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.

A235690 Semiprimes which have one or more occurrences of exactly two different digits.

Original entry on oeis.org

10, 14, 15, 21, 25, 26, 34, 35, 38, 39, 46, 49, 51, 57, 58, 62, 65, 69, 74, 82, 85, 86, 87, 91, 93, 94, 95, 115, 118, 119, 121, 122, 133, 141, 155, 161, 166, 177, 202, 221, 226, 262, 299, 303, 323, 334, 335, 339, 355, 377, 393, 411, 422, 445, 446, 447, 454
Offset: 1

Views

Author

Colin Barker, Jan 14 2014

Keywords

Comments

The first term having a repeated digit is 115.

Examples

			1000000000010101 is a term because it is made of the digits 0 and 1 and it is the product of the two primes 18463559 and 54160739.

Crossrefs

Cf. A235691-A235696.
Cf. A235154-A235161.

Programs

  • Mathematica
    Select[Range[454], Length@Union@ IntegerDigits[#] == 2 && Total[Last /@ FactorInteger[#]] == 2 &] (* Giovanni Resta, Jan 14 2014 *)
  • PARI
    list(lim)=my(v=List(), t); forprime(p=2, sqrt(lim), t=p; forprime(q=p, lim\t, listput(v, t*q))); vecsort(Vec(v)) \\ From A001358
b=list(10000); s=[]; for(n=1, #b, if(#vecsort(eval(Vec(Str(b[n]))),,8)==2, s=concat(s, b[n]))); s

A337313 a(n) is the number of n-digit positive integers with exactly three distinct base 10 digits.

Original entry on oeis.org

0, 0, 648, 3888, 16200, 58320, 195048, 625968, 1960200, 6045840, 18468648, 56068848, 169533000, 511252560, 1539065448, 4627812528, 13904670600, 41756478480, 125354369448, 376232977008, 1129038669000, 3387795483600, 10164745404648, 30496954122288, 91496298184200
Offset: 1

Views

Author

Stefano Spezia, Aug 22 2020

Keywords

Comments

a(n) is the number of n-digit numbers in A031962.

Examples

			a(1) = a(2) = 0 since the positive integers must have at least three digits;
a(3) = #{xyz in N | x,y,z are three different digits with x != 0} = 9*9*8 = 648;
a(4) = 3888 since #[9999] - #[999] - #(1111*[9]) - A335843(4) - #{xywz in N | x,y,w,z are four different digits with x != 0} = 9999 - 999 - 9 - 567 - 9*9*8*7 = 3888;
...

Links

Crossrefs

Cf. A000225, A000244, A000392, A031962, A048993, A335843, A337127.
Cf. A055642, A180599, A235155, A235691, A235718.

Programs

  • Mathematica
    LinearRecurrence[{6,-11,6},{0,0,648},26]
  • PARI
    concat([0,0],Vec(648*x^3/(1-6*x+11*x^2-6*x^3)+O(x^26)))

Formula

O.g.f.: 648*x^3/(1 - 6*x + 11*x^2 - 6*x^3).
E.g.f.: 108*(exp(x) - 1)^3.
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n > 3.
a(n) = 648*S2(n, 3) where S2(n, 3) = A000392(n).
a(n) = 324*(3^(n-1) - 2^n + 1).
a(n) ~ 108 * 3^n.
a(n) = 324*(A000244(n-1) - A000225(n)).
a(n) = A337127(n, 3).

A235692 Semiprimes which have one or more occurrences of exactly four different digits.

Original entry on oeis.org

1027, 1037, 1042, 1043, 1046, 1047, 1057, 1059, 1067, 1073, 1079, 1082, 1094, 1203, 1205, 1207, 1234, 1238, 1243, 1247, 1253, 1257, 1263, 1267, 1273, 1285, 1286, 1293, 1294, 1306, 1329, 1345, 1346, 1347, 1349, 1354, 1357, 1369, 1379, 1382, 1385, 1387, 1389
Offset: 1

Views

Author

Colin Barker, Jan 14 2014

Keywords

Comments

The first term having a repeated digit is 10027.

Crossrefs

Cf. A235690, A235691, A235693-A235696.
Cf. A235154-A235161.

Programs

  • PARI
    list(lim)=my(v=List(), t); forprime(p=2, sqrt(lim), t=p; forprime(q=p, lim\t, listput(v, t*q))); vecsort(Vec(v)) \\ From A001358
b=list(15000); s=[]; for(n=1, #b, if(#vecsort(eval(Vec(Str(b[n]))),,8)==4, s=concat(s, b[n]))); s

A337314 a(n) is the number of n-digit positive integers with exactly four distinct base 10 digits.

Original entry on oeis.org

0, 0, 0, 4536, 45360, 294840, 1587600, 7715736, 35244720, 154700280, 661122000, 2773768536, 11487556080, 47136955320, 192126589200, 779279814936, 3149513947440, 12695388483960, 51073849285200, 205172877726936, 823325141746800, 3301203837670200, 13228529919066000
Offset: 1

Views

Author

Stefano Spezia, Sep 26 2020

Keywords

Comments

a(n) is the number of n-digit numbers in A031969.

Examples

			a(1) = a(2) = a(3) = 0 since the positive integers must have at least four digits;
a(4) = #{wxyz in N | w,x,y,z are four different digits with w != 0} = A073531(4) = 4536;
a(5) = 45360 since #[99999] - #[9999] - #(11111*[9]) - A335843(5) - A337313(5) - #{vwxyz in N | v,w,x,y,z are five different digits with v != 0} = 99999 - 9999 - 9 - 1215 - 16200 - 9*9*8*7*6 = 45360;
...

Links

Crossrefs

Cf. A000079, A000244, A000302, A000453, A031969, A073531, A335843, A337313, A337127.
Cf. A055642, A180599, A235155, A235691, A235718.

Programs

  • Mathematica
    LinearRecurrence[{10,-35,50,-24},{0,0,0,4536},23]
  • PARI
    concat([0,0,0],Vec(4536*x^4/(1-10*x+35*x^2-50*x^3+24*x^4)+O(x^24)))

Formula

O.g.f.: 4536*x^4/(1 - 10*x + 35*x^2 - 50*x^3 + 24*x^4).
E.g.f.: 189*(exp(x) - 1)^4.
a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4) for n > 4.
a(n) = 4536*S2(n, 4) where S2(n, 4) = A000453(n).
a(n) = 189*(4^n - 4*3^n + 3*2^(n+1) - 4).
a(n) ~ 189 * 4^n.
a(n) = 189*(A000302(n) - 4*A000244(n) + 3*A000079(n+1) - 4).
a(n) = A337127(n, 4).
Showing 1-4 of 4 results.

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