A007091 -id:A007091 - cp's OEIS frontend

A037381 Numbers k such that every base-3 digit of k is a base-5 digit of k.

1, 2, 7, 27, 28, 30, 35, 40, 51, 54, 55, 60, 71, 121, 127, 132, 135, 136, 137, 138, 139, 142, 147, 152, 157, 160, 161, 175, 176, 177, 178, 179, 180, 185, 190, 195, 202, 210, 211, 212, 214, 227, 232, 235, 238, 239, 242, 251, 255

Offset: 1

Clark Kimberling

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007089, A007091.

Cf. A037380, A037382, A037383, A037384, A037385, A037386.

import Data.List ((\\), nub)
a037381 n = a037381_list !! (n-1)
a037381_list = filter f [1..] where
   f x = null $ nub (ds 3 x) \\ nub (ds 5 x)
   ds b x = if x > 0 then d : ds b x' else []  where (x', d) = divMod x b
-- Reinhard Zumkeller, May 30 2013

Select[Range[300],SubsetQ[IntegerDigits[#,5],IntegerDigits[#,3]]&] (* Harvey P. Dale, Dec 31 2017 *)

A037396 Numbers k such that every base-5 digit of k is a base-9 digit of k.

Original entry on oeis.org

1, 2, 3, 4, 11, 22, 93, 121, 124, 126, 156, 181, 199, 317, 362, 598, 750, 751, 752, 755, 756, 758, 768, 770, 771, 775, 776, 780, 781, 785, 796, 812, 831, 841, 843, 849, 859, 895, 900, 906, 907, 911, 912, 918, 922, 927, 931, 932

Offset: 1

Clark Kimberling

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007091, A007095.

Cf. A037393, A037394, A037395, A037397.

import Data.List ((\\), nub)
a037396 n = a037396_list !! (n-1)
a037396_list = filter f [1..] where
   f x = null $ nub (ds 5 x) \\ nub (ds 9 x)
   ds b x = if x > 0 then d : ds b x' else []  where (x', d) = divMod x b
-- Reinhard Zumkeller, May 30 2013

is(n)=#setminus(Set(digits(n,5)), Set(digits(n,9)))==0 \\ Charles R Greathouse IV, Feb 11 2017

A043279 Maximal run length in base 5 representation of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 3, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 3, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2

Offset: 1

Clark Kimberling

Winston de Greef, Table of n, a(n) for n = 1..10000

Cf. A007091.

Cf. A043276-A043290 for base-2 to base-16 analogs.

A043279[n_]:=Max[Map[Length,Split[IntegerDigits[n,5]]]];Array[A043279,100] (* Paolo Xausa, Sep 27 2023 *)

A043279(n, b=5)={my(m,c=1); while(n>0, n%b==(n\=b)%b && c++ && next; m=max(m, c); c=1); m} \\ M. F. Hasler, Jul 23 2013

A110603 Numbers n whose base 5 representations, interpreted as base 10 integers, are semiprimes.

Original entry on oeis.org

4, 5, 9, 11, 12, 18, 19, 31, 36, 37, 38, 43, 44, 46, 47, 48, 51, 52, 53, 58, 59, 61, 76, 77, 78, 84, 86, 88, 94, 96, 103, 106, 108, 112, 128, 131, 146, 147, 148, 151, 156, 159, 161, 168, 171, 172, 177, 178, 181, 184, 194, 196, 198, 208, 212, 218, 223, 226, 227, 228

Offset: 1

Jonathan Vos Post, Jul 30 2005

A108873 is the equivalent using base 3. A110602 is the equivalent using base 4. A110604 is the equivalent using base 6. A110605 is the equivalent using base 7. A110606 is the equivalent using base 8. A110607 is the equivalent using base 9.

			a(1) = 4 because 4 (base 5) = 4 and 4 (base 10) = 2 * 2, a semiprime (A001358).
a(2) = 5 because 5 (base 5) = 10 and 10 (base 10) = 2 * 5.
a(3) = 9 because 9 (base 5) = 14 and 14 (base 10) = 2 * 7.
a(4) = 11 because 11 (base 5) = 21 and 21 (base 10) = 3 * 7.
a(5) = 12 because 12 (base 5) = 22 and 22 (base 10) = 2 * 11.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001358, A007091, A108873, A110602, A110604, A110605, A110606, A110607.

