A009994 -id:A009994 - cp's OEIS frontend

A333955 Numbers k with digits in nondecreasing order and each digit greater than 1 such that the iterated product of digits of k is a prime.

2, 3, 5, 7, 26, 34, 35, 37, 57, 223, 278, 279, 299, 355, 359, 367, 369, 389, 447, 469, 557, 579, 666, 999, 2247, 2269, 2337, 2339, 2349, 2366, 2699, 2799, 3335, 3336, 3338, 3346, 3357, 3399, 3499, 3669, 3679, 3889, 3999, 4689, 4788, 5579, 5777, 6668, 22227, 22239, 22336

Offset: 1

David A. Corneth, Apr 11 2020

Primitive sequence of A028843. If k is in this sequence, then one can concatenate as many 1s as one likes, and/or permute the digits, to get terms of A028843 that are not in this sequence. For example, from 35 in this sequence, we can obtain 135, 1135, 11135, ... as well as 153, 315, 351, 1153, 1315, 1351, 1513, 1531, etc.

			For 35, we have 3 * 5 = 15 and then 1 * 5 = 5, which is a prime. Furthermore, the digits of 35 are nondecreasing and all digits of 35 are greater than 1, so 35 is in the sequence.
Likewise with 37, we see that 3 * 7 = 21 and 2 * 1 = 2, which is prime, and 3 < 7, so 37 is also in the sequence. The numbers 137, 1137, 11137, etc., are in A028843 but are not in this sequence of account of containing the digit 1.
With 43, we confirm that 4 * 3 = 12 and 1 * 2 = 2, which is prime, but 4 > 3, so 43 is not in the sequence.

Cf. A002473, A003001, A009994, A028843.

Select[Range[25000], Min[(d = IntegerDigits[#])] > 1 && (Length[d] < 2 || Min @ Differences[d] > -1) && PrimeQ[FixedPoint[IntegerDigits @ (Times @@ #)&, d][[1]]] &] (* Amiram Eldar, Apr 14 2020 *)

is(n) = my(d=digits(n), v); if(d!=(v=vecsort(d))||v[1]<2, return(0)); while(n>=10, n=vecprod(digits(n))); isprime(n)

def hasDigitsSorted(n: Int): Boolean = {
  val digSort = Integer.parseInt(n.toString.toCharArray.sorted.mkString)
  n == digSort
}
def iterDigitProd(n: Int): Int = n.toString.length match {
  case 1 => n
  case  => iterDigitProd(n.toString.toCharArray.map( - 48).scanRight(1)( * ).head)
}
val prelim = (1 to 20000).filter(hasDigitsSorted(_)).filter(n => List(2, 3, 5, 7).contains(iterDigitProd(n)))
prelim.filter(!.toString.startsWith("1")) // _Alonso del Arte, Apr 20 2020

A341009 Numbers whose sum of even digits and sum of odd digits differ by 8.

Original entry on oeis.org

8, 17, 26, 35, 44, 53, 62, 71, 80, 107, 129, 170, 192, 206, 219, 224, 237, 242, 255, 260, 273, 291, 305, 327, 349, 350, 372, 394, 404, 422, 439, 440, 457, 475, 493, 503, 525, 530, 547, 552, 569, 574, 596, 602, 620, 659, 677, 695, 701, 710, 723, 732, 745, 754, 767, 776, 789

Offset: 1

Eric Angelini and Carole Dubois, Feb 02 2021

Carole Dubois, Table of n, a(n) for n = 1..5001

Cf. A009994, A036301 (sums are equal), A341002 to A341010 (sums differ by 1 to 9).

Select[Range[1000], Abs[Plus @@ Select[(d = IntegerDigits[#]), OddQ] - Plus @@ Select[d, EvenQ]] == 8 &] (* Amiram Eldar, Feb 02 2021 *)

def eodiff(n):
  digs = list(map(int, str(n)))
  return abs(sum(d for d in digs if d%2==0)-sum(d for d in digs if d%2==1))
def aupto(lim): return [m for m in range(lim+1) if eodiff(m) == 8]
print(aupto(789)) # Michael S. Branicky, Feb 21 2021

A341011 a(n) is the smallest positive number m not yet in the sequence with the property that the sum of the even digits of m and the sum of the odd digits of m differ by n.

