A050288 -id:A050288 - cp's OEIS frontend

A175276 Base-4 pandigital primes: primes having at least one of each digit 0,1,2,3, when written in base 4.

283, 313, 331, 397, 419, 433, 457, 541, 557, 569, 587, 647, 653, 659, 709, 809, 929, 1051, 1063, 1069, 1123, 1163, 1171, 1181, 1187, 1201, 1213, 1249, 1259, 1291, 1307, 1319, 1327, 1423, 1427, 1459, 1481, 1483, 1543, 1549, 1559, 1567, 1571, 1579, 1583

Offset: 1

M. F. Hasler, May 27 2010

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Alonso Del Arte, Classifications of prime numbers - By representation in specific bases, OEIS Wiki as of Mar 19 2010.

Cf. A050288, A138837, A175271, A175272, A175273, A175274, A175275, A175277, A175278, A175279, A175280.

Select[Prime[Range[300]],Min[DigitCount[#,4]]>0&] (* Harvey P. Dale, Apr 15 2012 *)

base(n,b=4,s=0)={local(a=[n%b]);while(09,s,48)+a[i])),a)}
forprime(p=1,1999,#Set(base(p,4))==4 & print1(p","))

A175277 Base-5 pandigital primes: primes having at least one of each digit 0,1,2,3,4, when written in base 5.

Original entry on oeis.org

3319, 3323, 3347, 3469, 3491, 3539, 3547, 3559, 3571, 3607, 3613, 3617, 3691, 3823, 3847, 3863, 4019, 4079, 4139, 4327, 4423, 4483, 4493, 4519, 4523, 4603, 4759, 4903, 4951, 5039, 5059, 5107, 5113, 5147, 5179, 5227, 5273, 5279, 5351, 5477, 5507, 5527

Offset: 1

M. F. Hasler, May 27 2010

Terms in this sequence have at least 6 digits in base 5, i.e., are larger than 5^5, since sum(d_i 5^i) = sum(d_i) (mod 4), and 0+1+2+3+4 is divisible by 2. So the smallest ones should be of the form "10...." in base 5, where "...." is a permutation of "1234". By chance the identical permutation already yields a prime, i.e. a(1) = "101234" in base 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2500 from Harvey P. Dale)
Alonso Del Arte, Classifications of prime numbers - By representation in specific bases, OEIS Wiki as of Mar 19 2010.

Cf. A050288, A138837, A175271, A175272, A175273, A175274, A175275, A175276, A175278, A175279, A175280.

Select[Prime[Range[800]],Min[DigitCount[#,5]]>0&] (* Harvey P. Dale, Mar 10 2019 *)

base(n,b=5,s=0)={local(a=[n%b]);while(09,s,48)+a[i])),a)}
forprime(p=5^5,5^6,#Set(base(p,5))==5 & print1(p","))

A175278 Base-6 pandigital primes: primes having at least one of each digit 0,1,2,3,4,5 when written in base 6.

Original entry on oeis.org

48761, 50033, 50051, 50069, 50101, 50207, 50231, 50311, 50461, 51131, 51137, 51151, 51461, 51503, 51511, 51721, 52181, 52391, 52541, 52571, 52583, 53731, 53881, 54091, 54121, 55001, 57191, 58481, 58901, 60161, 62591, 62921, 63029

Offset: 1

M. F. Hasler, May 30 2010

Terms in this sequence have at least 7 digits in base 6, i.e., are larger than 6^6, since sum(d_i 6^i) = sum(d_i) (mod 5), and 0+1+2+3+4+5 is divisible by 5. So the smallest ones should be of the form "101...." in base 6, where "...." is a permutation of "2345". Actually there is only one such prime, cf. examples.

			The smallest base-6 pandigital prime is written "1013425" in base 6.
The next smallest such prime is "1023345"[6]; note that here the "3" is repeated, since there is no such prime of the form "102wxyz" with w=0, 1 or 2. (Using the same reasoning as in the comment, it follows that the (7-digit base-6 pandigital) number has the same parity as the repeated digit, which therefore must be odd to get a prime.)

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

Cf. A050288, A138837, A175271, A175272, A175273, A175274, A175275, A175276, A175277, A175279, A175280.

