A068670 -id:A068670 - cp's OEIS frontend

A086171 Continued fraction of sum(prime(n)/10^b(n)), where b(n) = 1 + the total number of digits of the first n-1 primes, A068670.

0, 4, 4, 6, 2, 5, 18, 1, 3, 4, 1, 2, 1, 2, 4, 7, 4, 1, 21, 2, 1, 3, 5, 2, 1, 1, 27, 1, 1, 5, 12, 1, 18, 1, 1, 1, 1, 1, 1, 9, 3, 1, 1, 1, 5, 1, 5, 2, 2, 1, 53, 1, 8, 1, 23, 6, 2, 2, 1, 1, 3, 1, 1, 25, 2, 2, 7, 1, 2, 3, 1, 4, 3, 12, 1, 2, 7, 1, 68, 1, 19, 1, 2, 2, 14, 4, 6, 2, 1, 2, 58, 2, 16, 1, 1, 1, 2, 2, 1

Offset: 1

Roger L. Bagula, Aug 26 2003

			r=0.235811317192329313741434753596167717...
= 2/10^1 + 3/10^2 + 5/10^3 + 7/10^4 + 11/10^5 + 13/10^7 + ..., the exponents being increased by the length of the previous prime. - _M. F. Hasler_, Oct 17 2013

(* number of powers of ten in the Primes as a sequence*) byte[n_Integer?Positive] := byte[n] =byte[n-1]+Floor[Log[Prime[n-1]]/Log[10]]+1 byte[0]=byte[1] = 1 b=Table[N[Prime[n]*10^(-byte[n]), Digits], {n, 1, Digits}] r=Apply[Plus, b]

r = sum_{n=1..infinity} prime(n)/10^b(n), where b(n+1)=b(n)+floor(log[10] prime(n))+1, b(1)=1. (Edited by M. F. Hasler, Oct 17 2013)

Edited by M. F. Hasler, Oct 17 2013

A019518 Smarandache-Wellin numbers: a(n) is the concatenation of first n primes (written in base 10).

Original entry on oeis.org

2, 23, 235, 2357, 235711, 23571113, 2357111317, 235711131719, 23571113171923, 2357111317192329, 235711131719232931, 23571113171923293137, 2357111317192329313741, 235711131719232931374143, 23571113171923293137414347

Offset: 1

R. Muller

			E.g. a(6) = 2_3_5_7_11_13 = 23571113.

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 72. [The 2002 printing states incorrectly that a(719) is prime. Cf. A046035.] This book uses the name "Smarandache-Wellin numbers", referring to a 1998 private communication from P. Wellin.
H. Ibstedt, A Few Smarandache Sequences, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 170-183.
M. Le, On Smarandache Concatenated Sequences I: Prime Power Sequences, Smarandache Notions Journal, Vol. 9, No. 1-2, 1998, 129-130.
S. Smarandoiu, Convergence of Smarandache continued fractions, Abstract 96T-11-195, Abstracts Amer. Math. Soc., 17 (No. 4, 1996), 680.

Reinhard Zumkeller, Table of n, a(n) for n = 1..300
M. Fleuren, Factoring of the Smarandache Concatenated Prime Sequence.
F. Smarandache, Collected Papers, Vol. II, Tempus Publ. Hse., Bucharest, Romania, 1996.
Eric Weisstein's World of Mathematics, Consecutive Number Sequences
Eric Weisstein's World of Mathematics, Copeland-Erdős Constant
Index entries for sequences related to Most Wanted Primes video

For the primes in this sequence see A069151. For where the primes occur see A046035.

Cf. A000040, A038394, A046284, A068670 (number of digits).

a019518 n = a019518_list !! (n-1)
a019518_list = map read $ scanl1 (++) $ map show a000040_list :: [Integer]
-- Reinhard Zumkeller, Mar 03 2014

[Seqint(Reverse(&cat[Reverse(Intseq(NthPrime(k))): k in [1..n]])): n in [1..20]]; // Vincenzo Librandi, Aug 23 2015

ConsecutivePrimes[n_] := FromDigits[Flatten[IntegerDigits /@ Prime[Range[n]]]] (* Eric W. Weisstein *)
Table[FromDigits[Flatten[IntegerDigits[Prime[Range[i]]]]],{i,15}] (* Jayanta Basu, May 30 2013 *)

s="";for(n=1,30,print1(s=Str(s,prime(n))",")) \\ Cino Hilliard; simplified by M. F. Hasler, Oct 06 2013

A019518(n)=eval(concat(concat([""],primes(n)))) \\ Faster than concat(apply(s->Str(s),primes(n))) or forprime(...s=Str(s,p)). - M. F. Hasler, Oct 06 2013

Definition edited by N. J. A. Sloane, Jul 02 2017

A033308 Decimal expansion of Copeland-Erdős constant: concatenate primes.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein

