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 1-10 of 15 results. Next

A256786 Numbers which are divisible by prime(d) for all digits d in their decimal representation.

Original entry on oeis.org

12, 14, 42, 55, 154, 222, 228, 714, 1122, 1196, 1212, 1414, 2112, 2142, 2262, 3355, 4144, 4242, 5335, 5544, 5555, 6162, 9499, 11112, 11144, 11214, 11424, 11466, 11622, 11818, 11914, 12222, 12882, 14112, 15554, 16666, 21216, 21222, 21252, 21888, 22122, 22212
Offset: 1

Views

Author

Eric Angelini and Reinhard Zumkeller, Apr 10 2015

Keywords

Comments

All terms are zerofree, cf. A052382;
there is no term containing digits 1 and 3 simultaneously;
a(n) contains at least one digit 1 iff a(n) is even, cf. A011531, A005843;
a(n) contains at least one digit 2 iff a(n) mod 3 = 0, cf. A011532, A008585;
a(n) contains at least one digit 3 iff a(n) mod 10 = 5, cf. A011533, A017329;
A020639(a(n)) <= 23.
The equivalent in base 2 is the empty sequence, in base 3 it is A191681\{0}; see A256874-A256879 for the base 4 - base 9 variant, and A256870 for a variant where digits 0 are allowed but divisibility by prime(d+1) is required instead. - M. F. Hasler, Apr 11 2015

Examples

			Smallest terms containing the nonzero decimal digits:
.  d | prime(d) |  n | a(n)
. ---+----------+--------------------------
.  1 |       2  |  1 |   12 = 2^2 * 3
.  2 |       3  |  1 |   12 = 2^2 * 3
.  3 |       5  | 16 | 3355 = 5 * 11 * 61
.  4 |       7  |  2 |   14 = 2 * 7
.  5 |      11  |  4 |   55 = 5 * 11
.  6 |      13  | 10 | 1196 = 2^2 * 13 * 23
.  7 |      17  |  8 |  714 = 2 * 3 * 7 * 17
.  8 |      19  |  7 |  228 = 2^2 * 3 * 19
.  9 |      23  | 10 | 1196 = 2^2 * 13 * 23 .

Links

Crossrefs

Cf. A000040, A005843, A008585, A011531, A011532, A011533, A017329, A020639, A052382, A256874-A256879, A256882-A256884, A256865-A256870.

Programs

  • Haskell
    a256786 n = a256786_list !! (n-1)
a256786_list = filter f a052382_list where
   f x = g x where
     g z = z == 0 || x `mod` a000040 d == 0 && g z'
           where (z', d) = divMod z 10
  • Mathematica
    Select[Range@22222,FreeQ[IntegerDigits[#],0]&&Total[Mod[#,Prime[IntegerDigits[#]]]]==0&] (* Ivan N. Ianakiev, Apr 11 2015 *)
  • PARI
    is_A256786(n)=!for(i=1,#d=Set(digits(n)),(!d[i]||n%prime(d[i]))&&return) \\ M. F. Hasler, Apr 11 2015
  • Python
    primes = [1, 2, 3, 5, 7, 11, 13, 17, 19, 23]
def ok(n):
    s = str(n)
    return "0" not in s and all(n%primes[int(d)] == 0 for d in s)
print([k for k in range(22213) if ok(k)]) # Michael S. Branicky, Dec 14 2021

A256879 Numbers divisible by prime(d) for each digit d in their base-9 representation, none of which may be zero.

Original entry on oeis.org

10, 30, 195, 275, 280, 364, 429, 546, 646, 820, 840, 1000, 1144, 1360, 1560, 1650, 2280, 2370, 2440, 2460, 2640, 2730, 3010, 3740, 4114, 4940, 5236, 5928, 6555, 7800, 8018, 8130, 8850, 8940, 8970, 9030, 9100, 9660, 9730, 9814, 10868, 11050, 11076, 14352, 14700, 14820, 15015, 15420, 18564, 20670, 21090, 21405, 22225
Offset: 1

Views

Author

M. F. Hasler, Apr 11 2015

Keywords

Comments

Base-9 analog of A256786. See A256874 - A256878 for the base-3, ..., base-8 analogs.
See A256869 for a variant where divisibility by prime(d+1) is required instead.