Select[Range[60000], Min @ DigitCount[#, 6] > 0 && PrimeQ[#] &] (* Amiram Eldar, Apr 13 2021 *)

base(n,b=6)={ local(a=[n%b]);while(0

Edited by Charles R Greathouse IV, Aug 02 2010

A171744 a(n) is the smallest exponent such that prime(n)^k is pandigital in base 10.

Original entry on oeis.org

68, 39, 19, 18, 23, 22, 14, 17, 14, 12, 11, 13, 11, 13, 12, 13, 11, 14, 10, 15, 14, 13, 9, 11, 13, 9, 15, 14, 13, 12, 11, 15, 10, 7, 12, 9, 12, 10, 11, 8, 11, 8, 12, 11, 13, 13, 10, 12, 10, 8, 11, 12, 9, 7, 6, 7, 8, 12, 8, 8, 7, 7, 10, 9, 9, 6, 9, 10, 9, 10

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Dec 17 2009

A pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once.

			2^68 = 295147905179352825856 (21 digits), 3^39 = 4052555153018976267 (19)
5^19 = 19073486328125 (14), 7^18 = 1628413597910449 (16), 11^23 = 895430243255237372246531 (24)
13^22 = 3211838877954855105157369 (25), 17^14 = 168377826559400929 (18)
19^17 = 5480386857784802185939 (22), 23^14 = 11592836324538749809 (20)
29^12 = 353814783205469041 (18), 31^11 = 25408476896404831 (17)
37^13 = 243569224216081305397 (21), 41^11 = 550329031716248441 (18)
43^13 = 1718264124282290785243 (22), 47^12 = 116191483108948578241 (21)
53^13 = 26036721925606486195973 (23), 59^11 = 30155888444737842659 (20)
61^14 = 9876832533361318095112441 (25), 67^10 = 1822837804551761449 (19)
71^15 = 5873205959385493353867330551 (28), 73^14 = 122045014039746588673695409 (23)
79^13 = 4668229371502258117133839 (25), 83^9 = 186940255267540403 (18)
89^11 = 2775173073766990340489 (22), 97^13 = 67302709016557486028618977 (26)
101^9 = 1093685272684360901 (19), 103^15 = 1557967416600764580522382952407 (31)
107^14 = 25785341502012466393542552649 (29), 109^13 = 306580461214335498944273629 (27)
113^12 = 4334523100191686738306881 (25), 127^11 = 138624799340320978519423 (24)

E.I. Ignatjew, Mathematische Spielereien, Urania Verlag Leipzig-Jena-Berlin, 2. Auflage 1982.
Helmut Kracke, Mathe-musische Knobelisken, Duemmler Bonn, 2. Auflage 1983.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A049363, A050278, A050288, A171132.

sepan[n_]:=Module[{p=Prime[n],k=1},While[Min[DigitCount[p^k]]==0,k++];k]; Array[sepan,100] (* Harvey P. Dale, Aug 03 2019 *)

a(n) = {my(k=1, p=prime(n)); while(#Set(digits(p^k))<10, k++); k; } \\ Jinyuan Wang, Aug 14 2020

Edited by Charles R Greathouse IV, Aug 02 2010

Corrected and extended by Harvey P. Dale, Aug 03 2019

A274328 a(n) is the sum of a sequence of multiples of the n-th prime such that it contains each of the digits from 0 to 9 exactly once and with the least sum possible, or 0 if there is no satisfying sequence.

Original entry on oeis.org

270, 135, 38475, 252, 1881, 702, 918, 684, 1656, 2349, 1953, 7326, 2952, 2322, 2961, 3339, 3717, 3843, 3015, 3195, 3285, 5688, 8217, 5607, 4365, 95445, 6489, 4815, 3924, 37629, 35433, 10611, 9864, 5004, 41571, 4077, 39564, 2934, 34569, 42039

Offset: 1

Claudio Meller, Jun 21 2016

From Ryan Hitchman, Sep 15 2017: (Start)
a(172) = 1023847569, prime(172) = 1021 is the first entry with one multiple.
a(1884) = 145953, prime(1884) = 16217 is last with more than one multiple.
a(10545) = 0, prime(10545) = 111119 is the first zero. (End)

			For n = 7, a(7) = 918 because prime(7) = 17, sequence 34, 85, 102, 697, sum 918.

