A019518 -id:A019518 - cp's OEIS frontend

A190480 Concatenation of first n digits in the concatenation of first n primes written in base 2.

1, 10, 101, 1011, 10111, 101110, 1011101, 10111011, 101110111, 1011101111, 10111011111, 101110111110, 1011101111101, 10111011111011, 101110111110111, 1011101111101111, 10111011111011110, 101110111110111101, 1011101111101111011

Offset: 1

Juri-Stepan Gerasimov, May 24 2011

The terms are 1, 2, 5, 11, 23, 46, 93, 185, 375, ... in decimal.

The terms are the first n digits of the concatenation 10//11//101//111//1011//1101//.. generated with A004676.

Cf. A019518.

With[{cc=Flatten[Table[IntegerDigits[p,2],{p,Prime[ Range[ 10]]}]]},Table[ FromDigits[Take[cc,n]],{n,Length[cc]}]] (* Harvey P. Dale, Jan 26 2021 *)

A103208 Numbers k such that 3 divides prime(1) + ... + prime(k).

Original entry on oeis.org

10, 16, 18, 20, 24, 26, 28, 30, 32, 34, 36, 40, 42, 44, 46, 52, 54, 57, 68, 70, 74, 76, 78, 80, 82, 84, 86, 88, 90, 97, 99, 103, 105, 107, 111, 113, 119, 121, 123, 125, 127, 129, 134, 136, 138, 161, 163, 166, 169, 175, 177, 179, 185, 187, 195, 197, 199, 203, 205, 207, 211, 213

Offset: 1

Robert G. Wilson v, Mar 19 2005

nonn

Also, numbers k such that 3 divides the concatenation of the first k primes (see A019518).

The first comment and the description are true whenever the number of primes congruent to 1 mod 6 exceeds the number of primes congruent to 5 mod 6 and the difference is congruent to 1 mod 3 or the number of primes congruent to 5 mod 6 exceeds the number of primes congruent to 1 mod 6 and the difference is congruent to 2 mod 3. - Roderick MacPhee, Oct 30 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Hisanori Mishima, Smarandache consecutive prime sequences (n = 1 to 100).

Cf. A007504, A019518, A104644, A111287.

Cf. A111318, A111319, A111320, A111321, A111322, A111323, A111324, A111325, A111326, A111327.

s1:=[2]; M:=1000; for n from 2 to M do s1:=[op(s1),s1[n-1]+ithprime(n)]; od: s1;
f:=proc(k) global M,s1; local t1,n; t1:=[]; for n from 1 to M do if s1[n] mod k = 0 then t1:=[op(t1),n]; fi; od: t1; end; f(3);

f[n_] := FromDigits[ Flatten[ Table[ IntegerDigits[ Prime[i]], {i, n}]]]; Select[ Range[ 206], Mod[f[ # ], 3] == 0 &]
Flatten[Position[Accumulate[Prime[Range[250]]],?(Divisible[#,3]&)]] (* _Harvey P. Dale, Jan 14 2016 *)

a=0;b=0;for(x=3,1000,if(prime(x)%6==1,a+=1,b+=1);if((a-b)%3==1 || (b-a)%3==2,print1(x","))) \\ Roderick MacPhee, Oct 30 2015

lista(nn) = { s=0; for(k=1, nn, s += prime(k); if(s % 3 == 0, print1(k, ", ")););} \\ Altug Alkan, Dec 04 2015

Entry revised by N. J. A. Sloane, Nov 09 2005

A069151 Concatenations of consecutive primes, starting with 2, that are also prime.

Original entry on oeis.org

2, 23, 2357

Offset: 1

Joseph L. Pe, Apr 08 2002

Primes in A019518.
The next term is the 355-digit number 2357111317192329313741434753...677683691701709719 which is too large to include here. See A046035, A046284.
The term after the 355-digit term has 499 digits, and the next two terms after that have 1171 and 1543 digits respectively. - Harvey P. Dale, Oct 03 2024

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, 2nd ed., Springer, NY, 2005; see p. 78. [The 2002 printing states incorrectly that 2357...5441 is prime.]