Crossrefs

Cf. A256786, A256874 - A256878, A256869, A256865 - A256870.

Programs

  • PARI
    is(n,b=9)=!for(i=1,#d=Set(digits(n,b)),(!d[i]||n%prime(d[i]))&&return)

A256884 Numbers divisible by prime(d+1) for each digit d of their base-4 representation.

Original entry on oeis.org

0, 10, 21, 40, 63, 84, 90, 105, 130, 140, 150, 160, 165, 170, 175, 210, 252, 276, 324, 330, 336, 345, 360, 390, 405, 420, 520, 560, 600, 630, 640, 650, 660, 680, 700, 735, 770, 840, 861, 910, 1008, 1044, 1050, 1092, 1104, 1110, 1170, 1260, 1284, 1290, 1296, 1320, 1344, 1350, 1365, 1380, 1407, 1410, 1440, 1470, 1533, 1560, 1620
Offset: 1

Views

Author

M. F. Hasler, Apr 11 2015

Keywords

Comments

The base-4 variant of A256882, A256883, A256865, ..., A256870 in bases 2, ..., 10.
A variant of A256874 where digits 0 are forbidden and divisibility by prime(d) is required.

Examples

			0 is divisible by prime(0+1)=2.
10 = 22_4 and is divisible by prime(2+1)=5.
n = 1, 2, 3 are not divisible by prime(n+1) = 3, 5, 7; nor are 4=10_4, 5=11_4, and 7=13_4 divisible by prime(1+1) = 3; nor are 6=12_4, 8=20_4, 9=21_4 divisible by prime(2+1)=5.

Links

Crossrefs

Cf. A256882, A256883, A256865-A256870, A256874-A256879, A256786.

Programs

  • Mathematica
    Select[Range[0,2000],AllTrue[#/(Prime[#+1]&/@IntegerDigits[#,4]),IntegerQ]&] (* Harvey P. Dale, Mar 13 2025 *)
  • PARI
    is(n,b=4)=!for(i=1,#d=Set(digits(n,b)),n%prime(d[i]+1)&&return)

A256865 Numbers divisible by prime(d+1) for each digit d of their base-5 representation.

Original entry on oeis.org

0, 6, 10, 30, 50, 60, 110, 126, 150, 156, 168, 180, 210, 231, 250, 260, 300, 310, 378, 550, 630, 660, 726, 750, 756, 780, 810, 840, 900, 930, 1008, 1050, 1250, 1260, 1300, 1310, 1320, 1380, 1410, 1500, 1510, 1530, 1550, 1560, 1680, 1760, 1870, 1890, 1960, 2016, 2268, 2310, 2331, 2618, 2750, 2860, 3124, 3126, 3150, 3156, 3180
Offset: 1

Views

Author

M. F. Hasler, Apr 11 2015

Keywords

Comments

The base-5 variant of A256882 - A256884, A256866 - A256870 in bases 2, ..., 10.
A variant of A256875 where digits 0 are forbidden and divisibility by prime(d) is required.

Examples

			0 is divisible by prime(0+1)=2.
6 = 11_5 and is divisible by prime(1+1)=3.
10 = 20_5 and is divisible by prime(0+1)=2 and by prime(2+1)=5.
n = 1, 2, 3, 4 are not divisible by prime(n+1) = 3, 5, 7, 11. 5=10_5 is not divisible by prime(0+1)=2; 7=12_5 is not divisible by prime(2+1)=5, etc.

Crossrefs

Cf. A256882, A256883, A256884, A256866-A256870, A256874-A256879, A256786.

Programs

  • PARI
    is(n,b=5)=!for(i=1,#d=Set(digits(n,b)),n%prime(d[i]+1)&&return)

A256870 Numbers divisible by prime(d+1) for each digit d of their base-10 representation.

Original entry on oeis.org

0, 20, 44, 111, 120, 171, 200, 210, 220, 290, 440, 520, 1020, 1110, 1113, 1200, 1710, 1914, 2000, 2010, 2020, 2030, 2100, 2145, 2200, 2220, 2310, 2420, 2900, 3220, 3381, 4004, 4048, 4400, 4444, 5200, 5525, 6120, 7220, 8280, 9338
Offset: 1

Views

Author

M. F. Hasler, Apr 11 2015

Keywords

Comments

A variant of A256786 where digits 0 are forbidden and divisibility by prime(d) is required.
See A256882 - A256884, A256866 - A256869 for the analog in bases 2, ..., 9.

