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

1,1

Comments

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.

Links

Table of n, a(n) for n=1..51.

Example

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.

Mathematica

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 *)

Prog

(PARI) 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)
(Scala) 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

Crossrefs

Cf. A002473, A003001, A009994, A028843.

Sequence in context: A085907 A024777 A002957 * A333960 A273726 A211660

Adjacent sequences: A333952 A333953 A333954 * A333956 A333957 A333958

Keyword

nonn,base

Author

David A. Corneth, Apr 11 2020

Status

approved