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 1-6 of 6 results.

A080152 Values of n corresponding to the primes given in A073640. The concatenation of the a(n) and a(n+1)-th primes for any n is also prime.

Original entry on oeis.org

1, 2, 4, 8, 11, 12, 18, 21, 31, 34, 46, 47, 50, 61, 78, 84, 85, 90, 99, 100, 101, 106, 110, 111, 114, 131, 133, 138, 141, 157, 159, 167, 169, 172, 175, 183, 197, 198, 211, 229, 231, 232, 233, 235, 260, 261, 269, 276, 285, 289, 295, 306, 322, 337, 339, 340, 347
Offset: 1

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 31 2003

Keywords

Examples

			E.g. For n=5: a(5)=11, a(6)=12 and the concatenation of 11th and 12th primes is 3137, which is also prime. The 5th and 6th terms in A073640 are 31 and 37.

Crossrefs

Cf. A000720, A073640.

Programs

  • Maple
    nout := [1]: for n from 2 to 1000 do: p := ithprime(n): d := parse(cat(pout[nops(pout)],p)): if (isprime(d)) then nout := [op(nout),n]: fi: od:

Formula

A073640(n) = prime(a(n)), with a(n) the n-th term in this sequence.
a(n) = A000720(A073640(n)). - Sean A. Irvine, Sep 05 2025

Extensions

Edited by Charles R Greathouse IV, Apr 26 2010

A080155 a(1)=2; a(n) for n>1 is the smallest prime number > a(n-1) such that the concatenation of all previous terms is also prime.

Original entry on oeis.org

2, 3, 11, 31, 47, 229, 251, 577, 857, 859, 911, 1123, 1223, 1297, 1571, 2161, 2417, 2551, 2879, 3319, 5273, 6121, 6947, 7603, 8273, 12721, 12953, 13291, 15683, 16453, 17207, 18133, 20399, 23743, 23909, 25849, 28277, 28879, 35291, 35461, 36107, 43573
Offset: 1

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 31 2003

Keywords

Comments

See A073640 for the sequence involving concatenation of 2 successive terms, A080153 for 3 successive terms. Primeness is established using Maple's isprime() function, so later terms should be regarded as "probable".

Examples

			E.g. a(5)=47 since this is the smallest prime>a(4) which, when concatenated with the concatenation of a(1) to a(4) (=231131), also yields a prime, in this case 23113147.

Links

Crossrefs

Cf. A073640, A080152, A080153, A080154, A080156.
Cf. A033679, A069603, A074338.

Programs

  • Maple
    with(numtheory): pout := [2]: nout := [1]: for n from 2 to 5000 do: p := ithprime(n): d := parse(cat(seq(pout[i],i=1..nops(pout)),p)): if (isprime(d)) then pout := [op(pout),p]: nout := [op(nout),n]: fi: od: pout;
  • Mathematica
    f[s_List] := Block[{p=NextPrime@s[[-1]], pp=FromDigits@Flatten[IntegerDigits/@s]}, While[!PrimeQ[pp*10^Floor[Log[10,p]+1]+p], p=NextPrime@p]; Append[s,p]]; Nest[f,{2},40]

Formula

For any n>1, a(n) is prime and a(n) > a(n-1). a(n) is the smallest prime for which a(1)//a(2)//...//a(n) is prime. // denotes concatenation.

A073841 LCM of the composite numbers between n and 2n (both inclusive).

Original entry on oeis.org

1, 4, 12, 24, 360, 360, 2520, 5040, 5040, 5040, 55440, 55440, 3603600, 10810800, 10810800, 21621600, 367567200, 367567200, 6983776800, 6983776800, 6983776800, 6983776800, 160626866400, 160626866400, 1124388064800, 1124388064800, 1124388064800, 1124388064800
Offset: 1

Views

Author

Amarnath Murthy, Aug 13 2002

Keywords

Comments

Also, smallest number divisible by all integers 1 through n as well as all composite numbers 1 through 2n. - J. Lowell, Jul 16 2008 [Definition of A140813, that is a duplicate of this sequence]
Not a subsequence of A002182: a(79) = 10703173554082014360835514860858032000 is the smallest term that is not in A002182. [Klaus Brockhaus, Aug 28 2008]

Examples

			a(6) = lcm(6,8,9,10,12) = 360.
The primes <= 10 are 2, 3, 5 and 7. Their highest powers below 2 * 10 = 20 are 16, 9, 5 and 7 respectively. Therefore, a(10) = 16 * 9 * 5 * 7 = 5040. - _David A. Corneth_, Mar 19 2018

Links

Crossrefs

Cf. A073839, A073640.

Programs

  • Maple
    for n from 1 to 100 do l := 1:for j from n to 2*n do if not isprime(j) then l := lcm(l,j):fi:od:a[n] := l:od: seq(a[j],j=1..100);
  • Mathematica
    Table[ Apply[ LCM, Select[Range[n, 2n], !PrimeQ[ # ] & ]], {n, 2, 26}]
  • PARI
    iscomposite(x) = (x!=1) && !isprime(x);
a(n) = lcm(select(x->iscomposite(x), vector(n+1, k, n+k-1))); \\ Michel Marcus, Mar 18 2018
  • PARI
    a(n) = my(res = 1); forprime(p = 2, n, res *= p^(logint(n<<1, p))); res \\ David A. Corneth, Mar 19 2018

Extensions

Edited by Robert G. Wilson v, Sascha Kurz and Labos Elemer, Aug 14 2002
a(1) changed to 1 by Alois P. Heinz, Mar 18 2018

A080153 a(1)=2, a(2)=3; a(n) for n>2 is the first prime > a(n-1) such that the concatenation of a(n-1), a(n-2) and a(n) is also prime.