The number .23571113171923.... was proved normal in base 10 by Copeland and Erdős but is not known to be normal in other bases. - Jeffrey Shallit, Mar 14 2008
Could be read (with indices 1, 2, ...) as irregular table whose n-th row lists the A097944(n) digits of the n-th prime A000040(n). - M. F. Hasler, Oct 25 2019
Named after the American mathematician Arthur Herbert Copeland (1898-1970) and the Hungarian mathematician Paul Erdős (1913-1996). - Amiram Eldar, May 29 2021
This constant is irrational but it is not (yet) known to be transcendental. - Charles R Greathouse IV, Feb 03 2025

			0.235711131719232931374143475359616771737983899710110310710911312...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.9, p. 442.
Glyn Harman, One hundred years of normal numbers, in M. A. Bennett et al., eds., Number Theory for the Millennium, II (Urbana, IL, 2000), A K Peters, Natick, MA, 2002, pp. 149-166.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.

T. D. Noe, Table of n, a(n) for n = 0..2000
John M. Campbell, The prime-counting Copeland-Erdős constant, arXiv:2309.13520 [math.NT], 2023.
A. H. Copeland and P. Erdős, Note on Normal Numbers, Bull. Amer. Math. Soc., Vol. 52, No. 10 (1946), pp. 857-860.
Mikołaj Morzy, Tomasz Kajdanowicz, and Przemysław Kazienko, On Measuring the Complexity of Networks: Kolmogorov Complexity versus Entropy, Complexity, Volume 2017 (2017), Article ID 3250301, p. 5.
Simon Plouffe, Copeland-Erdos constant, the primes concatenated.
Simon Plouffe, Copeland-Erdos constant, the primes concatenated.
Eric Weisstein's World of Mathematics, Copeland-Erdős Constant.

Cf. A030168 (continued fraction), A072754 (numerators of convergents), A072755 (denominators of convergents).
Cf. A000040 (primes), A097944 (row lengths if this is read as table), A228355 (digits of the primes listed in reversed order).
Cf. A033307 (Champernowne constant: analog for positive integers instead of primes), A007376 (digits of the integers, considered as infinite word or table), A066716 (decimals of the binary Champernowne constant).
Cf. A165450, A019518, A074721, A073034, A191232, A129808.
Cf. A066747 and A191232: binary Copeland-Erdős constant: decimals and binary digits.
See also A338072.

a033308 n = a033308_list !! (n-1)
a033308_list = concatMap (map (read . return) . show) a000040_list :: [Int]
-- Reinhard Zumkeller, Mar 03 2014

N[Sum[Prime[n]*10^-(n + Sum[Floor[Log[10, Prime[k]]], {k, 1, n}]), {n, 1, 40}], 100] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 12 2006 *)
N[Sum[Prime@n*10^-(n + Sum[Floor[Log[10, Prime@k]], {k, n}]), {n, 45}], 106] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 12 2006 *)
IntegerDigits //@ Prime@Range@45 // Flatten (* Robert G. Wilson v Oct 03 2006 *)

default(realprecision, 2080); x=0.0; m=-1; forprime (p=2, 4000, n=1+floor(log(p)/log(10)); x=p+x*10^n; m+=n; ); x=x/10^m; for (n=0, 2000, d=floor(x); x=(x-d)*10; write("b033308.txt", n, " ", d)); \\ Harry J. Smith, Apr 30 2009

concat( apply( {row(n)=digits(prime(n))},  [1..99] )) \\ Yields this sequence; row(n) then yields the digits of prime(n) = n-th row of the table, cf. comments. - M. F. Hasler, Oct 25 2019

Equals Sum_{n>=1} prime(n)*10^(-A068670(n)). - Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 12 2006

Equals Sum_{i>=1} (p_i * 10^(-(Sum_{j=1..i} 1 + floor(log_10(p_j))) )) or Sum_{i>=1} (p_i * 10^(-( i + Sum_{j=1..i} floor(log_10(p_j))) )) or Sum_{i>=1} (p_i * 10^(-( Sum_{j=1..i} ceiling(log_10(1 + p_j))) )). - Daniel Forgues, Mar 26-28 2014

A317903 a(n) = A038394(n)^^A038394(n) (mod 10^len(A038394(n))), where ^^ indicates tetration or hyper-4 (e.g., 3^^4=3^(3^(3^3))).

