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.

A235154 Primes which have one or more occurrences of exactly two different digits.

Original entry on oeis.org

13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 113, 131, 151, 181, 191, 199, 211, 223, 227, 229, 233, 277, 311, 313, 331, 337, 353, 373, 383, 433, 443, 449, 499, 557, 577, 599, 661, 677, 727, 733, 757, 773, 787, 797, 811
Offset: 1

Views

Author

Colin Barker, Jan 04 2014

Keywords

Comments

The first term having a repeated digit is 101.
a(3402) > 10^10.

Links

Crossrefs

Cf. A034845, A235155-A235161, A030291.

Programs

  • PARI
    s=[]; forprime(n=10, 1000, if(#vecsort(eval(Vec(Str(n))),,8)==2, s=concat(s, n))); s
  • PARI
    is(n)=isprime(n) && #Set(digits(n))==2 \\ Charles R Greathouse IV, Feb 23 2017
  • PARI
    \\ See Corneth link
  • Python
    from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
from itertools import count, islice, combinations_with_replacement, product
def agen():
    for digits in count(2):
        s = set()
        for pair in product("0123456789", "1379"):
            if pair[0] == pair[1]: continue
            for c in combinations_with_replacement(pair, digits):
                if len(set(c)) < 2 or sum(int(ci) for ci in c)%3 == 0:
                    continue
                for p in multiset_permutations(c):
                    if p[0] == "0": continue
                    t = int("".join(p))
                    if isprime(t):
                        s.add(t)
        yield from sorted(s)
print(list(islice(agen(), 100))) # Michael S. Branicky, Jan 23 2022

A235156 Primes which have one or more occurrences of exactly four different digits.

Original entry on oeis.org

1039, 1049, 1063, 1069, 1087, 1093, 1097, 1237, 1249, 1259, 1279, 1283, 1289, 1297, 1307, 1327, 1367, 1409, 1423, 1427, 1429, 1439, 1453, 1459, 1483, 1487, 1489, 1493, 1523, 1543, 1549, 1567, 1579, 1583, 1597, 1607, 1609, 1627, 1637, 1657, 1693, 1697, 1709
Offset: 1

Views

Author

Colin Barker, Jan 04 2014

Keywords

Comments

The first term having a repeated digit is 10037.

Links

Crossrefs

Cf. A235154, A235155, A235157-A235161.
Cf. A074673.

Programs

  • PARI
    s=[]; forprime(n=1000, 2000, if(#vecsort(eval(Vec(Str(n))),,8)==4, s=concat(s, 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).

A057879 Primes with 3 distinct digits that remain prime (no leading zeros allowed) after deleting all occurrences of any one of its distinct digits.

Original entry on oeis.org

137, 173, 179, 197, 317, 431, 617, 719, 1531, 1831, 1997, 2113, 2131, 2237, 2273, 2297, 2311, 2797, 3137, 3371, 4337, 4373, 4733, 4919, 7297, 7331, 7573, 7873, 8191, 8311, 8831, 8837, 33413, 33713, 34313, 37313, 41117, 41999, 44417, 49199, 73331
Offset: 1

Views

Author

Patrick De Geest, Oct 15 2000

Keywords

Comments

Numbers in A057876 with exactly 3 distinct digits.

Crossrefs

Intersection of A057876 and A235155.
Cf. A057876-A057883, A051362, A034302-A034305.

Extensions

Offset changed by Andrew Howroyd, Aug 14 2024

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 1-5 of 5 results.

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