A050278 -id:A050278 - cp's OEIS frontend

A115940 Pandigital (meaning every digit appears exactly once) triangular numbers.

1062489753, 1239845706, 1256984730, 1520843976, 1539264870, 1597283460, 1684930275, 1952843760, 1957346028, 1978236450, 2197480365, 2367098415, 2418079653, 2503948761, 2634980715, 2718609453, 2735891406, 2750483196

Offset: 1

Giovanni Resta, Feb 06 2006

There are 82 such numbers, the largest being T(138959)=9654871320.

The sequence of pandigital binomial coefficients C(m,k) with k>1 contains 84 numbers, these 82 triangular terms of the form C(m,2) and only two other ones C(595,4) = 5169738420 and C(253,5) = 8301429675 (see link). - Bernard Schott, Apr 15 2022

			T(46097)=1062489753.

Zak Seidov, Table of n, a(n) for n = 1..82 (full sequence)
Fun With Num3ers, Pandigital Binomial coefficient | When C(n, k) is pandigital.

Cf. A050278, A115939.

A116670 Numbers with all but one decimal digit.

Original entry on oeis.org

102345678, 102345679, 102345687, 102345689, 102345697, 102345698, 102345768, 102345769, 102345786, 102345789, 102345796, 102345798, 102345867, 102345869, 102345876, 102345879, 102345896, 102345897, 102345967, 102345968, 102345976, 102345978, 102345986

Offset: 1

Rick L. Shepherd, Feb 22 2006

More precisely, each term has exactly nine distinct decimal digits any of which may occur more than once. Leading zeros are not permitted.

			a(1) = 102345678 as all decimal digits but 9 appear and there is no smaller number with only one missing digit.

Index entries for 10-automatic sequences.

Cf. A050278 (pandigital numbers).

Select[Range[10^8, 10^8 + 3000000], Length[Union[IntegerDigits[#]]] == 9 &] (* T. D. Noe, Dec 05 2012 *)

Offset corrected by Charles R Greathouse IV, Feb 15 2017

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

A187565 Numbers divisible by at least eight of their digits, different and >1.

Original entry on oeis.org

1234759680, 1234857960, 1234895760, 1234958760, 1235487960, 1235679480, 1235976840, 1236795840, 1237569480, 1237589640, 1237594680, 1237695480, 1237894560, 1238549760, 1238574960, 1238597640, 1238975640, 1239547680, 1239567840, 1239756840, 1239784560, 1239847560, 1239857640, 1243579680, 1243589760, 1243879560, 1243957680, 1245378960, 1245973680, 1245983760

Offset: 1

Zak Seidov, Mar 11 2011

First 11460 terms are pandigital numbers (A050278).
Asymptotic density 1/2520 = 0.000396.... - Charles R Greathouse IV, Mar 11 2011
11460 terms up to 10^10, 299275 terms up to 10^11, 6224794 terms up to 10^12. - Charles R Greathouse IV, Mar 11 2011

David A. Corneth, Table of n, a(n) for n = 1..11460 (terms <= 10^10)

Cf. A050278, A187398, A187516, A187238, A187533, A187534.

Subsequence of A187551 (numbers divisible by at least seven ...).

numdig = 8; Select[Range[1245983760], Length[(u = Union[Select[IntegerDigits[#], #1 > 1 &]])] >= numdig && Plus @@ (Boole@Divisible[#, u]) >= numdig &] (* Amiram Eldar, Aug 30 2020 *)

s(n) = my(res=Set(digits(n)));select(x->x>1,res)
is(n) = my(d=s(n));if(#d < 8, return(0)); sum(i=1, #d, n%d[i]==0) >= 8 \\ David A. Corneth, Aug 30 2020

A218019 Integers in which the number of distinct base-10 digits is 7.

Original entry on oeis.org

1023456, 1023457, 1023458, 1023459, 1023465, 1023467, 1023468, 1023469, 1023475, 1023476, 1023478, 1023479, 1023485, 1023486, 1023487, 1023489, 1023495, 1023496, 1023497, 1023498, 1023546, 1023547, 1023548, 1023549, 1023564, 1023567, 1023568, 1023569

Offset: 1

Jonathan Vos Post, Dec 04 2012

This is to A031969 as 7 is to 4. This is the 7th row of the array A(k,n) = n-th number in which the number of distinct base-10 digits is k. A031969 is the 4th row. A220063 is the 5th row. A220076 is the 6th row. Pandigital numbers A050278 is the 10th row. The subsequence of primes begins: 1023467, 1023487.

Cf. A031969, A050278, A220063, A220076.

Select[Range[1000000, 1030000], Length[Union[IntegerDigits[#]]] == 7 &] (* T. D. Noe, Dec 04 2012 *)

A116667 Greatest digit not used in n (or 10 if n is pandigital).

