A014950 -id:A014950 - cp's OEIS frontend

A348150 a(n) is the smallest Niven (or Harshad) number with exactly n digits and not containing the digit 0.

1, 12, 111, 1116, 11112, 111114, 1111112, 11111112, 111111111, 1111111125, 11111111112, 111111111126, 1111111111116, 11111111111114, 111111111111114, 1111111111111122, 11111111111111112, 111111111111111132, 1111111111111111119, 11111111111111111121, 111111111111111111117

Offset: 1

Bernard Schott, Oct 03 2021

This sequence is inspired by a problem, proposed by Argentina during the 39th International Mathematical Olympiad in 1998 at Taipei, Taiwan, but not used for the competition.
The problem asked for a proof that, for each positive integer n, there exists a n-digit number, not containing the digit 0 and that is divisible by the sum of its digits (see links: Diophante in French and Kalva in English).
This sequence lists the smallest such n-digit integer.

			111114 has 6 digits, does not contain 0 and is divisible by 1+1+1+1+1+4 = 9 (111114 = 9*12346), while 111111, 111112, 111113 are not respectively divisible by sum of their digits: 6, 7, 8; hence, a(6) = 111114.

Michael S. Branicky, Table of n, a(n) for n = 1..1000
Diophante, Bon souvenir de Buenos-Aires.
Kalva, 39th IMO 1998 shortlisted problems, problem N7.
Index to sequences related to Olympiads.

Cf. A002275, A005349, A217973.

hQ[n_] := ! MemberQ[(d = IntegerDigits[n]), 0] && Divisible[n, Plus @@ d]; a[n_] := Module[{k = (10^n - 1)/9}, While[! hQ[k], k++]; k]; Array[a, 30] (* Amiram Eldar, Oct 03 2021 *)

a(n) = for(k=(10^n-1)/9, 10^n-1, if (vecmin(digits(k)) && !(k % sumdigits(k)), return (k)));  \\ Michel Marcus, Oct 03 2021

def niven(n):
    s = str(n)
    return '0' not in s and n%sum(map(int, s)) == 0
def a(n):
    k = int("1"*n)
    while not niven(k): k += 1
    return k
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Oct 09 2021

a(n) = A002275(n) = R_n iff n is in A014950.

More terms from Amiram Eldar, Oct 03 2021

A014959 Integers k such that k divides 22^k - 1.

Original entry on oeis.org

1, 3, 7, 9, 21, 27, 39, 49, 63, 81, 117, 147, 189, 243, 273, 343, 351, 441, 507, 567, 729, 819, 1029, 1053, 1143, 1323, 1521, 1701, 1911, 2187, 2401, 2457, 2943, 3081, 3087, 3159, 3429, 3549, 3969, 4401, 4563, 5103, 5733, 6561, 6591, 7203, 7371

Offset: 1

Olivier Gérard

nonn

Also, numbers n such that n divides s(n), where s(1)=1, s(k)=s(k-1)+k*22^(k-1) (cf. A014940).

Max Alekseyev, Table of n, a(n) for n = 1..69065

Integers n such that n divides b^n - 1: A067945 (b=3), A014945 (b=4), A067946 (b=5), A014946 (b=6), A067947 (b=7), A014949 (b=8), A068382 (b=9), A014950 (b=10), A068383 (b=11), A014951 (b=12), A116621 (b=13), A014956 (b=14), A177805 (b=15), A014957 (b=16), A177807 (b=17), A128358 (b=18), A125000 (b=19), A128360 (b=20), A014960 (b=24).

nxt[{n_,s_}]:={n+1,s+(n+1)*22^n}; Transpose[Select[NestList[nxt,{1,1},7500], Divisible[ Last[#],First[#]]&]][[1]] (* Harvey P. Dale, Jan 27 2015 *)

Edited by Max Alekseyev, Nov 16 2019

A275945 Numbers n such that the average of different permutations of digits of n is an integer.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 20, 22, 24, 26, 28, 31, 33, 35, 37, 39, 40, 42, 44, 46, 48, 51, 53, 55, 57, 59, 60, 62, 64, 66, 68, 71, 73, 75, 77, 79, 80, 82, 84, 86, 88, 91, 93, 95, 97, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111

Offset: 1

Altug Alkan, Aug 29 2016

Complement of A273492.
Permutations with a first digit of 0 are included in the average (i.e. 0010 is taken to be 10, 01 is taken to be 1, etc.).
From Robert Israel, Sep 01 2016: (Start)
n such that A002275(A055642(n))*A007953(n) is divisible by A055642(n).
In particular, contains all k-digit numbers if k is in A014950. (End)

			97 is a term because (97+79) is divisible by 2.
100 is a term because (1+10+100) is divisible by 3.
123 is a term because (123+132+213+231+312+321) is divisible by 6.
1001 is not a term because (11+101+110+1001+1010+1100) is not divisible by 6.