Original entry on oeis.org

4, 76, 176, 4176, 314176, 91314176, 891314176, 80891314176, 88080891314176, 5288080891314176, 705288080891314176, 10705288080891314176, 2410705288080891314176, 912410705288080891314176, 42912410705288080891314176, 9242912410705288080891314176, 989242912410705288080891314176

Offset: 1

Marco Ripà, Aug 10 2018

For any n >= 2, a(n) (mod 10^len(A038394(n))) == a(n + 1) (mod 10^len(A038394(n))), where len(k) := number of digits in k. Assuming len(a(n))>1, this is a general property of every concatenated sequence with fixed rightmost digits (such as A014925 or A092447), as shown in Ripà's book "La strana coda della serie n^n^...^n".

			For n = 6, a(6) = 13117532^^13117532 (mod 10^8) == 91314176.

Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, page 60. ISBN 978-88-6178-789-6

Marco Ripà, On the Convergence Speed of Tetration, ResearchGate (2018).

Cf. A038394, A068670, A171882 (tetration), A317824.

tmod(b, n) = {if (b % n == 0, return (0)); if (b % n == 1, return (1)); if (gcd(b, n)==1, return (lift(Mod(b, n)^tmod(b, lift(znorder(Mod(b, n))))))); lift(Mod(b, n)^(eulerphi(n) + tmod(b, eulerphi(n))));}
f(n) = fromdigits(concat([digits(p) | p<-Vecrev(primes(n))])); \\ A038394
a(n) = if (n==1, 4, my(x=f(n)); tmod(x, 10^#Str(x))); \\ Michel Marcus, Sep 12 2021

a(n) = (p(n)p(n-1)_p(n-2)...3_2)^^(p(n)_p(n-1)_p(n-2)...3_2) (mod 10^len(p(n)_p(n-1)_p(n-2)..._3_2)), where len(k) := number of digits in k.

More terms from Jinyuan Wang, Aug 30 2020

A097944 Number of digits in n-th prime.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Cino Hilliard, Sep 05 2004

For primes p <= n sum(a(n)) -> n/2 and n-> inf.

			The first 4 primes are 2,3,5,7. These are 1-digit numbers so the first 4 entries in the table are 1's.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A060417, A068670 (partial sums).

a097944 = a055642 . a000040 -- Reinhard Zumkeller, Apr 08 2012

a[n_]:=StringLength[ToString[Prime[n]]]; (* Vladimir Joseph Stephan Orlovsky, Dec 03 2008 *)
IntegerLength[Prime[Range[110]]] (* Harvey P. Dale, Oct 04 2012 *)

PARI
```
a(n)=#Str(prime(n))
```

A097944(n)=logint(prime(n),10)+1 \\ M. F. Hasler, Oct 24 2019

a(n) = (log n + log log n)/(log 10) + O(1).
a(n) = A055642(A000040(n)). - Reinhard Zumkeller, Apr 08 2012
a(n) = A068670(n) - A068670(n-1). - M. F. Hasler, Oct 24 2019

A121242 Number of 2's in first n primes.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 7, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 24, 24, 24, 24, 24

Offset: 1

Zak Seidov, Aug 22 2006

In the first 10^m (m=3,4,5) primes there are 339, 4070, 55213 2's, among all 3803, 48982, 610484 digits, which gives the frequencies {0.08914, 0.08309, 0.09044}.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A068670 Number of digits in the first n primes.

Partial sums of A085976.

ListTools:-PartialSums([seq(numboccur(2,convert(ithprime(i),base,10)),i=1..100)]); # Robert Israel, Jun 13 2018

Accumulate[DigitCount[Prime[Range[90]],10,2]] (* Harvey P. Dale, Nov 09 2013 *)

A155498 Number of odd digits in the concatenation of first n primes.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 9, 11, 12, 13, 15, 17, 18, 19, 20, 22, 24, 25, 26, 28, 30, 32, 33, 34, 36, 38, 40, 42, 44, 47, 49, 52, 55, 58, 60, 63, 66, 68, 70, 73, 76, 78, 81, 84, 87, 90, 92, 93, 94, 95, 97, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 116, 119, 122, 125, 128

Offset: 1

Juri-Stepan Gerasimov, Jan 23 2009

Partial sums of A104638. [R. J. Mathar, Feb 13 2009]

			If n=1, concatenate first 1 prime = 2, then number of odd digits = 0 = a(1).
If n=2, concatenate first 2 primes = 23, then number of odd digits = 1 = a(2).
If n=3, concatenate first 3 primes = 235, then number of odd digits = 2 = a(3).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A068670.

ListTools:-PartialSums([seq(nops(select(type, convert(ithprime(i), base, 10), odd)), i=1..100)]); # Robert Israel, Jul 15 2018

Accumulate[Table[Count[IntegerDigits[p],?OddQ],{p,Prime[Range[100]]}]] (* _Harvey P. Dale, Apr 05 2025 *)

a(n) = sum(k=1, n, #select(x->(x%2), digits(prime(k)))); \\ Michel Marcus, Jul 16 2018

Replaced 53 by 55. R. J. Mathar, Feb 13 2009

A278959 Length of the string that is generated by the concatenation of all the prime numbers < n (where n >= 0).