Jens Kruse Andersen, Table of n, a(n) for n = 1..5
M. Fleuren, Smarandache Concatenated Primes
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Eric Weisstein's World of Mathematics, Smarandache-Wellin Number
Eric Weisstein's World of Mathematics, Smarandache-Wellin Prime
Wikipedia, Smarandache-Wellin number
Index entries for sequences related to Most Wanted Primes video

Cf. A019518.
Cf. A046035 (Numbers n such that the concatenation of the first n primes is prime)
Cf. A046284 (Primes p such that concatenation of primes from 2 through p is a prime).
Cf. A030997 (Smallest prime which is a concatenation of n consecutive primes).

Cases[FromDigits /@ Rest[FoldList[Join, {}, IntegerDigits[Prime[ Range[10^3]]]]], ?PrimeQ] (* _Eric W. Weisstein, Oct 30 2015 *)
Select[Table[FromDigits[Flatten[IntegerDigits/@Prime[Range[n]]]],{n,500}],PrimeQ] (* Harvey P. Dale, Oct 03 2024 *)

s=""; for(n=1, 200, s=concat(s, prime(n)); if(ispseudoprime( eval(s)), print1(s", "))) \\ Jens Kruse Andersen, Jun 26 2014

from sympy import isprime, nextprime
def afind(terms, verbose=False):
  n, p, pstr = 0, 2, "2"
  while n < terms:
    if isprime(int(pstr)): n += 1; print(n, int(pstr))
    p = nextprime(p); pstr += str(p)
afind(5) # Michael S. Branicky, Feb 23 2021

Edited by Robert G. Wilson v, Apr 11 2002

Entry revised Jan 18 2004

A091762 Last n digits of concatenation of first n primes.

Original entry on oeis.org

2, 23, 235, 2357, 35711, 571113, 7111317, 11131719, 113171923, 1317192329, 31719232931, 171923293137, 7192329313741, 19232931374143, 923293137414347, 2329313741434753, 32931374143475359, 293137414347535961, 9313741434753596167, 31374143475359616771, 137414347535961677173, 3741434753596167717379

Offset: 1

Reinhard Zumkeller, Feb 04 2004

a(n) = A019518(n) mod 10^n; a(n) mod 10^A055642(A000040(n)) = A000040(n); for the primes in this sequence see A091763.

			The first 5 primes are 2, 3, 5, 7, 11. The last 5 digits concatenated are 35711 so a(5) = 35711. - _David A. Corneth_, Sep 15 2019

David A. Corneth, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Consecutive Number Sequences

Table[FromDigits[Take[Flatten[IntegerDigits/@Prime[Range[n]]],-n]],{n,20}] (* Harvey P. Dale, Sep 15 2019 *)

a(n) = {my(p = prime(n), v = digits(p)); while(#v < n, p = precprime(p - 1); v = concat(digits(p), v)); fromdigits(vector(n, i, v[#v - n + i]))} \\ David A. Corneth, Sep 15 2019

More terms from David A. Corneth, Sep 15 2019

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

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154

Offset: 0

Eugene McDonnell (eemcd(AT)mac.com), Jan 18 2004

Partial sums of A097944. - Lekraj Beedassy, Aug 26 2008

			a(5) is 6 because concatenating the first five primes gives 235711, which has six digits.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000. (Initial 0 added by _M. F. Hasler_, Nov 02 2019.)
Eric Weisstein's World of Mathematics, Copeland-Erdos Constant

Cf. A019518, A097944 (number of decimal digits of the primes).

Cf. A033308 (decimal expansion of the Copeland-Erdos constant).

a068670:=func< n | n + &+[ Floor(Log(10, NthPrime(k))): k in [1..n] ] >; [ a068670(n): n in [1..70] ];

Table[n + Sum[Floor[Log[10, Prime[k]]], {k, 1, n}], {n, 1, 90}] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 12 2006 *)
Accumulate[IntegerLength[Prime[Range[70]]]] (* Harvey P. Dale, Jul 01 2012 *)

A68670=List(0); A068670(n)={for(N=#A68670,n, listput(A68670, A68670[N] + A097944(N))); A68670[n+1]} \\ M. F. Hasler, Oct 24 2019

from sympy import sieve
from itertools import accumulate, chain
def f(, n): return  + len(str(n))
def agen(): yield from accumulate(chain((0,), (p for p in sieve)), f)
print(list(islice(agen(), 62))) # Michael S. Branicky, Feb 03 2023

a(n) = Sum_{i=1..n} ceiling(log_10(1 + prime(i))). - Daniel Forgues, Apr 02 2014

Extended to a(0) = 0 by M. F. Hasler, Oct 24 2019

A089933 Concatenate the first n odd primes.

