A009994 -id:A009994 - cp's OEIS frontend

A304392 Numbers without a digit 1 with digits in nondecreasing order and the product of digits is a power of 6.

6, 23, 49, 66, 229, 236, 334, 389, 469, 666, 2233, 2269, 2349, 2366, 2899, 3338, 3346, 3689, 4499, 4669, 6666, 22239, 22336, 22499, 22669, 23334, 23389, 23469, 23666, 26899, 33368, 33449, 33466, 34899, 36689, 44699, 46669, 66666, 88999, 222299, 222333, 222369, 223349

Offset: 1

David A. Corneth, Jun 20 2018

Applying any of the following to terms in this sequence in any order gives a term from A276038: - Prepend a 1. - Permute digits. - Do nothing.

Subsequence of A276038.

			229 is in the sequence because it has digits in nondecreasing order, no digit 1 and a product of digits 2*2*9 = 36 which is a power of 6.

Michael S. Branicky, Table of n, a(n) for n = 1..10656 (all terms with <= 23 digits)

Cf. A009994, A276038.

Select[Range[10^6], And[FreeQ[#, 1], AllTrue[Differences@ #, # > -1 &], IntegerQ@ Log[6, Times @@ #]] &@ IntegerDigits@ # &] (* Michael De Vlieger, Jun 30 2018 *)

is(n) = my(d = digits(n), p = prod(i = 1, #d, d[i])); d[1] >= 2 && vecsort(d) == d && 6^logint(p, 6) == p

from math import prod
from sympy.utilities.iterables import multiset_combinations
def auptod(maxdigs):
  n, digs, alst, targets = 0, 1, [], set(6**i for i in range(1, maxdigs*3))
  for digs in range(1, maxdigs+1):
    mcstr = "".join(str(d)*digs for d in "234689")
    for mc in multiset_combinations(mcstr, digs):
      if prod(map(int, mc)) in targets: alst.append(int("".join(mc)))
  return alst
print(auptod(6)) # Michael S. Branicky, Jun 23 2021

A318275 Numbers with digits in nondecreasing order and with multiplicative digital root > 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 26, 27, 28, 29, 33, 34, 35, 36, 37, 38, 39, 44, 46, 47, 48, 49, 57, 66, 67, 68, 77, 79, 88, 89, 99, 111, 112, 113, 114, 115, 116, 117, 118, 119, 122, 123, 124, 126, 127, 128, 129, 133, 134

Offset: 1

David A. Corneth, Aug 23 2018

This sequence is a primitive sequence of A277061, it has digits in nondecreasing order. Terms in A277061 can be found by permuting digits of terms of this sequence.

Cf. A009994, A277061, A299690.

Select[Range@ 134, And[FixedPoint[Times @@ IntegerDigits@ # &, #] != 0, AllTrue[Differences@ IntegerDigits@ #, # >= 0 &]] &] (* Michael De Vlieger, Aug 25 2018 *)

A340125 Numbers whose sum of even digits and sum of odd digits are equal and whose digits are in nondecreasing order.

Original entry on oeis.org

0, 112, 134, 156, 178, 336, 358, 1223, 1245, 1267, 1289, 1447, 1469, 2334, 2356, 2378, 2558, 3445, 3467, 3489, 3669, 4556, 4578, 5667, 5689, 6778, 7889, 11114, 11136, 11158, 11338, 12225, 12247, 12269, 12449, 22233, 22345, 22367, 22389, 22556, 22578, 23447, 23469, 24455

Offset: 1

David A. Corneth, Feb 21 2021

			1223 is in the sequence as the sum of the odd digits is 1 + 3 = 4 and the sum of the even digits is 2 + 2 = 4 are equal.

David A. Corneth, Table of n, a(n) for n = 1..10001

Intersection of A009994 and A036301.

seodQ[n_]:=With[{idn=IntegerDigits[n]},Total[Select[idn,EvenQ]]==Total[Select[idn,OddQ]]&&Min[Differences[idn]]>=0]; Select[Range[0,25000],seodQ] (* Harvey P. Dale, Dec 30 2024 *)

is(n) = { my(d = digits(n), w = vector(2)); if(d != vecsort(d), return(0)); for(i = 1, #d, w[d[i]%2 + 1] += d[i] ); w[1] == w[2] }

def ok(n):
  digs = list(map(int, str(n)))
  return sorted(digs) == digs and sum(d*(-1)**d for d in digs) == 0
def aupto(lim): return [m for m in range(lim+1) if ok(m)]
print(aupto(24455)) # Michael S. Branicky, Feb 21 2021

A342034 a(n) is the number of numbers k with n digits where k has digits in nondecreasing order and satisfies k < (product of digits of k) * (sum of digits of k).