Select[Range[228], Plus @@ Last /@ FactorInteger[FromDigits[IntegerDigits[ #, 5]]] == 2 &] (* Ray Chandler, Aug 05 2005 *)
Select[Range[300],PrimeOmega[FromDigits[IntegerDigits[#,5],10]]==2&] (* Harvey P. Dale, Aug 14 2023 *)

Corrected and extended by Ray Chandler, Aug 05 2005

A262102 Pseudoprimes to base 5, written in base 5.

Original entry on oeis.org

4, 444, 1332, 4221, 11111, 22131, 23404, 30031, 42241, 112443, 133321, 134421, 140122, 140411, 202401, 214244, 222223, 224104, 241121, 304011, 323131, 331401, 402201, 404041, 411313, 421411, 434411, 1001001, 1001331, 1010142, 1032032, 1140421, 1212131, 1224103, 1233321, 1302302, 1302401, 1414331, 1421124, 1440143

Offset: 1

Abdul Gaffar Khan, Sep 11 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007091 (numbers in base 5), A005936 (pseudoprimes to base 5).

Cf. A258189, A262101, A262103, A262104, A262105, A262154.

base = 5; t = {}; n = 1;
While[Length[t] < 40, n++;
If[! PrimeQ[n] && PowerMod[base, n - 1, n] == 1,
  AppendTo[t, FromDigits@IntegerDigits[n, 5]]]]; t

lista(nn, b=5) = {for (n=1, nn, if (Mod(b, n)^(n-1)==1 && !ispseudoprime(n) && n>1, print1(subst(Pol(digits(n,b), x), x, 10), ", ");););} \\ Michel Marcus, Sep 30 2015

a(n) = A007091(A005936(n)).

A293658 Base-5 circular primes that are not base-5 repunits.

Original entry on oeis.org

7, 11, 13, 17, 19, 23, 167, 211, 239, 283, 359, 431, 547, 571, 1069, 1249, 1733, 2221, 2417, 2713, 2749, 3049, 3109, 3121

Offset: 1

Felix Fröhlich, Oct 28 2017

Conjecture: The sequence is finite, with 3121 being the last term (see A293142).
Written in base 5 (A007091), the terms are 12, 21, 23, 32, 34, 43, 1132, 1321, 1424, 2113, 2414, 3211, 4142, 4241, 13234, 14444, 23413, 32341, 34132, 41323, 41444, 44144, 44414, 44441. - Antti Karttunen, Nov 26 2017
a(25), if it exists, must be larger than prime(10^6) = 15485863, an 11-digit quinary number. - Michael De Vlieger, Nov 26 2017

			1069 written in base 5 is 13234. The base-5 numbers 13234, 32341, 23413, 34132, 41323 written in base 10 are 1069, 2221, 1733, 2417, 2713, respectively and all those numbers are prime, so 1069, 1733, 2221, 2417 and 2713 are terms of the sequence.

Cf. A007091, A293142.

Cf. base-b nonrepunit circular primes: A293657 (b=4), A293659 (b=6), A293660 (b=7), A293661 (b=8), A293662 (b=9), A293663 (b=10).

Select[Array[Map[If[Union@ # == {1}, 0, FromDigits[#, 5]] &, NestList[RotateLeft, #, Length@ # - 1]] &@ IntegerDigits[Prime@ #, 5] &, 10^5, 4], AllTrue[#, PrimeQ] &][[All, 1]] (* Michael De Vlieger, Nov 26 2017 *)

rot(n) = if(#Str(n)==1, v=vector(1), v=vector(#n-1)); for(i=2, #n, v[i-1]=n[i]); u=vector(#n); for(i=1, #n, u[i]=n[i]); v=concat(v, u[1]); v
decimal(v, base) = my(w=[]); for(k=0, #v-1, w=concat(w, v[#v-k]*base^k)); sum(i=1, #w, w[i])
is_circularprime(p, base) = my(db=digits(p, base), r=rot(db), i=0); if(vecmin(db)==0, return(0), while(1, dec=decimal(r, base); if(!ispseudoprime(dec), return(0)); r=rot(r); if(r==db, return(1))))
forprime(p=1, , if(vecmin(digits(p, 5))!=vecmax(digits(p, 5)), if(is_circularprime(p, 5), print1(p, ", "))))

A353113 Base-5 representation of A000422(n).

Original entry on oeis.org

1, 41, 2241, 114241, 3214241, 131414241, 3424414241, 134414414241, 4010314414241, 140000314414241, 121200300314414241, 111333240300314414241, 102224302240300314414241, 43323231102240300314414241, 40130121131102240300314414241

Offset: 1

Seiichi Manyama, Apr 24 2022

Cf. A000422, A353110, A353111, A353112, A353114, A353115, A353116, A353117.