Original entry on oeis.org

3, 35, 357, 35711, 3571113, 357111317, 35711131719, 3571113171923, 357111317192329, 35711131719232931, 3571113171923293137, 357111317192329313741, 35711131719232931374143, 3571113171923293137414347, 357111317192329313741434753, 35711131719232931374143475359

Offset: 1

Cino Hilliard, Jan 11 2004

Except for a(1) = 3, these numbers are not prime for n < 10000. See A089948.

Robert Israel, Table of n, a(n) for n = 1..298

Cf. A019518, A086043, A089948.

[Seqint(Reverse(&cat[Reverse(Intseq(NthPrime(k+1))): k in [1..n]])): n in [1..17]]; // Vincenzo Librandi, Nov 30 2015

a[1]:= 3;
for n from 2 to 100 do
  p:= ithprime(n+1);
  a[n]:= 10^(1+ilog10(p))*a[n-1]+p
od:
seq(a[i],i=1..100); # Robert Israel, Aug 25 2016

Table[FromDigits[Flatten[IntegerDigits[Prime[Range[n] + 1]]]],{n, 20}] (* Vincenzo Librandi, Nov 30 2015 *)

concatprime(n) = { y=""; forprime(x=3,n, y=concat(Str(y),Str(x)); z=eval(y); print1(z",") ) }

Offset changed by Robert Israel, Aug 25 2016

A046284 Primes p such that concatenation of primes from 2 through p is a prime.

Original entry on oeis.org

2, 3, 7, 719, 1033, 2297, 3037, 11927

Offset: 1

Patrick De Geest, Jun 15 1998

"w_n = (P_1)(P_2) ... (P_n) [A019518], by which notation we mean that w_n is constructed in decimal by simple concatenation of digits [much like the Almost Natural numbers (A007376)]. For example, the first few w_n are 2, 23, 235, 2357, 235711, ... ." - Crandall and Pomerance

			7 is a member, since 2357 is a prime.

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 72. [The 2002 printing states incorrectly that 5441 is a term.]

Eric Weisstein's World of Mathematics, Consecutive Number Sequences.
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Eric Weisstein's World of Mathematics, Smarandache-Wellin Number

Cf. A019518, A033308, A069151. a(n) = prime(A046035(n)).

a = ""; Do[a = StringJoin[a, ToString[ Prime[n]]]; If[ PrimeQ[ ToExpression[a]], Print[n]], {n, 1, 1429}]

Additional comments from Robert G. Wilson v, Sep 10 2001

A171154 Smallest prime whose decimal expansion begins with concatenation of first n primes in descending order.

Original entry on oeis.org

2, 3203, 5323, 75323, 11753221, 131175329, 171311753203, 19171311753229, 231917131175321, 292319171311753231, 3129231917131175327, 3731292319171311753239, 41373129231917131175321, 43413731292319171311753233, 4743413731292319171311753269, 534743413731292319171311753223

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Dec 04 2009

Sequence is conjectured to be infinite.
a(n) = "prime(n)...prime(1) R(n)".
R(n) for n>1: 03, 3, 3, 21, 9, 03, 29, 1, 31, 7, 39, 1, 33, 69, 23, 3, 59, 27, ...
It is conjectured that R(n)=1 for infinite many n.

			a(1) = 2 = prime(1) is the exceptional case, because no R(1).
a(2) = 3203 = prime(453) = "32 03", R(2)="03".
a(5) = 11753221 = prime(772902) = "prime(5)...prime(1) 21", R(5)=21.

Leonard E. Dickson, History of the Theory of numbers, vol. I, Dover Publications 2005.