Original entry on oeis.org

2, 3, 11, 23, 31, 41, 59, 79, 97, 107, 113, 151, 163, 179, 197, 223, 227, 241, 257, 271, 337, 383, 433, 439, 467, 491, 547, 619, 773, 797, 853, 883, 887, 911, 967, 977, 1069, 1129, 1187, 1223, 1291, 1297, 1409, 1483, 1489, 1523, 1559, 1567, 1579, 1607, 1619
Offset: 1

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 31 2003

Keywords

Examples

			E.g. a(3) is the smallest prime > a(2)=3 which, when concatenated to 23 (which is the concatenation of a(1) and a(2)) gives a prime. Thus a(3)=11 because 235 and 237 are composite.

Crossrefs

Cf. A073640.

Programs

  • Maple
    with(numtheory): pout := [2,3]: nout := [1,2]: for n from 3 to 1000 do: p := ithprime(n): d := parse(cat(pout[nops(pout)-1],pout[nops(pout)],p)): if (isprime(d)) then pout := [op(pout),p]: nout := [op(nout),n]: fi: od: pout;
  • Mathematica
    a[1] = 2; a[2] = 3; a[n_] := a[n] = SelectFirst[Prime@ Range[#, 10^3 + #] &[PrimePi@ a[n - 1] + 1], PrimeQ@ FromDigits@ Join[IntegerDigits@ a[n - 2], IntegerDigits@ a[n - 1], IntegerDigits@ #] &]; Array[a, 51] (* Version 10, or *)
a[1] = 2; a[2] = 3; a[n_] := a[n] = Block[{p = PrimePi@ a[n - 1] + 1},
While[! PrimeQ@ FromDigits@ Join[IntegerDigits@ a[n - 2], IntegerDigits@ a[n - 1], IntegerDigits@ p], p = NextPrime@ p]; p]; Array[a, 51] (* Michael De Vlieger, Aug 15 2016 *)

Extensions

Edited by Charles R Greathouse IV, Apr 26 2010
Edited by Zak Seidov, Aug 15 2016

A080156 Values of n corresponding to the terms in sequence A080155. For any k, the concatenation of the a(1) to a(k)-th primes is prime and each value of k is the smallest for which this is true.

Original entry on oeis.org

1, 2, 5, 11, 15, 50, 54, 106, 148, 149, 156, 188, 200, 211, 248, 326, 359, 374, 417, 467, 699, 798, 891, 966, 1038, 1519, 1542, 1578, 1831, 1908, 1982, 2079, 2305, 2640, 2660, 2845, 3078, 3145, 3760, 3777, 3835, 4538, 4630, 4991, 5019, 5554, 5658, 5827
Offset: 1

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 31 2003

Keywords

Links

Crossrefs

Cf. A000720, A073640, A080152, A080153, A080154, A080155.

Programs

  • Maple
    with(numtheory): pout := [2]: nout := [1]: for n from 2 to 5000 do: p := ithprime(n): d := parse(cat(seq(pout[i],i=1..nops(pout)),p)): if (isprime(d)) then pout := [op(pout),p]: nout := [op(nout),n]: fi: od: nout;

Formula

a(n) = primepi(A080155(n)) = A000720(A080155(n)).

Extensions

Edited by Charles R Greathouse IV, Apr 30 2010

A080154 Values of n corresponding to the terms in sequence A080153.

Original entry on oeis.org

1, 2, 5, 9, 11, 13, 17, 22, 25, 28, 30, 36, 38, 41, 45, 48, 49, 53, 55, 58, 68, 76, 84, 85, 91, 94, 101, 114, 137, 139, 147, 153, 154, 156, 163, 165, 180, 189, 195, 200, 210, 211, 223, 235, 237, 241, 246, 247, 249, 253, 256, 272, 274, 286, 289, 293, 296, 306, 321, 323
Offset: 1

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 31 2003

Keywords

Examples

			E.g. the first three terms are 1, 2 and 5 because the concatenation of the first, 2nd and 11th primes is 2311 and this is prime. Also, the 5th prime is the first one after 3 for which this concatenation is prime.

Crossrefs

Cf. A000720, A073640, A080153.

Programs

  • Maple
    with(numtheory): pout := [2,3]: nout := [1,2]: for n from 3 to 600 do: p := ithprime(n): d := parse(cat(pout[nops(pout)-1],pout[nops(pout)],p)): if (isprime(d)) then pout := [op(pout),p]: nout := [op(nout),n]: fi: od: nout;

Formula

For each term a(n) in this sequence, A080153(n) = prime(a(n)).
a(n) = A000720(A080153(n)). - Sean A. Irvine, Sep 05 2025

Extensions

Edited by Charles R Greathouse IV, Apr 30 2010
Showing 1-6 of 6 results.

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