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 1-5 of 5 results.

A219743 Number for which the number of distinct base 10 digits is 8.

Original entry on oeis.org

10234567, 10234568, 10234569, 10234576, 10234578, 10234579, 10234586, 10234587, 10234589, 10234596, 10234597, 10234598, 10234657, 10234658, 10234659, 10234675, 10234678, 10234679, 10234685, 10234687, 10234689, 10234695, 10234697, 10234698
Offset: 1

Views

Author

Jonathan Vos Post, Dec 05 2012

Keywords

Links

Crossrefs

Cf. A010785 (1 digits), A031955 (2 digits), A031962 (3 digits), A031969 (4 digits), A031987 (5 digits), A220076 (6 digits), A218019 (7 digits), A116670 (9 digits), A171102 (10 digits).

Programs

  • Mathematica
    Select[Range[10^7, 10^7 + 1000000], Length[Union[IntegerDigits[#]]] == 8 &] (* T. D. Noe, Dec 05 2012 *)

Extensions

Corrected and extended by T. D. Noe, Dec 05 2012

A337127 Table with 10 columns read by rows: T(n, k) is the number of n-digit positive integers with exactly k distinct base 10 digits (0 < k <= 10).

Original entry on oeis.org

9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 81, 0, 0, 0, 0, 0, 0, 0, 0, 9, 243, 648, 0, 0, 0, 0, 0, 0, 0, 9, 567, 3888, 4536, 0, 0, 0, 0, 0, 0, 9, 1215, 16200, 45360, 27216, 0, 0, 0, 0, 0, 9, 2511, 58320, 294840, 408240, 136080, 0, 0, 0, 0, 9, 5103, 195048, 1587600, 3810240, 2857680, 544320, 0, 0, 0
Offset: 1

Views

Author

Stefano Spezia, Aug 17 2020

Keywords

Examples

			The table T(n, k) begins:
9     0      0       0       0       0  0  0  0  0
9    81      0       0       0       0  0  0  0  0
9   243    648       0       0       0  0  0  0  0
9   567   3888    4536       0       0  0  0  0  0
9  1215  16200   45360   27216       0  0  0  0  0
9  2511  58320  294840  408240  136080  0  0  0  0
...

Links

Crossrefs

Cf. A010785, A031955, A031962, A031969, A031987, A116670, A171102, A218019, A219743, A220076.
Cf. A010734, A048993, A052268 (row sums), A073531 (diagonal), A180599 (k = 1), A335843 (k = 2), A337313 (k = 3).

Programs

  • Mathematica
    T[n_,k_]:=9Pochhammer[11-k,k-1]/k!*n!*Coefficient[Series[(Exp[x]-1)^k,{x,0,n}],x,n]; Table[T[n,k],{n,7},{k,10}]//Flatten

Formula

T(n, k) = 9*Pochhammer(11-k, k-1)*n! * [x^n] (exp(x) - 1)^k/k!.
T(n, k) = 9*Pochhammer(11-k, k-1) * [x^n] x^k/Product_{j=1..k} (1-j*x).
T(n, k) = 9*Pochhammer(11-k, k-1)*S2(n, k) where S2(n, k) = A048993(n, k) are the Stirling numbers of the 2nd kind.

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).

A303504 Integers whose digits, together with a single supplementary digit, cannot be reordered to form a base-10 palindrome number.

Original entry on oeis.org

102, 103, 104, 105, 106, 107, 108, 109, 120, 123, 124, 125, 126, 127, 128, 129, 130, 132, 134, 135, 136, 137, 138, 139, 140, 142, 143, 145, 146, 147, 148, 149, 150, 152, 153, 154, 156, 157, 158, 159, 160, 162, 163, 164, 165, 167, 168, 169, 170, 172, 173, 174, 175, 176, 178, 179, 180, 182, 183
Offset: 1

Views

Author

Eric Angelini and Jean-Marc Falcoz, Apr 25 2018

Keywords

Comments

This is the complement of A303502.
Starts with the 648 terms of A031962. - Georg Fischer, Oct 08 2018

Examples

			a(1) = 102 together with a single 0 can form 1002, 1020, 1200, 2001, 2010 and 2100, but none of these are palindromes;
a(1) = 102 together with a single 1 can form 1012, 1021, 1102, 1120, 1201, 1210, 2011, 2101 and 2110, but none of these are palindromes;
etc.

Links

Crossrefs

Cf. A002113 (Palindromes in base 10), A303502 (complement of this sequence).

A288040 Integers whose number of distinct decimal digits is prime.

Original entry on oeis.org

10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100, 101
Offset: 1

Views

Author

Jonathan Frech, Jun 04 2017

Keywords

Comments

Differs from A139819 (which contains, for example, 1234, a number with 4 distinct decimal digits). - R. J. Mathar, Jun 14 2017

Crossrefs

Union of A031955 and A031962 and ....

Programs

  • Mathematica
    Select[Range@ 101, PrimeQ@ Count[DigitCount[#], ?(# != 0 &)] &] (* _Michael De Vlieger, Jun 06 2017 *)
  • PARI
    isok(m) = isprime(#Set(digits(m))); \\ Michel Marcus, May 10 2020
  • Python
    from sympy import isprime
print([n for n in range(1, 100) if isprime(len(set(str(n))))])
Showing 1-5 of 5 results.

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