Original entry on oeis.org

112, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 119, 39, 139, 59, 159, 79, 179, 99, 199, 488, 399, 688, 599, 888, 799, 1799, 999, 1999, 11999, 3999, 13999, 5999, 15999, 7999, 17999, 9999, 19999, 68888, 39999, 88888, 59999, 159999, 79999, 179999, 99999, 199999, 1199999, 399999, 1399999, 599999, 1599999, 799999, 1799999, 999999

Offset: 0

Carole Dubois and Eric Angelini, Feb 02 2021

This is the lexicographically earliest sequence of distinct integers > 0 having this property.

Indices of terms not congruent to 9 (mod 10): 0, 1, 2, 3, 4, 5, 6, 7, 8, 20, 22, 24, 38, 40, 56, .... - Robert G. Wilson v, Feb 21 2021

			a(19) = 199 since 199 is the smallest number such that the sum of even digits (0) and the sum of odd digits (19) differ by n = 19;
a(20) = 488 since 488 is the smallest number such that the sum of even digits (20) and the sum of odd digits (0) differ by n = 20; etc.

Robert G. Wilson v, Table of n, a(n) for n = 0..8983 (terms 1 to 100 from Carole Dubois)

Cf. A009994, A036301, A341002, A341003, A341004, A341005, A341006, A341007, A341008, A341009, A341010.

del[n_] := Abs[Plus @@ Select[(d = IntegerDigits[n]), OddQ] - Plus @@ Select[d, EvenQ]]; m = 54; s = Table[0, {m}]; c = n = 0; While[c < m, n++; i = del[n]; If[i > 0 && i <= m && s[[i]] == 0, c++; s[[i]] = n]]; s (* Amiram Eldar, Feb 02 2021 *)
f[n_] := Block[{b, c, d, e, o}, d = 0; c = Floor[n/9]; b = 10^c -1; While[n != (Plus @@ IntegerDigits[d*10^c + b]), If[ OddQ@ d, d += 2, d++]]; o = d*10^c + b;
d = 0; c = Floor[n/8]; b = 8(10^c -1)/9; While[n != (Plus @@ IntegerDigits[d*10^c + b]), If[ OddQ@ d, d++, d += 2]]; e = d*10^c + b; Min[o, e]]; f[0] = 112; (* Robert G. Wilson v, Feb 21 2021 *)

a(0) added by Robert G. Wilson v, Feb 21 2021

A343403 Numbers k such that the product of the digits of k is not the product of digits of any earlier term in the sequence.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 25, 26, 27, 28, 29, 35, 37, 38, 39, 45, 47, 48, 49, 55, 56, 57, 58, 59, 67, 68, 69, 77, 78, 79, 88, 89, 99, 255, 256, 257, 258, 259, 267, 268, 269, 277, 278, 279, 288, 289, 299, 355, 357, 358, 359, 377, 378, 379, 388, 389, 399, 455

Offset: 1

Collin King, Apr 14 2021

Observations:
The digits 0 and 1 appear only in terms 0 and 1, respectively.
Terms cannot contain two 2s, two 3s, a 2 and a 3, a 2 and a 4, a 3 and a 4, or a 4 and a 6.
Digits in each term appear in ascending order (A009994).

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A343403

Cf. A003001, A007954, A009994 (ascending digits), A031346, A068189, A343160.

# product of digits
A007954 := proc(n::integer)
    if n = 0 then
        0;
    else
        mul( d, d=convert(n, base, 10)) ;
    end if;
end proc:
hit:=Array(0..10000,-1);
a:=[0];
hit[0]:=0;
for n from 1 to 50000 do p:=A007954(n);
   if p>0 and hit[p]=-1 then hit[p]:=n; a:=[op(a),n]; fi; od:
a; # N. J. A. Sloane, Apr 14 2021

PARI
```
See Links section.
```

A348318 Number of n-digit Niven (or Harshad) numbers not containing the digit 0.

Original entry on oeis.org

9, 14, 108, 710, 4978, 35724, 273032, 2097356, 16674554, 135091242, 1112325268, 9296413365, 77991481271, 654495034497, 5420117473932

Offset: 1

Bernard Schott, Oct 17 2021

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

This sequence gives the number of such n-digit Niven numbers not containing the digit 0.

			a(2) = 14 because there are 14 integers {12, 18, 21, 24, 27, 36, 42, 45, 48, 54, 63, 72, 81, 84} in the set of Niven numbers with 2 digits and not containing the digit 0.

Diophante, Bon souvenir de Buenos-Aires.
John Scholes, 39th IMO 1998 shortlisted problems, problem N7.
Index to sequences related to Olympiads.