Original entry on oeis.org

8, 41, 140, 367, 789, 1432, 2276, 3280, 4326, 5350, 6254, 7009, 7588, 7970, 8175, 8210, 8120, 7923, 7633, 7272, 6877, 6445, 6013, 5555, 5122, 4693, 4298, 3901, 3534, 3189, 2872, 2562, 2285, 2029, 1789, 1576, 1376, 1194, 1037, 893, 759, 654, 548, 454, 384, 315, 254, 210, 168, 127, 97, 79, 56, 39, 31, 21, 12, 8, 4

Offset: 1

David A. Corneth, Mar 05 2021

As A066310 is finite there exists m such that a(n) = 0 for all n > m.

a(n) = 0 for n >= 85 since 9^n*9n <= 10^(n-1) for n >= 85. This may occur as early as n = 60, as 9^n*9n <= 10^n-1 for n >= 60. But a(59) > 0 since 10^59-1 < 9^59*9*59. - Michael S. Branicky, Mar 05 2021

			a(1) = 8 as there are 8 one-digit numbers k as described in name. Those are {2, 3, 4, 5, 6, 7, 8, 9}.

David A. Corneth, PARI program

Cf. A009994, A066310.

PARI
```
See PARI link
```

from math import prod
from itertools import combinations_with_replacement as cwr
def c(digs): return int("".join(map(str, digs))) < prod(digs) * sum(digs)
def a(n): return sum(1 for u in cwr(range(1, 10), n) if c(u))
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Mar 05 2021

a(27)-a(41) from Michael S. Branicky, Mar 05 2021

A342226 Numbers k with digits in nondecreasing order and satisfying k < (product of digits of k) * (sum of digits of k).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 16, 17, 18, 19, 23, 24, 25, 26, 27, 28, 29, 33, 34, 35, 36, 37, 38, 39, 44, 45, 46, 47, 48, 49, 55, 56, 57, 58, 59, 66, 67, 68, 69, 77, 78, 79, 88, 89, 99, 127, 128, 129, 136, 137, 138, 139, 145, 146, 147, 148, 149, 155, 156, 157, 158

Offset: 1

David A. Corneth, Mar 06 2021

This sequence contains 165623 terms.

			39 is in the sequence as 39 has digits in nondecreasing order and 39 < (3 + 9) * (3*9).

Intersection of A009994 and A066310.

A342944 Numbers m such that d(1)^0 + d(2)^1 + ... + d(k)^(k-1) = d(1)! + d(2)! + ... + d(k)!, where d(i), i=1..k, are the digits of m.

Original entry on oeis.org

1, 11, 12, 111, 121, 133, 202, 1020, 1111, 1211, 1331, 1403, 2021, 2030, 2120, 2220, 2305, 2413, 3012, 3102, 3115, 3202, 3215, 3322, 3335, 4033, 4123, 4223, 4434, 10165, 10201, 10210, 10300, 10533, 11065, 11111, 11200, 12065, 12111, 12200, 13050, 13265, 13311

Offset: 1

Carole Dubois, Mar 30 2021

			2413 is in this sequence because 2^0 + 4^1 + 1^2 + 3^3 = 2! + 4! + 1! + 3! = 33.

Cf. A009994, A342945, A342826, A178354.

Select[Range@20000,Total[(a=IntegerDigits@#)^Range[0,Length@a-1]]==Total[a!]&] (* Giorgos Kalogeropoulos, Mar 30 2021 *)

is(n) = my(d = digits(n)); sum(i = 1, #d, d[i]!) == sum(i = 1, #d, d[i]^(i-1)) \\ David A. Corneth, Mar 30 2021

from math import factorial
def digfac(s): return sum(factorial(int(d)) for d in s)
def digpow(s): return sum(int(d)**i for i, d in enumerate(s))
def aupto(limit):
  alst = []
  for k in range(1, limit+1):
    s = str(k)
    if digpow(s) == digfac(s): alst.append(k)
  return alst
print(aupto(14000)) # Michael S. Branicky, Mar 30 2021

A344842 Numbers with digits in nondecreasing order whose digit sum is prime and whose digit product is a perfect square > 0.

Original entry on oeis.org

11, 14, 49, 111, 119, 122, 128, 133, 155, 166, 188, 199, 229, 236, 289, 368, 449, 559, 779, 1114, 1334, 1444, 1466, 1477, 1499, 2249, 2489, 3349, 4559, 4889, 4999, 11111, 11119, 11122, 11128, 11144, 11155, 11177, 11188, 11236, 11339, 11368, 11449, 11669, 11999, 12233

Offset: 1

David A. Corneth, May 29 2021

Primitive sequence of A344825.

			49 is in the sequence as its product of digits is 36 which is a perfect square > 0 and its sum of digits is 13 which is prime.