Ryan Hitchman, Table of n, a(n) for n = 1..10544
Claudio Meller Blog, Multiples with all the digits, June 13 2016.

Cf. A180489 for n>1884. Superset of A050288. - Ryan Hitchman, Sep 15 2017

(m = Select[#*Range[10000], Max[DigitCount[#]] == 1 &];
   Total[m*LinearProgramming[m, Thread[DigitCount /@ m],
      ConstantArray[{1, 0}, 10], 0, Integers]]) & /@ Prime[Range[40]] (* Ryan Hitchman, Sep 15 2017 *)

Terms a(9) and beyond, zero case from Ryan Hitchman, Sep 15 2017

A253172 Numbers n = p * q, where n, p, and q together contain all 10 digits at least once.

Original entry on oeis.org

15628, 15678, 16038, 17082, 17820, 19084, 20457, 20748, 20754, 21658, 24507, 24587, 25704, 26910, 26970, 27096, 27504, 27690, 28156, 28651, 29076, 29370, 29670, 29706, 29730, 30956, 30972, 30976, 32890, 32970, 34056, 34902, 34986, 35046, 35074, 35096, 35496, 35690, 36092, 36490, 36508, 36950, 36970, 36972, 37092, 37096, 37290, 37590, 37690, 37908, 38870, 39026, 39720, 39760, 40587, 40596

Offset: 1

Randy L. Ekl, Dec 28 2014

All pandigital numbers (cf. A171102) belong to this sequence; therefore A050288(1) = 10123457689 is the smallest prime term. - Reinhard Zumkeller, Dec 29 2014

			a(1) is 15628 = 4 * 3907, using all 10 digits.
a(8) is 20748 = 13 * 1596 (note duplicate 1, which is ok in this sequence).
a(3) is 16038 = 27 * 594, and also 16038 = 54 * 297; two different solutions for a(3).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A195814, which restricts sequence terms along with their factors to exactly 10 digits, and thus has a finite number of terms.

Cf. A027750, subsequences: A050278, A171102, A050288.

import Data.List (nub, sort)
a253172 n = a253172_list !! (n-1)
a253172_list = filter f [2..] where
   f x = g divs $ reverse divs where
         g (d:ds) (q:qs) = d <= q &&
           (sort (nub $ xs ++ show d ++ show q) == decs || g ds qs)
         xs = show x
         divs = a027750_row x
   decs = "0123456789"
-- Reinhard Zumkeller, Dec 29 2014

isokpq(n) = {fordiv(n, d, digs = digits(n); if ( d <= sqrtint(n), digs = concat(digs, digits(d)); digs = concat(digs, digits(n/d)); if (#Set(digs) == 10, return(1));););}
lista(nn) = {for(n=2, nn, if (isokpq(n), print1(n, ", ")););} \\ Michel Marcus, Dec 29 2014

A030083 Primes p such that all digits of p^2 appear in p.

Original entry on oeis.org

1049, 17923, 49261, 81619, 94583, 100469, 100549, 102953, 107251, 110923, 125789, 149827, 184903, 256169, 279863, 285101, 289573, 298327, 370421, 406951, 459521, 471923, 472963, 492671, 493621, 497521, 499423, 502261, 504821, 569423, 582139, 597823, 631927

Offset: 1

Patrick De Geest

Robert Israel, Table of n, a(n) for n = 1..10000

Contains A050288.

Cf. A030084.

R:= NULL: count:= 0: p:= 2:
while count < 100 do
 p:= nextprime(p);
 if convert(convert(p^2,base,10),set) subset convert(convert(p,base,10),set) then
    count:= count+1; R:= R,p
 fi
od:
R; # Robert Israel, Nov 01 2023

Select[Prime[Range[52000]],SubsetQ[IntegerDigits[#],IntegerDigits[#^2]]&] (* Harvey P. Dale, Aug 12 2025 *)

from sympy import isprime
def ok(n): return isprime(n) and set(str(n)) >= set(str(n**2))
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Nov 01 2023

Offset corrected by Robert Israel, Nov 01 2023

A030084 Primes p such that all digits of p^3 appear in p.