Cf. A005349, A009994, A140866, A217973, A348150, A348316, A348317.

q[n_] := ! MemberQ[(d = IntegerDigits[n]), 0] && Divisible[n, Plus @@ d]; a[n_] := Module[{c = 0, k = (10^n - 1)/9}, While[k < 10^n, If[q[k], c++]; k++]; c]; Array[a, 6] (* Amiram Eldar, Oct 17 2021 *)

from itertools import product
def a(n): return sum(1 for p in product("123456789", repeat=n) if int("".join(p))%sum(map(int, p)) == 0)
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, Oct 17 2021

a(6)-a(10) from Amiram Eldar, Oct 17 2021

a(11)-a(15) from Martin Ehrenstein, Oct 22 2021

A377948 Numbers that have at least 1 repeated decimal digit and whose decimal digits are nondecreasing as place value decreases.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 112, 113, 114, 115, 116, 117, 118, 119, 122, 133, 144, 155, 166, 177, 188, 199, 222, 223, 224, 225, 226, 227, 228, 229, 233, 244, 255, 266, 277, 288, 299, 333, 334, 335, 336, 337, 338, 339, 344, 355, 366, 377, 388, 399

Offset: 1

Michael De Vlieger, Nov 14 2024

Intersection of A009994 and A109303.

Does not intersect either A009993 or A009995.

Michael De Vlieger, Table of n, a(n) for n = 1..10938 (includes all numbers up to 7 digits long)

Cf. A009993, A009994, A009995, A109303, A178788.

Select[Range[10^6], And[CountDistinct[#] != Length[#], AllTrue[Differences[#], # >= 0 &]] &[IntegerDigits[#]] &]
(* More efficient program: *)
b = 10; mm = b - 1; nn = 14;
s = Table[Map[Position[#, 1][[All, 1]] &,
  Permutations@ Join[ConstantArray[1, r], ConstantArray[0, mm - r] ] ],
    {r, Min[mm, nn]}];
Union@ Flatten@ Table[
  w = Apply[Join, Permutations /@ IntegerPartitions[n, Min[mm, n - 1] ] ];
  Reap[Do[
    Sow[Table[FromDigits[Flatten@
      MapIndexed[ConstantArray[m[[First[#2] ]], #1] &,
      w[[i]]], b], {m, s[[Length[w[[i]]] ]]} ] ],
    {i, Length[w]}] ][[-1, 1]], {n, 2, nn}]

A178788(a(n)) = 0.

A110069 Numbers n such that n = (d_1 + d_2 + ... + d_k)prime(d_1d_2...d_k) where d_1 d_2 ... d_k is the decimal expansion of n.

Original entry on oeis.org

188217, 216925, 329319, 22146969, 236256594, 269226639788

Offset: 1

Farideh Firoozbakht, Jul 17 2005

There is no further term up to 660000000.
This sequence is finite since (d_1+d_2+...+d_k)*prime(d_1*d_2*...*d_k) <= 9k * prime(9^k) << 9^k * k^2 << n. The bound can be made effective using the results of Dusart or others; for example, a(n) < 10^150. These can be improved with more work, but completing the sequence seems hard. - Charles R Greathouse IV, May 07 2011
a(7) > 7*10^14, if it exists. - Giovanni Resta, Jun 01 2020
If it exists, a(7) > 10^18. - Max Alekseyev, Jan 28 2024

			236256594 is in the sequence because 236256594 = (2 + 3 + 6 + 2 + 5 + 6 + 5 + 9 +4)*prime(2*3*6*2*5*6*5*9*4).

Cf. A009994, A097640.

Do[h = IntegerDigits[m]; l = Length[h]; If[Min[h] > 0 && m == Sum[h[[k]], {k, l}]*(Prime[Product[h[[k]], {k, l}]]), Print[m]], {m, 655000000}]

a(6) from Giovanni Resta, Jun 01 2020

A245017 Numbers k such that (product of digits of k) + 1 and (product of digits of k)^2 + 1 are both prime.

Original entry on oeis.org

1, 2, 4, 6, 11, 12, 14, 16, 21, 22, 23, 25, 28, 32, 41, 44, 49, 52, 58, 61, 66, 82, 85, 94, 111, 112, 114, 116, 121, 122, 123, 125, 128, 132, 141, 144, 149, 152, 158, 161, 166, 182, 185, 194, 211, 212, 213, 215, 218, 221, 224, 229, 231, 236, 242, 245, 251, 254, 263, 279, 281, 292

Offset: 1

Derek Orr, Jul 12 2014

A number k is a term of this sequence iff A007954(k) and A007954(k)^2 are both in A006093.

