cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A185689 Final prime adjoined in the smallest term of A019518 divisible by 53^n.

Original entry on oeis.org

1153, 5519, 1288769, 608227163, 3059326339
Offset: 1

Views

Author

James G. Merickel, Feb 05 2011

Keywords

Comments

Associated with A019518(i) at i = 191, 729, 99242, ...

Crossrefs

Cf. A019518, A185656, A185659, A185662, A185665, A185677.

A185692 Final prime adjoined in the smallest term of A019518 divisible by 59^n.

Original entry on oeis.org

149, 409, 2216519, 45510791, 11543761309
Offset: 1

Views

Author

James G. Merickel, Feb 05 2011

Keywords

Comments

Associated with A019518(i) at i=35, 80, 163796, 2747287,...

Crossrefs

Cf. A185656, A185659, A185662, A185665, A185671, A185677, A185680.

A185701 Final prime adjoined in the smallest term of A019518 divisible by 71^n.

Original entry on oeis.org

173, 54851, 4340267, 822527543
Offset: 1

Views

Author

James G. Merickel, Feb 05 2011

Keywords

Comments

Associated with A019518(i) at i = 40, 5578, 305467,...

Crossrefs

Cf. A185656, A185659, A185662, A185665, A185671, A185677, A185680.

A185707 Final prime adjoined in the smallest term of A019518 divisible by 79^n.

Original entry on oeis.org

173, 52237, 19366099, 207681823
Offset: 1

Views

Author

James G. Merickel, Feb 05 2011

Keywords

Comments

Associated with A019518(i) at i = 40, 5340, 1232852, 11480591,..

Crossrefs

Cf. A185656, A185659, A185662, A185665, A185671, A185677, A185680, A185704.

A185710 Final prime adjoined in the smallest term of A019518 divisible by 83^n.

Original entry on oeis.org

281, 25561, 4014239, 670182949
Offset: 1

Views

Author

James G. Merickel, Feb 05 2011

Keywords

Comments

Associated with A019518(i) at i = 60, 2814, 284086,...

Crossrefs

Cf. A185656, A185659, A185662, A185665, A185671, A185677, A185680, A185704.

A185713 Final prime adjoined in the smallest term of A019518 divisible by 89^n.

Original entry on oeis.org

347, 11867, 15827689, 165429863
Offset: 1

Views

Author

James G. Merickel, Feb 05 2011

Keywords

Comments

Associated with A019518(i) at i = 69, 1423, 1020690, 9261551,...

Crossrefs

Cf. A185656, A185659, A185662, A185665, A185671, A185677, A185680, A185704.

A185716 Final prime adjoined in the smallest term of A019518 divisible by 97^n.

Original entry on oeis.org

191, 43801, 3620161, 427245967
Offset: 1

Views

Author

James G. Merickel, Feb 05 2011

Keywords

Comments

Associated with A019518(i) at i = 43, 4564, 258100, 22709459,...

Crossrefs

Cf. A185656, A185659, A185662, A185665, A185671, A185677, A185680, A185704.

A185719 Final prime of the first member of A019518 that is divisible by 43^n.

Original entry on oeis.org

211, 4051, 398261, 35912593, 5355284791
Offset: 1

Views

Author

James G. Merickel, Feb 05 2011

Keywords

Examples

			The number 2357...199211 is the first term in A019518 divisible by 43^1, therefore a(1) = 211.

Crossrefs

Cf. A019518, A183194-A183196, A185656.

Programs

  • PARI
    a(n,m=43)={ my(s=10,p=2); n=Mod(0,m^n); while(n=n*s+p,(p=nextprime(p+1))>s & s*=10) ;p}  \\ M. F. Hasler, Feb 08 2011

Formula

a(n) = Min_{ prime(k) | A019518(k) = 0 (mod 43^n) }. - M. F. Hasler, Feb 08 2011

A276201 Largest prime < A019518(n).

Original entry on oeis.org

19, 233, 2351, 235699, 23571103, 2357111297, 235711131709, 23571113171891, 2357111317192243, 235711131719232929, 23571113171923293127, 2357111317192329313727, 235711131719232931374061, 23571113171923293137414303, 2357111317192329313741434739, 235711131719232931374143475271
Offset: 2

Views

Author

Ilya Gutkovskiy, Aug 24 2016

Keywords

Comments

The complement of A074366.

Examples

			a(5) = 235699, because this is the largest prime less than 235711 (concatenation of first 5 primes, written in decimal system).

Links

Crossrefs

Cf. A000040, A000720, A019518, A074366, A151799.

Programs

  • Mathematica
    Table[NextPrime[FromDigits[Flatten[IntegerDigits[Prime[Range[n]]]]], -1], {n, 2, 17}]

Formula

a(n) = A151799(A019518(n)).
a(n) = A000040(A000720(A019518(n)-1)).

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

Views

Author

Eric W. Weisstein

Keywords

Comments

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

Examples

			0.235711131719232931374143475359616771737983899710110310710911312...

References

  • 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.

Links

Crossrefs

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.

Programs

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

Formula

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
Previous Showing 21-30 of 84 results. Next

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