Michael S. Branicky, Table of n, a(n) for n = 1..298

Cf. A000040, A066065, A019518, A089710, A053546.

from sympy import isprime, primerange, prime
def a(n):
    if n == 1: return 2
    c = int("".join(map(str, [p for p in primerange(2, prime(n)+1)][::-1])))
    pow10 = 10
    while True:
        c *= 10
        for b in range(1, pow10, 2):
            if b%5 == 0: continue
            if isprime(c+b):
                return c+b
        pow10 *= 10
print([a(n) for n in range(1, 17)]) # Michael S. Branicky, Mar 12 2022

a(14) and beyond from Michael S. Branicky, Mar 12 2022

A038394 Concatenate first n primes in reverse order.

Original entry on oeis.org

2, 32, 532, 7532, 117532, 13117532, 1713117532, 191713117532, 23191713117532, 2923191713117532, 312923191713117532, 37312923191713117532, 4137312923191713117532, 434137312923191713117532, 47434137312923191713117532, 5347434137312923191713117532

Offset: 1

M. I. Petrescu (mipetrescu(AT)yahoo.com)

F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.

Reinhard Zumkeller, Table of n, a(n) for n = 1..300
Mihaly Bencze [Beneze] and L. Tutescu, editors, Some Notions and Questions in Number Theory, Vol. II.
M. Fleuren, Smarandache Back Concatenated Primes.
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.

Cf. A000040, A000422, A019518.

a038394 n = a038394_list !! (n-1)
a038394_list = f "" a000040_list where
   f xs (q:qs) = (read ys :: Integer) : f ys qs
     where ys = show q ++ xs
-- Reinhard Zumkeller, Mar 03 2014

Join[{s = 2}, Table[s = FromDigits[Flatten[IntegerDigits[{Prime[n], s}]]], {n, 2, 13}]] (* Jayanta Basu, Jul 14 2013 *)

a(n) = fromdigits(concat([digits(p) | p<-Vecrev(primes(n))])); \\ Andrew Howroyd, Aug 29 2020

Offset corrected by Reinhard Zumkeller, Mar 03 2014

A038395 Concatenation of the first n odd numbers in reverse order.

Original entry on oeis.org

1, 31, 531, 7531, 97531, 1197531, 131197531, 15131197531, 1715131197531, 191715131197531, 21191715131197531, 2321191715131197531, 252321191715131197531, 27252321191715131197531, 2927252321191715131197531, 312927252321191715131197531

Offset: 1

M. I. Petrescu (mipetrescu(AT)yahoo.com)

a(n) starts with the digits of 2n-1. Indices of prime or probable prime terms are 1,2,37,62,409,...: see also A089922. - M. F. Hasler, Apr 13 2008

If n == 0 (mod 3), so is a(n). - Sergey Pavlov, Mar 29 2017

Mihaly Bencze [Beneze] and L. Tutescu, Some Notions and Questions in Number Theory, Sequence 3.

Florentin Smarandache, Sequences of Numbers Involved in Unsolved Problems, arXiv:math/0604019 [math.GM], 2006.

Cf. A089922, A109837, A038394-A038399, A019518, A019519.

Table[FromDigits[Flatten[IntegerDigits/@Join[Reverse[Range[1,n,2]]]]], {n,1,29,2}] (* Harvey P. Dale, Jun 02 2011 *)

t=""; for( n=1,10^3, ( t=eval( Str( 2*n-1,t))) & print(n" "t)) \\ M. F. Hasler, Apr 13 2008

def a(n): return int("".join(map(str, range(2*n-1, 0, -2))))
print([a(n) for n in range(1, 17)]) # Michael S. Branicky, Jan 31 2021

Edited and extended by M. F. Hasler, Apr 13 2008

Edited by T. D. Noe, Oct 30 2008

cp's OEIS Frontend

A190480 Concatenation of first n digits in the concatenation of first n primes written in base 2.

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A103208 Numbers k such that 3 divides prime(1) + ... + prime(k).

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A069151 Concatenations of consecutive primes, starting with 2, that are also prime.

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A091762 Last n digits of concatenation of first n primes.

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

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Magma

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A089933 Concatenate the first n odd primes.

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A046284 Primes p such that concatenation of primes from 2 through p is a prime.

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