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

Cf. A002275, A014950, A045876, A055642, A066642, A007953, A273492.

f:= proc(n) local L, d, s;
  L:= convert(n, base, 10);
  d:= nops(L);
  s:= convert(L, `+`);
  evalb(s*(10^d-1)/9 mod d = 0)
end proc:
select(f, [$1..10000]); # Robert Israel, Sep 01 2016

Select[Range@ 111, IntegerQ@ Mean@ Map[FromDigits, Permutations@ #] &@ IntegerDigits@ # &] (* Michael De Vlieger, Aug 29 2016 *)

A055642(n) = #Str(n);
A007953(n) = sumdigits(n);
for(n=1, 2000, if((((10^A055642(n)-1)/9)*A007953(n)) % A055642(n) == 0, print1(n, ", ")));

A348317 a(n) = A348150(n) - A002275(n) where A002275(n) = R_n is the repunit with n times digit 1.

Original entry on oeis.org

0, 1, 0, 5, 1, 3, 1, 1, 0, 14, 1, 15, 5, 3, 3, 11, 1, 21, 8, 10, 6, 5, 1, 3, 2, 12, 0, 17, 25, 14, 5, 13, 6, 74, 1, 54, 41, 12, 8, 14, 4, 105, 41, 55, 63, 33, 25, 13, 5, 103, 3, 33, 40, 63, 3, 52, 15, 23, 40, 21, 20, 10, 21, 11, 25, 33, 41, 47, 45, 14, 1, 171

Offset: 1

Bernard Schott, Oct 12 2021

a(n) measures the gap between the smallest n-digit number not containing the digit 0 and the smallest n-digit Niven number not containing the digit 0.
For more informations and links, see A348150.
a(n) = 0  iff n is in A014950.

			A348150(4) = 1116 since it is the smallest 4-digit integer not containing the digit 0 that is divisible by the sum of its digits:1116 = (1+1+1+6) * 124;  A002275(4) = R_4 = 1111, hence a(4) = 1116 - 1111 = 5.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A002275, A005349, A014950, A217973, A348150.

hQ[n_] := ! MemberQ[(d = IntegerDigits[n]), 0] && Divisible[n, Plus @@ d]; a[n_] := Module[{m= (10^n - 1)/9,k=0}, While[! hQ[m+k], k++]; k]; Array[a, 30] (* Amiram Eldar, Oct 13 2021 *)

a(n) = my(r=(10^n-1)/9); for(k=r, 10^n-1, if (vecmin(digits(k)) && !(k % sumdigits(k)), return (k-r))); \\ Michel Marcus, Oct 13 2021

def a(n):
    s, k = "1"*n, int("1"*n)
    while '0' in s or k%sum(map(int, s)): k += 1; s = str(k)
    return k - int("1"*n)
print([a(n) for n in range(1, 73)]) # Michael S. Branicky, Oct 12 2021

a(n) = A348150(n) - A002275(n).

a(23) and beyond from Michael S. Branicky, Oct 12 2021

A273492 Numbers n such that the average of different permutations of digits of n is not an integer.

Original entry on oeis.org

10, 12, 14, 16, 18, 21, 23, 25, 27, 29, 30, 32, 34, 36, 38, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 61, 63, 65, 67, 69, 70, 72, 74, 76, 78, 81, 83, 85, 87, 89, 90, 92, 94, 96, 98, 1000, 1001, 1002, 1004, 1005, 1006, 1008, 1009, 1010, 1011, 1013, 1014, 1015, 1017, 1018, 1019, 1020, 1022, 1023, 1024

Offset: 1

Altug Alkan, Aug 29 2016

Complement of A275945.
Permutations with a first digit of 0 are included in the average (i.e. 0010 is taken to be 10, 01 is taken to be 1, etc.).
From Robert Israel, Sep 01 2016: (Start)
n such that A002275(A055642(n))*A007953(n) is not divisible by A055642(n).
In particular, contains no k-digit numbers if k is in A014950. (End)

			12 is a term because (12+21) = 33 is not divisible by 2.
1000 is a term because (1+10+100+1000) = 1111 is not divisible by 4.
123 is not a term because (123+132+213+231+312+321) is divisible by 6.
1001 is a term because (11+101+110+1001+1010+1100) is not divisible by 6.