Examples

			0 is divisible by prime(0+1)=2.
n = 1,...,9 are not divisible by prime(n+1) = 3, 5, ..., 29, respectively.
20 is divisible by prime(2+1)=5 and by prime(0+1)=2. The same is true for any other 2...20...0 =  2*10^k*(10^m-1)/9; k >= 1, m >= 0.
44 is divisible by prime(4+1)=11.

Crossrefs

Cf. A256882, A256883, A256884, A256865 - A256869, A256874 - A256879, A256786.

Programs

  • PARI
    is(n,b=10)=!for(i=1,#d=Set(digits(n,b)),n%prime(d[i]+1)&&return)

A256882 Numbers divisible by prime(d+1) for each digit d of their base-2 representation.

Original entry on oeis.org

0, 3, 6, 12, 15, 18, 24, 30, 36, 42, 48, 54, 60, 63, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 216, 222, 228, 234, 240, 246, 252, 255, 258, 264, 270, 276, 282, 288, 294, 300, 306, 312, 318, 324, 330, 336, 342, 348, 354, 360, 366, 372, 378, 384, 390, 396, 402
Offset: 1

Views

Author

M. F. Hasler, Apr 11 2015

Keywords

Comments

Also, numbers divisible by 3 which are even or have no digit zero in their base-2 representation.
The base-2 variant of A256883 - A256884, A256865 - A256870 in bases 3,...,10.

Examples

			6 = 110_2 (in base 6), and 6 is divisible by prime(1+1)=3 and by prime(0+1) = 2.
9 = 101_2 is not in the sequence because it is odd but has a binary digit 0.

Crossrefs

Cf. A256883 - A256884, A256865 - A256870, A256786, A256874 - A256879.

Programs

  • PARI
    is(n,b=2)=!for(i=1,#d=Set(digits(n,b)),n%prime(d[i]+1)&&return)

A256883 Numbers divisible by prime(d+1) for each digit d of their base-3 representation.

Original entry on oeis.org

0, 12, 20, 30, 36, 60, 80, 84, 90, 108, 120, 150, 170, 180, 210, 240, 246, 252, 270, 282, 300, 324, 330, 336, 354, 360, 390, 420, 450, 480, 510, 540, 560, 570, 600, 630, 650, 660, 690, 710, 720, 732, 738, 750, 756, 768, 780, 810, 822, 840, 846, 870, 900, 930, 960, 972, 984, 990, 1002, 1008, 1020, 1050, 1056, 1062, 1080, 1092, 1110
Offset: 1

Views

Author

M. F. Hasler, Apr 11 2015

Keywords

Comments

The base-3 variant of A256882, A256884, A256865, ..., A256870 in bases 2, ..., 10.

Examples

			0 is divisible by prime(0+1)=2.
12 = 110_3 (in base 3), and is divisible by prime(1+1)=3 and by prime(0+1)=2.

Crossrefs

Cf. A256882 - A256884, A256865 - A256870, A256874 - A256879, A256786.

Programs

  • PARI
    is(n,b=3)=!for(i=1,#d=Set(digits(n,b)),n%prime(d[i]+1)&&return)

A256866 Numbers divisible by prime(d+1) for each digit d of their base-6 representation.

Original entry on oeis.org

0, 6, 21, 36, 42, 126, 216, 222, 252, 258, 273, 300, 480, 510, 525, 560, 693, 735, 756, 770, 777, 880, 924, 1001, 1012, 1296, 1302, 1320, 1326, 1332, 1338, 1380, 1482, 1512, 1518, 1548, 1554, 1560, 1590, 1638, 1716, 1740, 1770, 1785, 1800, 1995, 2058, 2079, 2310, 2418
Offset: 1

Views

Author

M. F. Hasler, Apr 11 2015

Keywords

Comments

The base-6 variant of A256882 - A256884, A256866 - A256870 in bases 2, ..., 10.
A variant of A256876 where digits 0 are forbidden and divisibility by prime(d) is required.