Original entry on oeis.org

1035743, 1045573, 1215397, 1396247, 1642309, 2031487, 2149573, 2363149, 2385961, 2458019, 2569751, 2815973, 2857319, 2986301, 3109867, 3349517, 3482461, 3530467, 3865079, 4332871, 4387291, 4631489, 4893617, 5170283

Offset: 1

Patrick De Geest

Robert Israel, Table of n, a(n) for n = 1..2000

Includes A050288.

Cf. A030083.

filter:= proc(p) isprime(p) and convert(convert(p^3,base,10),set) subset convert(convert(p,base,10),set) end proc:
select(filter, [seq(i,i=3..3*10^6,2)]); # Robert Israel, Nov 01 2023

Offset corrected by Robert Israel, Nov 01 2023

A108418 Primes with at least one of each odd digit and no even digits.

Original entry on oeis.org

13597, 13759, 15739, 15937, 15973, 17359, 17539, 19753, 31957, 37159, 37591, 37951, 39157, 51973, 53197, 53719, 53791, 53917, 57139, 57193, 71359, 71593, 73951, 75193, 75391, 75913, 75931, 79153, 79531, 91573, 91753, 95317, 95713, 95731

Offset: 1

Rick L. Shepherd, Jun 02 2005

This is a subsequence of A030096.

No even digits are allowable. Otherwise the first missing terms would be 105379, 105397, 109357, 109537. - Zak Seidov, Nov 24 2013

Georg Fischer, Table of n, a(n) for n = 1..3000 [recomputed Jul 08 2022; first 390 terms from Carmine Suriano]

Cf. A030096 (Primes whose digits are all odd), A050288 (Pandigital primes), A108386 (Primes p such that p's set of distinct digits is {1, 3, 7, 9}).

Cf. A232447 (even digits are allowable). - Zak Seidov, Nov 24 2013

Select[Table[Prime[n],{n,10000}],!ContainsAny[IntegerDigits[#],{0,2,4,6,8}]&&ContainsAll[IntegerDigits[#],{1,3,5,7,9}]&] (* James C. McMahon, Mar 05 2024 *)

from sympy import isprime
from itertools import count, islice, product
def agen():
    for d in count(5):
        for p in product("13579", repeat=d):
            if set(p) != set("13579"): continue
            t = int("".join(p))
            if isprime(t): yield t
print(list(islice(agen(), 40))) # Michael S. Branicky, Jul 08 2022

Added missing last term with 5 different digits, Carmine Suriano, Jan 14 2011

A159474 Pandigital primes with least digit sum (46) starting with the largest and listed in descending order.

Original entry on oeis.org

98765421103, 98765410231, 98765401321, 98765321401, 98765320411, 98765301241, 98765241103, 98765240131, 98765240113, 98765234011, 98765204311, 98765201341, 98765140231, 98765134021, 98765124103, 98765124013, 98765121043, 98765120431, 98765120413, 98765120143

Offset: 1

Lekraj Beedassy, Apr 13 2009

The last term is a(1251724) = 10123457689. - Jinyuan Wang, Aug 14 2020

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

Cf. A050288, A159473.

Select[Reverse[Sort[FromDigits[Join[{9,8,7,6},#]]&/@Permutations[{5,4,3,2,1,0,1}]]], PrimeQ] (* The program generates the first 150 terms of the sequence *) (* Harvey P. Dale, May 05 2016 *)

Corrected and extended by Ray Chandler, Apr 16 2009

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A175276 Base-4 pandigital primes: primes having at least one of each digit 0,1,2,3, when written in base 4.

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A175277 Base-5 pandigital primes: primes having at least one of each digit 0,1,2,3,4, when written in base 5.

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A175278 Base-6 pandigital primes: primes having at least one of each digit 0,1,2,3,4,5 when written in base 6.

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A171744 a(n) is the smallest exponent such that prime(n)^k is pandigital in base 10.

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A274328 a(n) is the sum of a sequence of multiples of the n-th prime such that it contains each of the digits from 0 to 9 exactly once and with the least sum possible, or 0 if there is no satisfying sequence.

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A253172 Numbers n = p * q, where n, p, and q together contain all 10 digits at least once.

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A030083 Primes p such that all digits of p^2 appear in p.

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A030084 Primes p such that all digits of p^3 appear in p.

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