Cf. A007091, A353105.

def A(k, n)
  (1..n).map{|i| (1..i).to_a.reverse.join.to_i.to_s(k).to_i}
end
p A(5, 20)

a(n) = A007091(A000422(n)).

A004053 For m=2,3,..., write m in bases 2,3,..,m.

Original entry on oeis.org

10, 11, 10, 100, 11, 10, 101, 12, 11, 10, 110, 20, 12, 11, 10, 111, 21, 13, 12, 11, 10, 1000, 22, 20, 13, 12, 11, 10, 1001, 100, 21, 14, 13, 12, 11, 10, 1010, 101, 22, 20, 14, 13, 12, 11, 10, 1011, 102, 23, 21, 15, 14, 13, 12, 11, 10, 1100, 110, 30, 22, 20, 15, 14, 13, 12, 11, 10

Offset: 2

Johan Boye (johbo(AT)ida.liu.se)

			Triangle begins:
    10;
    11,  10;
   100,  11, 10;
   101,  12, 11, 10;
   110,  20, 12, 11, 10;
   111,  21, 13, 12, 11, 10;
  1000,  22, 20, 13, 12, 11, 10;
  1001, 100, 21, 14, 13, 12, 11, 10;
  ...

Alois P. Heinz, Rows n = 2..20, flattened (row 21 would require digits > 9)

Cf. A007088, A007090, A007091, A007092, A007093, A007094, A007095.

FromDigits/@Flatten[Table[IntegerDigits[m,b],{m,2,20},{b,2,m}],1] (* Harvey P. Dale, Dec 01 2024 *)

T(n, k) = fromdigits(digits(n, k), 10);
tabl(nn) = for (n=2, nn, for (b=2, n, print1(T(n, b), ", "))); \\ Michel Marcus, Aug 30 2019

A037387 Numbers k such that every base-4 digit of k is a base-5 digit of k.

Original entry on oeis.org

1, 2, 3, 5, 10, 15, 21, 28, 37, 38, 42, 58, 63, 76, 80, 86, 132, 136, 137, 138, 142, 152, 160, 167, 178, 183, 190, 191, 202, 204, 205, 210, 213, 214, 215, 217, 220, 221, 222, 223, 238, 240, 256, 257, 258, 261, 266, 276, 277, 278

Offset: 1

Clark Kimberling

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007090, A007091.

Cf. A037388, A037389, A037390, A037391, A037392.

import Data.List ((\\), nub)
a037387 n = a037387_list !! (n-1)
a037387_list = filter f [1..] where
   f x = null $ nub (ds 4 x) \\ nub (ds 5 x)
   ds b x = if x > 0 then d : ds b x' else []  where (x', d) = divMod x b
-- Reinhard Zumkeller, May 30 2013

Select[Range[300],SubsetQ[IntegerDigits[#,5],IntegerDigits[#,4]]&] (* Harvey P. Dale, Mar 27 2019 *)

A037393 Numbers k such that every base-5 digit of k is a base-6 digit of k.

Original entry on oeis.org

1, 2, 3, 4, 6, 12, 18, 24, 31, 46, 56, 62, 75, 81, 87, 90, 91, 92, 93, 94, 99, 118, 124, 145, 150, 157, 226, 232, 243, 245, 291, 300, 306, 307, 308, 311, 312, 314, 322, 326, 332, 336, 337, 338, 341, 362, 372, 374, 378, 411, 416, 418

Offset: 1

Clark Kimberling

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007091, A007092.

Cf. A037394, A037395, A037396, A037397.

import Data.List ((\\), nub)
a037393 n = a037393_list !! (n-1)
a037393_list = filter f [1..] where
   f x = null $ nub (ds 5 x) \\ nub (ds 6 x)
   ds b x = if x > 0 then d : ds b x' else []  where (x', d) = divMod x b
-- Reinhard Zumkeller, May 30 2013

cp's OEIS Frontend

A037381 Numbers k such that every base-3 digit of k is a base-5 digit of k.

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A037396 Numbers k such that every base-5 digit of k is a base-9 digit of k.

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A043279 Maximal run length in base 5 representation of n.

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Mathematica

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A110603 Numbers n whose base 5 representations, interpreted as base 10 integers, are semiprimes.

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Mathematica

Extensions

A262102 Pseudoprimes to base 5, written in base 5.

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A293658 Base-5 circular primes that are not base-5 repunits.

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A353113 Base-5 representation of A000422(n).

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A004053 For m=2,3,..., write m in bases 2,3,..,m.

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A037387 Numbers k such that every base-4 digit of k is a base-5 digit of k.

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