Original entry on oeis.org

9, 9, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 7, 8, 8, 8, 8, 8, 8, 8, 8, 7, 8, 9, 9, 9, 9, 9

Offset: 0

Rick L. Shepherd, Feb 22 2006

a(A050278(1)) = a(1023456789) = 10, the first term with that value, as 1023456789 is the first base 10 pandigital number.

			a(89) = 7 because decimal digits 8 and 9 are both used in 89.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A067898 (least digit not used in n), A050278 (pandigital numbers).

def A116667(n):
    s = set(str(n))
    for i in range(9,-1,-1):
        if str(i) not in s:
            return i
    return 10 # Chai Wah Wu, Apr 13 2024

A178775 Smallest prime factors of zerofull restricted pandigital numbers.

Original entry on oeis.org

3, 2, 3, 3, 2, 3, 3, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 2, 2, 3, 2, 2, 3, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 2, 3, 2, 3, 3, 3, 3, 2, 3, 2, 2, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 2, 2, 2, 3, 2, 3, 3, 3, 3, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 2, 3, 2, 2

Offset: 1

Reinhard Zumkeller, Jun 11 2010

a(n) = A020639(A050278(n)); 2 <= a(n) <= 3.

a(n) is 2 1653120 times and 3 1612800 times, making the average value 202/81 = 2.493.... [Charles R Greathouse IV, Sep 09 2011]

			A050278(1)=1023456789=3*3*3*3*2221*5689 --> a(1)=3;
A050278(10)=1023457896=2*2*2*3*3*3*59*80309 --> a(10)=2;
A050278(100)=1023495786=2*3*3*739*76943 --> a(100)=2;
A050278(1000)=1024658793=3*3*113850977 --> a(1000)=3.

a(n) = 2 + A000035(A050278(n)).

Edited by Charles R Greathouse IV, Aug 02 2010

A252490 Numbers whose set of digits is simply connected, with 9 and 0 considered as neighbors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 21, 22, 23, 32, 33, 34, 43, 44, 45, 54, 55, 56, 65, 66, 67, 76, 77, 78, 87, 88, 89, 90, 98, 99, 100, 101, 102, 109, 110, 111, 112, 120, 121, 122, 123, 132, 190, 201, 210, 211, 212, 213, 221, 222, 223, 231, 232, 233, 234, 243, 312, 321, 322, 323, 324, 332, 333, 334, 342, 343, 344, 345, 354, 423, 432

Offset: 1

M. F. Hasler, Dec 24 2014

The set of digits must consist of a single run without "holes", but for a cyclic topology where 9 and 0 are seen as neighbors.

A superset of A134336. Namely, numbers in A134336 or such that the complement of their digits in {0,...,9} satisfies the criterion of A134336.

Danny Rorabaugh, Table of n, a(n) for n = 1..10000

Cf. A032981, A050278, A033075 (a subsequence), A010785, A108965, A134336 (a subsequence).

is(n)=vecmax(if((d=Set(digits(n)))[1],d,d=setminus(vector(9,i,i),d)))-vecmin(d)==#d-1

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

A266279 Nonnegative integers where all occurring digits occur with equal frequency.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67

Offset: 1

Felix Fröhlich, Dec 26 2015

The set of terms of this sequence is a superset of each of the sets of terms of A050278 and A082810.

Except for i = 1, A261315(i) gives the number of existing n such that A055642(a(n)) = i.

			1122 is a term of the sequence, since all occurring digits occur twice.

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Cf. A050278, A082810, A261315.

Select[Range[0, 67], Length@ Union[DigitCount[#] /. 0 -> Nothing] == 1 &] (* Michael De Vlieger, Jan 06 2016 *)

digitcount(n) = my(d=digits(n), v=vector(10)); for(x=0, 9, for(k=1, #d, if(d[k]==x, v[x+1]++))); v
is(n) = my(c=digitcount(n), k=0); for(i=1, #c, if(k==0 && c[i]!=0, k=c[i]); if(k!=0 && c[i]!=0, if(k!=c[i], return(0)))); return(1)

cp's OEIS Frontend

A115940 Pandigital (meaning every digit appears exactly once) triangular numbers.

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A116670 Numbers with all but one decimal digit.

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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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A187565 Numbers divisible by at least eight of their digits, different and >1.

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A218019 Integers in which the number of distinct base-10 digits is 7.

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A116667 Greatest digit not used in n (or 10 if n is pandigital).

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A178775 Smallest prime factors of zerofull restricted pandigital numbers.

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A252490 Numbers whose set of digits is simply connected, with 9 and 0 considered as neighbors.

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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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