This sequence is infinite. With any number a(n), you can add infinitely many 1's to its decimal representation. E.g., 82 is in this sequence, so 821, 812, 1182, 18112, 81211, etc. are also terms of this sequence.

			(9*4) + 1 = 37 is prime and (9*4)^2 + 1 = 1297 is prime. Thus 94 is a term of this sequence.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A007954, A009994, A081988, A244748.

bpQ[n_]:=Module[{c=Times@@IntegerDigits[n]},AllTrue[{c+1,c^2+1},PrimeQ]]; Select[Range[300],bpQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 09 2019 *)

for(n=1, 10^3, d=digits(n); p=prod(i=1, #d, d[i]); if(ispseudoprime(p+1) && ispseudoprime(p^2 + 1), print1(n,", ")))

A256242 Numbers having digits in nondecreasing order and repeatedly setting n := A066308(n) yields a constant nonzero n.

Original entry on oeis.org

1, 89, 135, 139, 144, 233, 1224, 1367, 11249, 12222, 111126, 111266, 111338, 112229, 112337, 1111119, 1111134, 1111137, 1111177, 1111333, 1111346, 11111117, 11111119, 11111223, 11112236, 111111119, 111111139, 111111299, 111112334, 1111111169, 1111122233, 11111111118, 11111111133, 11111111369, 111111111133

Offset: 1

David A. Corneth, Mar 20 2015

Intersection of A009994 and A256240. All digits differ from 0.

Permutations of all numbers of the elements in the table give the first 56622402 elements from A256240 (unsorted).

David A. Corneth, Table of n, a(n) for n = 1..166 (all elements below 10^100).

Cf. A009994, A256240.

isok(n) = {d = digits(n); if (vecsort(d,,2) == d, ok = 1; while (ok, newn = sum(k=1, #d, d[k])*prod(k=1,#d, d[k]); if (! newn, return (0)); if (newn == n, return (1)); n = newn; d = digits(n);););} \\ Michel Marcus, Mar 27 2015

A302438 Numbers with digits in nondecreasing order and even digital sum (in base 10) whose digits can be partitioned in two multisets with equal digital sum.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 112, 123, 134, 145, 156, 167, 178, 189, 224, 235, 246, 257, 268, 279, 336, 347, 358, 369, 448, 459, 1111, 1113, 1122, 1124, 1133, 1135, 1144, 1146, 1155, 1157, 1166, 1168, 1177, 1179, 1188, 1199, 1223, 1225, 1234, 1236, 1245, 1247

Offset: 1

David A. Corneth, Apr 08 2018

			a(5000) = 11222699 is in this sequence as it has digits in nondecreasing order and an even digital sum, which is 32. The digits can be partitioned in two multisets with equal sum, for example as {1, 6, 9} and {1, 2, 2, 2, 9}, each having sum 16.

David A. Corneth, Table of n, a(n) for n = 1..11671
David A. Corneth, PARI prog

Cf. A009994, A054683.

PARI
```
\\ See PARI link.
```

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A333955 Numbers k with digits in nondecreasing order and each digit greater than 1 such that the iterated product of digits of k is a prime.

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A341009 Numbers whose sum of even digits and sum of odd digits differ by 8.

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A341011 a(n) is the smallest positive number m not yet in the sequence with the property that the sum of the even digits of m and the sum of the odd digits of m differ by n.

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A343403 Numbers k such that the product of the digits of k is not the product of digits of any earlier term in the sequence.

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A348318 Number of n-digit Niven (or Harshad) numbers not containing the digit 0.

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A377948 Numbers that have at least 1 repeated decimal digit and whose decimal digits are nondecreasing as place value decreases.

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A110069 Numbers n such that n = (d_1 + d_2 + ... + d_k)*prime(d_1*d_2*...*d_k) where d_1 d_2 ... d_k is the decimal expansion of n.

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A245017 Numbers k such that (product of digits of k) + 1 and (product of digits of k)^2 + 1 are both prime.

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A110069 Numbers n such that n = (d_1 + d_2 + ... + d_k)prime(d_1d_2...d_k) where d_1 d_2 ... d_k is the decimal expansion of n.