n)-th composite number.

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A300397 a(n) = number of primes that end in 1 among the first 10^n primes.

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A300398 a(n) = number of primes that end in 3 among the first 10^n primes.

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A300399 a(n) = number of primes that end in 7 among the first 10^n primes.

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A300400 a(n) = number of primes that end in 9 among the first 10^n primes.

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A335331 a(n) = prime(k) where k is the n-th 7-smooth number.

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A055680 (5^n)-th prime.

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