David A. Corneth, Table of n, a(n) for n = 1..29793 (terms <= 10^15)

Cf. A009994, A344825.

ndoQ[n_]:=Module[{id=IntegerDigits[n]},FreeQ[id,0]&&Min[ Differences[id]]> = 0&&PrimeQ[Total[id]]&&IntegerQ[Sqrt[Times@@id]]]; Select[Range[ 12500],ndoQ] (* Harvey P. Dale, Jun 18 2021 *)

uptoqdigits(n) = { my(res = List()); for(j = 2, n, forvec(x = vector(j, i, [1,9]), if(issquare(vecprod(x)) && isprime(vecsum(x)), listput(res, fromdigits(x)) ) , 1 ) ); res }

from math import prod
from sympy import isprime, integer_nthroot
from itertools import combinations_with_replacement as mc
def ok(s):
  d = list(map(int, s))
  return '0' not in s and isprime(sum(d)) and integer_nthroot(prod(d), 2)[1]
def auptod(digits): return [int("".join(p)) for d in range(2, digits+1) for p in mc("123456789", d) if ok("".join(p))]
print(auptod(5)) # Michael S. Branicky, May 29 2021

Definition (Name) corrected by Harvey P. Dale, Jun 18 2021

A350574 Primes p such that p, asc(p), and desc(p) are all distinct primes (where asc(p) and desc(p) are the digits of p in ascending and descending order, respectively) and p is the minimum of all permutations of its digits with this property.

Original entry on oeis.org

131, 197, 373, 419, 571, 593, 617, 839, 919, 1297, 1327, 1429, 1879, 1949, 1993, 2129, 2213, 2591, 3539, 4337, 4637, 4639, 5519, 6619, 8389, 8933, 11491, 11519, 11527, 11597, 11897, 11969, 12757, 12829, 12979, 13649, 13879, 14537, 14737, 14741, 14891

Offset: 1

William Riley Barker, Jan 06 2022

All terms in this sequence are zero-free: If p contained 0 as a digit, then desc(p) would end with the zero digit, making it divisible by 10 (thus composite).

			131 is prime, and so are asc(131) = 113 and desc(131) = 311. Further, these are all distinct primes.
419, asc(419) = 149, and desc(419) = 941 are all distinct primes, and 419 is the smallest permutation of the digits {1,4,9} with this property (149 is not included because asc(149) = 149, so these would not be distinct; 491 is not included because 419 < 491 with the same set of digits).

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A009994.

Subsequence of A038618.

d:= n-> convert(n, base, 10):
g:= (n, r)-> parse(cat(sort(d(n), r)[])):
f:= n-> (s-> nops(s)=3 and andmap(isprime, s))({n, g(n, `<`), g(n, `>`)}):
q:= n-> f(n) and n=min(select(f, map(x-> parse(cat(x[])),
        combinat[permute](d(n))))):
select(q, [$1..15000])[];  # Alois P. Heinz, Jan 15 2022

Select[Prime@Range@2000,AllTrue[FromDigits/@{s=Sort[d=IntegerDigits@#],Reverse@s},PrimeQ]&&Min@Most@Rest@Sort@Select[FromDigits/@Permutations[d],PrimeQ]==#&] (* Giorgos Kalogeropoulos, Jan 16 2022 *)

import numpy as np
# preliminary functions we will use in building our list
def is_prime(n):
    for d in range(2,int(np.sqrt(n))+1):
        if n % d == 0:
            return False
    return True
def asc(n): # returns integer with digits of n in ascending order
    n_list = [int(digit) for digit in str(n)] # separate digits of n into a list
    n_list.sort() # rearrange numbers in ascending order
    asc_n = int(''.join([str(digit) for digit in n_list])) # concatenate the sorted digits
    return asc_n
def desc(n): # returns integer with digits of n in descending order
    n_list = [int(digit) for digit in str(n)]
    n_list.sort(reverse=True)
    desc_n = int(''.join([str(digit) for digit in n_list]))
    return desc_n
N = 5
# get list of integers n such that n, asc(n), and desc(n) are all distinct primes
condition_1 = [n for n in range(2,10**N)
               if is_prime(n)
               and is_prime(asc(n))
               and is_prime(desc(n))
               and n not in (asc(n),desc(n))]
# refine so that our list includes only the minimum permutation of a given set of digits satisfying condition 1
condition_2 = []
for num in condition_1:
    if asc(num) not in [asc(n) for n in condition_2]:
        condition_2.append(num)
print(condition_2)

from itertools import count, islice, combinations_with_replacement
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A350574_gen(): # generator of terms
    for l in count(1):
        rlist = []
        for a in combinations_with_replacement('123456789',l):
            s = ''.join(a)
            p, q = int(s), int(s[::-1])
            if p != q and isprime(p) and isprime(q):
                for b in multiset_permutations(a):
                    r = int(''.join(b))
                    if p < r < q and isprime(r):
                        rlist.append(r)
                        break
        yield from sorted(rlist)
A350574_list = list(islice(A350574_gen(),50)) # Chai Wah Wu, Feb 13 2022

A354466 Numbers k such that the decimal expansion of the sum of the reciprocals of the digits of k starts with the digits of k in the same order.

