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.

Showing 1-4 of 4 results.

A030458 Primes formed by concatenating n with n+1.

Original entry on oeis.org

23, 67, 89, 1213, 3637, 4243, 5051, 5657, 6263, 6869, 7879, 8081, 9091, 9293, 9697, 102103, 108109, 120121, 126127, 138139, 150151, 156157, 180181, 186187, 188189, 192193, 200201, 216217, 242243, 246247, 252253, 270271, 276277, 278279, 300301, 308309, 312313, 318319
Offset: 1

Views

Author

Patrick De Geest

Keywords

Comments

Primes in A030656.

Links

Crossrefs

Cf. A052089, A052087, A052088.

Programs

  • Magma
    [m: n in [2..270 by 2] | IsPrime(m) where m is Seqint(Intseq(n+1) cat Intseq(n))];  // Bruno Berselli, Jun 18 2011
  • Mathematica
    Select[Table[FromDigits[Join[Flatten[IntegerDigits[{n,n+1}]]]],{n,270}],PrimeQ] (* Jayanta Basu, May 16 2013 *)
  • PARI
    forstep(n=2,1e3,2,if(isprime(k=eval(Str(n,n+1))),print1(k", "))) \\ Charles R Greathouse IV, Jun 18 2011
  • Python
    from sympy import isprime
from itertools import count, islice
def agen(): yield from filter(isprime, (int(str(k)+str(k+1)) for k in count(2, 2)))
print(list(islice(agen(), 38))) # Michael S. Branicky, Aug 05 2022

A030655 Pair up the numbers.

Original entry on oeis.org

12, 34, 56, 78, 910, 1112, 1314, 1516, 1718, 1920, 2122, 2324, 2526, 2728, 2930, 3132, 3334, 3536, 3738, 3940, 4142, 4344, 4546, 4748, 4950, 5152, 5354, 5556, 5758, 5960, 6162, 6364, 6566, 6768, 6970, 7172, 7374, 7576, 7778, 7980, 8182, 8384, 8586
Offset: 1

Views

Author

Maghraoui Abdelkader

Keywords

Comments

That is, a(n) is the concatenation of 2n-1 and 2n. - Charles R Greathouse IV, Aug 07 2012

Links

Crossrefs

Cf. A030656.
Subsequence of A005843.

Programs

  • Magma
    [Seqint(Intseq(n+1) cat Intseq(n)): n in [1..86 by 2]];  // Bruno Berselli, Jun 18 2011
  • Mathematica
    FromDigits[Flatten[IntegerDigits/@{#}]]&/@Partition[Range[90],2] (* Harvey P. Dale, Sep 28 2011 *)
  • PARI
    a(n)=eval(concat(Str(2*n-1),2*n)) \\ Charles R Greathouse IV, Jun 18 2011

Extensions

More terms from Erich Friedman.

A137233 Number of n-digit even numbers.

Original entry on oeis.org

5, 45, 450, 4500, 45000, 450000, 4500000, 45000000, 450000000, 4500000000, 45000000000, 450000000000, 4500000000000, 45000000000000, 450000000000000, 4500000000000000, 45000000000000000, 450000000000000000, 4500000000000000000, 45000000000000000000, 450000000000000000000
Offset: 1

Views

Author

Ctibor O. Zizka, Mar 08 2008

Keywords

Comments

From Kival Ngaokrajang, Oct 18 2013: (Start)
a(n) is also the total number of double rows identified numbers in n digit.
For example:
n = 1: 01 23 45 67 89 = 5 double rows;
n = 2: 1011 1213 1415 1617 1819...9899 = 45 double rows;
n = 3: 100101 102103 104105...998999 = 450 double rows;
The number of double rows is also A030656. (End)
a(n) is also the number of n-digit integers with an even number of even digits (A356929); a(5) = 45000 is the answer to the question 2 of the Olympiade Mathématique Belge in 2004 (link). - Bernard Schott, Sep 06 2022
a(n) is also the number of n-digit integers with an odd number of odd digits (A054684). - Bernard Schott, Nov 07 2022

Examples

			a(2) = 45 because there are 45 2-digit even numbers.