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

Cf. A002275, A014950, A055642, A066642, A007953, A045876, A055642, A275945.

f:= proc(n) local L,d,s;
  L:= convert(n,base,10);
  d:= nops(L);
  s:= convert(L,`+`);
  evalb(s*(10^d-1)/9 mod d = 0)
end proc:
remove(f, [$1..10000]); # Robert Israel, Sep 01 2016

Select[Range[2^10], ! IntegerQ@ Mean@ Map[FromDigits, Permutations@ #] &@ IntegerDigits@ # &] (* Michael De Vlieger, Aug 29 2016 *)

A055642(n) = #Str(n);
A007953(n) = sumdigits(n);
for(n=1, 2000, if((((10^A055642(n)-1)/9)*A007953(n)) % A055642(n) != 0, print1(n, ", ")));

A348316 a(n) is the largest Niven (or Harshad) number with exactly n digits and not containing the digit 0.

Original entry on oeis.org

9, 84, 999, 9963, 99972, 999984, 9999966, 99999966, 999999999, 9999999828, 99999999898, 999999999853, 9999999999936, 99999999999783, 999999999999984, 9999999999999858, 99999999999999939, 999999999999999831, 9999999999999999951, 99999999999999999922, 999999999999999999687

Offset: 1

Bernard Schott, Oct 11 2021

This sequence is inspired by a problem, proposed by Argentina during the 39th International Mathematical Olympiad in 1998 at Taipei, Taiwan, but not used for the competition.
The problem asked for a proof that, for each positive integer n, there exists a n-digit number, not containing the digit 0 and that is divisible by the sum of its digits (see links: Diophante in French and Kalva in English).
This sequence lists the largest such n-digit integer.

			9963 has 4 digits, does not contain 0 and is divisible by 9+9+6+3 = 27 (9963 = 27*369), while there is no integer k with 9964 <= k <= 9999 that is divisible by sum of its digits, hence a(4) = 9963.

Michael S. Branicky, Table of n, a(n) for n = 1..1000
Diophante, Bon souvenir de Buenos-Aires.
Kalva, 39th IMO 1998 shortlisted problems, problem N7.
Index to sequences related to Olympiads.

Cf. A002283, A005349, A014950, A217973, A348150, A348317, A348318.

hQ[n_] := ! MemberQ[(d = IntegerDigits[n]), 0] && Divisible[n, Plus @@ d]; a[n_] := Module[{k = 10^n}, While[! hQ[k], k--]; k]; Array[a, 20] (* Amiram Eldar, Oct 11 2021 *)

def a(n):
    s, k = "9"*n, int("9"*n)
    while '0' in s or k%sum(map(int, s)): k -= 1; s = str(k)
    return k
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Oct 11 2021

a(n) = A002283(n) = 10^n - 1 iff n is in A014950 (compare with A348150 formula).

More terms from Amiram Eldar, Oct 11 2021

A014962 Odd numbers k that divide 25^k - 1.

Original entry on oeis.org

1, 3, 9, 21, 27, 63, 81, 93, 147, 171, 189, 243, 279, 441, 513, 567, 609, 651, 729, 837, 903, 1029, 1197, 1323, 1539, 1701, 1827, 1953, 2187, 2511, 2667, 2709, 2883, 2943, 3087, 3249, 3591, 3969, 4263, 4401, 4557, 4617, 5103, 5301, 5481, 5859, 6321

Offset: 1

Olivier Gérard

nonn

Also, numbers k such that k divides s(k), where s(1)=1, s(j) = s(j-1) + j*25^(j-1).

Equivalently, numbers k that divide ((24*k - 1)*25^k + 1) / 24^2 (cf. A014943).

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

Cf. A014943, A014945, A014946, A014949, A014950, A014951, A014956, A014957, A128358, A128360, A014959, A014960.

select(t -> 25 &^ t - 1 mod t = 0, [seq(i,i=1..10^4,2)]); # Robert Israel, Oct 04 2020

Edited by Max Alekseyev, Nov 16 2019

A098034 Numbers that are divisible both by the sum and by the product of the squares of their digits.