Examples

			0 is divisible by prime(0+1)=2.
6 = 10_6 in base 6, and is divisible by prime(1+1)=3 and by prime(0+1)=2.
21 = 33_6, and is divisible by prime(3+1)=7.
n = 1, 2, 3, 4, 5 are not divisible by prime(n+1) = 3, 5, 7, 11, 13, respectively. 7=11_6, 8=12_6, 10=13_6, ... are not divisible by prime(1+1)=3; 9=12_7 is not divisible by prime(2+1)=5, etc.

Crossrefs

Cf. A256882, A256883, A256884, A256865-A256870, A256874-A256879, A256786.

Programs

  • PARI
    is(n,b=6)=!for(i=1,#d=Set(digits(n,b)),n%prime(d[i]+1)&&return)

A256869 Numbers divisible by prime(d+1) for each digit d of their base-9 representation.

Original entry on oeis.org

0, 20, 84, 90, 180, 252, 273, 455, 510, 738, 756, 810, 816, 840, 900, 1224, 1275, 1460, 1470, 1620, 1640, 2090, 2185, 2268, 2450, 2541, 3080, 3289, 3553, 4199, 4590, 5434, 6570, 6642, 6660, 6804, 7290, 7326, 7344, 7380, 7395, 7470, 7560, 7866, 8100, 8160, 8190, 8778, 8841, 8925, 9282
Offset: 1

Views

Author

M. F. Hasler, Apr 11 2015

Keywords

Comments

The base-9 variant of A256882 - A256884, A256866 - A256870 in bases 2, ..., 10.
A variant of A256879 where digits 0 are forbidden and divisibility by prime(d) is required.
From Robert Israel, Aug 01 2019: (Start)
If n is in the sequence and is even, then 9*n is in the sequence.
If n is in the sequence and 9^k > n, then (9^k+1)*n is in the sequence.
All multiples of 223092870 are in the sequence.
(End)

Links

Crossrefs

Cf. A256882, A256883, A256884, A256865-A256870, A256874-A256879, A256786.

Programs

  • Maple
    P:= [seq(ithprime(i),i=1..9)]:
filter:= proc(n) local L;
  L:= convert(convert(n,base,9),set);
  L:= map(t -> P[t+1],L);
  n mod convert(L,`*`) = 0
end proc:
select(filter, [$0..10000]); # Robert Israel, Aug 01 2019
  • PARI
    is(n,b=9)=!for(i=1,#d=Set(digits(n,b)),n%prime(d[i]+1)&&return)

Formula

a(n) ~ 223092870*n. - Robert Israel, Aug 01 2019

A256875 Numbers divisible by prime(d) for each digit d in their base-5 representation, none of which may be zero.

Original entry on oeis.org

6, 12, 36, 156, 162, 186, 282, 306, 312, 546, 784, 786, 906, 912, 924, 936, 1246, 1536, 1806, 2674, 2814, 2856, 3906, 3912, 3936, 4032, 4056, 4062, 4074, 4284, 4536, 4656, 4662, 4686, 4746, 4914, 5796, 5922, 6174, 7032, 7056, 7062, 7182, 7434, 7656, 7662, 7686, 7782, 7806, 7812, 8064, 8106, 8946, 9072, 9114, 9282, 9324
Offset: 1

Views

Author

M. F. Hasler, Apr 11 2015

Keywords

Comments

Base-5 analog of A256786. See A256874 - A256879 for the base-3, ..., base-9 analogs.
See A256865 for a variant where divisibility by prime(d+1) is required instead.
From Robert Israel, Apr 04 2024: (Start)
Since digit 0 is not allowed, terms can't be divisible by 5, so digit 3 is also not allowed.
All terms are even. (End)

Links

Crossrefs

Cf. A256786, A256874 - A256879, A256882 - A256884, A256865 - A256870.

Programs

  • Maple
    P:= [seq(ithprime(i),i=1..4)]:
filter:= proc(n) local S,s;
  S:= convert(convert(n,base,5),set);
  if member(0,S) then return false fi;
  n mod mul(P[s],s=S) = 0
end proc:
select(filter, [$1..10^4]); # Robert Israel, Apr 04 2024
  • PARI
    is(n,b=5)=!for(i=1,#d=Set(digits(n,b)),(!d[i]||n%prime(d[i]))&&return)
Showing 1-10 of 15 results. Next

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