Links

Crossrefs

Cf. A000422, A002275, A002276, A011577, A014923, A014925, A016313, A019518, A037487, A053052, A057138, A090843, A097166, A099914, A099915.
Cf. A030656, A052268, A054684, A356929.

Programs

Formula

a(n) = 9*10^(n-1)/2 if n > 1. - R. J. Mathar, May 23 2008
From Elmo R. Oliveira, Jul 23 2025: (Start)
G.f.: 5*x*(1-x)/(1-10*x).
E.g.f.: (-9 + 10*x + 9*exp(10*x))/20.
a(n) = 10*a(n-1) for n > 2.
a(n) = A052268(n)/2 for n >= 2. (End)

Extensions

Corrected and extended by R. J. Mathar, May 23 2008
More terms from Elmo R. Oliveira, Jul 23 2025

A275238 a(n) = n*(10^floor(log_10(n)+1) + 1) + (-1)^n.

Original entry on oeis.org

1, 10, 23, 32, 45, 54, 67, 76, 89, 98, 1011, 1110, 1213, 1312, 1415, 1514, 1617, 1716, 1819, 1918, 2021, 2120, 2223, 2322, 2425, 2524, 2627, 2726, 2829, 2928, 3031, 3130, 3233, 3332, 3435, 3534, 3637, 3736, 3839, 3938, 4041, 4140, 4243, 4342, 4445, 4544, 4647, 4746, 4849, 4948, 5051, 5150, 5253, 5352, 5455, 5554
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 21 2016

Keywords

Comments

Concatenation of n with n+(-1)^n (A004442).
Subsequence of A248378.
Primes in this sequence: 23, 67, 89, 1213, 3637, 4243, 5051, 5657, 6263, 6869, 7879, 8081, 9091, 9293, 9697, 102103, ... (A030458).
Numbers n such that a(n) is prime: 2, 6, 8, 12, 36, 42, 50, 56, 62, 68, 78, 80, 90, 92, 96, 102, 108, 120, 126, 138, ... (A030457).

Examples

			a(0) =  0 + 1 = 1;
a(1) = 11 - 1 = 10;
a(2) = 22 + 1 = 23;
a(3) = 33 - 1 = 32;
a(4) = 44 + 1 = 45;
a(5) = 55 - 1 = 54, etc.
or
a(0) =  1 -> concatenation of 0 with 0 + (-1)^0 = 1;
a(1) = 10 -> concatenation of 1 with 1 + (-1)^1 = 0;
a(2) = 23 -> concatenation of 2 with 2 + (-1)^2 = 3;
a(3) = 32 -> concatenation of 3 with 3 + (-1)^3 = 2;
a(4) = 45 -> concatenation of 4 with 4 + (-1)^4 = 5;
a(5) = 54 -> concatenation of 5 with 5 + (-1)^5 = 4, etc.
........................................................
a(2k) = 1, 23, 45, 67, 89, 1011, 1213, 1415, 1617, 1819, ...

Crossrefs

Cf. A001704, A002285, A004442, A020338, A030457, A030458, A030656, A033999, A064834, A248378.

Programs

  • Mathematica
    Table[n (10^Floor[Log[10, n] + 1] + 1) + (-1)^n, {n, 0, 55}]
  • PARI
    a(n) = if(n, n*(10^(logint(n,10)+1) + 1) + (-1)^n, 1) \\ Charles R Greathouse IV, Jul 21 2016

Formula

a(n) = A020338(n) + A033999(n).
a(2k) = A030656(k).
A064834(a(n)) > 0, for n > 0.
a(n) ~ 10*n*10^floor(c*log(n)), where c = 1/log(10) = 0.4342944819... = A002285.
Showing 1-4 of 4 results.

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