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 6, 6, 8, 8, 8, 8, 10, 10, 12, 12, 12, 12, 14, 14, 14, 14, 14, 14, 16, 16, 18, 18, 18, 18, 18, 18, 20, 20, 20, 20, 22, 22, 24, 24, 24, 24, 26, 26, 26, 26, 26, 26, 28, 28, 28, 28, 28, 28, 30

Offset: 0

Indranil Ghosh, Dec 02 2016

In the following Python program, the algorithm based on the sieve of Eratosthenes is used to generate the primes.

			For n=15, the primes < n are 2,3,5,7,11,13. So the concatenated string is "23571113", which has length=8. a(n)=8.

Indranil Ghosh, Table of n, a(n) for n = 0..100000
Wikipedia, Sieve Of Eratosthenes

Cf. A000040, A068670, A097944.

Join[{0},Accumulate[Table[If[PrimeQ[n],IntegerLength[n],0],{n,0,60}]]] (* Harvey P. Dale, Mar 04 2023 *)

def p(n):
    if n<=2:
        return 0
    s=1
    l = [True] * n
    for i in range(3,int(n**0.5)+1,2):
        if l[i]:
            l[i*i::2*i]=[False]*((n-i*i-1)//(2*i)+1)
    for i in range(3,n,2):
            if l[i]:
                s+=len(str(i))
    return s
for i in range(0, 100001):
    print(f'{i} {p(i)}')

A304652 Digits of the Copeland-Erdős constant taken in groups of five digits.

Original entry on oeis.org

23571, 35711, 57111, 71113, 11131, 11317, 13171, 31719, 17192, 71923, 19232, 92329, 23293, 32931, 29313, 93137, 31374, 13741, 37414, 74143, 41434, 14347, 43475, 34753, 47535, 75359, 53596, 35961, 59616, 96167, 61677, 16771, 67717, 77173, 71737, 17379, 73798, 37983, 79838, 98389

Offset: 1

Alonso del Arte, May 16 2018

This sequence is presented by the Foo Bar Challenge, a recruitment tool for Google, as the challenge to renumber some entities with 5-digit ID numbers taken from the first 10004 places of the Copeland-Erdős constant.
The ID numbers are to be left zero-padded as needed to ensure they all consist of five digits. Of course here in the OEIS leading zeros will just be dropped.
The Foo Bar Challenge statement of the problem says nothing of the fact that there are duplicates among these well before 10000.

Cf. A033308, A068670.

numCopelandErdos = Flatten[IntegerDigits[Prime[Range[1000]]]]; Table[FromDigits[numCopelandErdos[[n ;; (n + 4)]]], {n, Length[numCopelandErdos] - 4}]

A306890 a(n) is the number of prime digits used in writing out all primes up to and including the n-th prime.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 6, 6, 8, 9, 10, 12, 12, 13, 14, 16, 17, 17, 18, 19, 21, 22, 23, 23, 24, 24, 25, 26, 26, 27, 29, 30, 32, 33, 33, 34, 36, 37, 38, 40, 41, 41, 41, 42, 43, 43, 44, 47, 50, 52, 55, 57, 58, 60, 63, 65, 66, 68, 71, 72, 74, 76, 78, 79

Offset: 1

Lekraj Beedassy, Mar 15 2019

			We have a(10) = 9 since all primes up to the 10th (2, 3, 5, 7, 11, 13, 17, 19, 23, 29) use the 9 prime digits 2, 3, 5, 7, 3, 7, 2, 3, 2.

Partial sums of A109066. Cf. A068670.

With[{s = IntegerDigits[Prime@ Range@ 64]}, Array[Count[Flatten[s[[1 ;; #]] ], ?PrimeQ] &, Length@ s]] (* _Michael De Vlieger, Mar 27 2019 *)

a(n) = sum(k=1, prime(n), if (isprime(k), #select(x->isprime(x), digits(k)))); \\ Michel Marcus, Mar 23 2019

cp's OEIS Frontend

A086171 Continued fraction of sum(prime(n)/10^b(n)), where b(n) = 1 + the total number of digits of the first n-1 primes, A068670.

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A019518 Smarandache-Wellin numbers: a(n) is the concatenation of first n primes (written in base 10).

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A033308 Decimal expansion of Copeland-Erdős constant: concatenate primes.

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A317903 a(n) = A038394(n)^^A038394(n) (mod 10^len(A038394(n))), where ^^ indicates tetration or hyper-4 (e.g., 3^^4=3^(3^(3^3))).

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A097944 Number of digits in n-th prime.

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A121242 Number of 2's in first n primes.

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Maple

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A155498 Number of odd digits in the concatenation of first n primes.

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