Original entry on oeis.org

111, 11112, 1122112, 111111111, 122121216, 1111112112, 1111211136, 1116122112, 1211162112, 11111113116, 11111121216, 11112122112, 11121114112, 11132111232, 11133122112, 11213111232, 11311322112, 12111213312, 21111311232, 31111221312

Offset: 1

Lekraj Beedassy, Sep 10 2004

Also called "uncommon numbers". Sequence contains the repunits R_m, where m=A014950,m>1. - Lekraj Beedassy, Jul 14 2008

Called "insolite numbers" in the paper by J.-M. De Koninck and N. Doyon, which contains a list of insolite numbers below 10^18. Their list lacks a(63)=112264112111616. They conjectured that 1111111111131111131111111111175 is the smallest insolite number containing the digit '5'. I verified this claim and found a further such number, 1111111111111111117111111111911111375, which may not be the second one. - Giovanni Resta, Oct 19 2012

			1122112 is in the sequence because 1^2 + 1^2 + 2^2 + 2^2 + 1^2 + 1^2 + 2^2=16,(1*1*2*2*1*1*2)^2=64 and we have 1122112=16*70132=64*17533.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 111, p. 39, Ellipses, Paris 2008.

Lekraj Beedassy and Giovanni Resta, Table of n, a(n) for n = 1..428 (terms < 10^20, first 39 terms from Lekraj Beedassy)
J.-M. De Koninck and N. Doyon, On a very thin sequence of integers, Annales Rolando Eötvös Nominatae, 20:157-177 (2001)

More terms from Lekraj Beedassy, Jul 14 2008

Missing term a(11) inserted, b-file corrected and extended to terms < 10^20 by Giovanni Resta, Oct 19 2012

A119888 Numbers n such that (n,R_n)>1, where R_n=(10^n -1)/9=A002275(n).

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 22, 24, 27, 30, 33, 36, 39, 42, 44, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 88, 90, 93, 96, 99, 102, 105, 108, 110, 111, 114, 117, 120, 123, 126, 129, 132, 134, 135, 138, 141, 144, 150, 153, 154

Offset: 1

Lekraj Beedassy, Aug 04 2006

nonn

Includes all multiples of 3 (A008585) and multiples of 11 with coefficients A063450. A special subsequence is A014950.

Cf. A119905.

A122787 a(n) is the smallest prime p such that the multiplicative order of 10 modulo p is 3^n.

Original entry on oeis.org

3, 37, 333667, 757, 163, 411361786890737698932559, 313471, 2558791, 618846643, 2238862519, 396319276163359, 34720813

Offset: 0

Farideh Firoozbakht, Oct 06 2006

For n>0, a(n) is the smallest prime p>3 such that 3^n*p but not 3^(n-1)*p is a solution to 10^x==1 (mod x). A014950 gives solutions of this equation. It's obvious that if n is a term of A014950 then 3n is also a term of A014950. So according to the definition of a(n), for each m>n-1, 3^m*a(n) is in the sequence A014950.
a(n) is the smallest prime divisor of \Phi_{3^n}(10)/3, where \Phi_k(x) is k-th cyclotomic polynomial. a(n) is congruent to 1 modulo 3^n and 1, 3, 9, 13, 27, 31, 37, or 39 modulo 40. - Max Alekseyev, Nov 18 2014
a(12)>10^17, a(13)=796884087799, a(14)=86093443, a(15)=70367039929, a(16)>8*10^18, a(17)=662489036191, a(18)>10^19, a(19)>10^19, a(20)=38180289190951, a(21)=28305715767319, a(22)>10^20, a(23)>10^20, a(24)=63829075244707. - Ray Chandler, Dec 25 2013

			p=333667 is the smallest prime such that multiplicative order of 10 modulo p is 3^2, so a(2)=333667.

J. Brillhart et al., Factorizations of b^n +- 1, b = 2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 2002.

Cf. A014950, A066364.

a(n) = factor(polcyclo(3^n,10)/3)[1,1] \\ Max Alekseyev, Nov 18 2014

Revised and extended by Max Alekseyev, Apr 25 2009

a(13), a(15), a(17), a(20), a(21), a(24) from Ray Chandler, Dec 25 2013

cp's OEIS Frontend

A348150 a(n) is the smallest Niven (or Harshad) number with exactly n digits and not containing the digit 0.

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A014959 Integers k such that k divides 22^k - 1.

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A275945 Numbers n such that the average of different permutations of digits of n is an integer.

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Maple

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A348317 a(n) = A348150(n) - A002275(n) where A002275(n) = R_n is the repunit with n times digit 1.

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A273492 Numbers n such that the average of different permutations of digits of n is not an integer.

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Maple

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PARI

A348316 a(n) is the largest Niven (or Harshad) number with exactly n digits and not containing the digit 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A014962 Odd numbers k that divide 25^k - 1.

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