Original entry on oeis.org

1, 13, 145, 153, 1825, 15789, 16666, 21583, 216666, 2416666, 28428571, 265833333, 3194444444, 3333333333, 9111111111, 35333333333, 3166666666666, 3819444444444, 26666666666666, 34166666666666, 527857142857142, 3944444444444444, 6135714285714285, 615833333333333333

Offset: 1

Metin Sariyar, Jun 01 2022

The sequence is infinite because all numbers of the form 10^(10^n-6) + 6*(10^(10^n-6)-1)/9, (n>0) are terms.
All terms are zeroless since 1/0 is undefined.
If n gives a sum < 1 then that sum is taken as 0.xyz.. but n does not start with 0, so not a term.

			28428571 is a term because 1/2 + 1/8 + 1/4 + 1/2 + 1/8 + 1/5 + 1/7 + 1/1 = 2.8428571...
825 is not a term since 1/8 + 1/2 + 1/5 = 0.825.

Kevin Ryde, Table of n, a(n) for n = 1..382
Michael S. Branicky, Python program
Kevin Ryde, PARI/GP Code

Cf. A009994, A034708, A337904.

Do[If[FreeQ[IntegerDigits[n], 0]&&Floor[Total[1/IntegerDigits[n]]*10^(IntegerLength[n]-IntegerLength[Floor[Total[1/IntegerDigits[n]]]])]==n&&Floor[Total[1/IntegerDigits[n]]]>0, Print[n]], {n, 1, 216666}]

PARI
```
\\ See links.
```
Python
```
# See links.
```

a(12)-a(24) from Michael S. Branicky, Jun 03 2022

A360443 Smallest integer m > n such that the multiset of nonzero decimal digits of m is exactly the same as the multiset of nonzero decimal digits of n.

Original entry on oeis.org

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 101, 21, 31, 41, 51, 61, 71, 81, 91, 200, 102, 202, 32, 42, 52, 62, 72, 82, 92, 300, 103, 203, 303, 43, 53, 63, 73, 83, 93, 400, 104, 204, 304, 404, 54, 64, 74, 84, 94, 500, 105, 205, 305, 405, 505, 65, 75, 85, 95, 600

Offset: 1

Marc LeBrun and M. F. Hasler, Feb 22 2023

Equivalently: a(n) is the number whose decimal digits are the next larger permutation of those of n, allowing any number of leading zeros.

M. F. Hasler, Table of n, a(n) for n = 1..1000
Wikipedia, Permutation: Generation in lexicographic order, as of Feb. 2023.

Cf. A057168 (analog for base 2), A354049 (digits of a(n) contain those of n as sub-multiset).

Cf. A009994 (numbers with digits in nondecreasing order: don't appear in this sequence).

A360443(n)={forperm(concat(0,digits(n)),p,n||return(fromdigits(Vec(p))); n=0)} \\ M. F. Hasler, Feb 23 2023; similar idea also suggested by Ruud H.G. van Tol.

# From Arthur O'Dwyer, edited by M. F. Hasler, Feb 22 2023
def A360443(n):
    s = '0' + str(n)
    i = next(i for i in range(len(s) - 1, 0, -1) if s[i-1] < s[i])
    tail = s[i-1:]
    j = min((ch, j) for j, ch in enumerate(tail) if s[i-1] < ch)[1]
    s = s[:i-1] + tail[j] + ''.join(sorted(tail[:j] + tail[j+1:]))
    return int(s)
for n in range(1, 100): print(n, A360443(n))

cp's OEIS Frontend

A304392 Numbers without a digit 1 with digits in nondecreasing order and the product of digits is a power of 6.

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A318275 Numbers with digits in nondecreasing order and with multiplicative digital root > 0.

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A340125 Numbers whose sum of even digits and sum of odd digits are equal and whose digits are in nondecreasing order.

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A342034 a(n) is the number of numbers k with n digits where k has digits in nondecreasing order and satisfies k < (product of digits of k) * (sum of digits of k).

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A342226 Numbers k with digits in nondecreasing order and satisfying k < (product of digits of k) * (sum of digits of k).

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A342944 Numbers m such that d(1)^0 + d(2)^1 + ... + d(k)^(k-1) = d(1)! + d(2)! + ... + d(k)!, where d(i), i=1..k, are the digits of m.

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PARI

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A344842 Numbers with digits in nondecreasing order whose digit sum is prime and whose digit product is a perfect square > 0.

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Mathematica

PARI

Python

Extensions

A350574 Primes p such that p, asc(p), and desc(p) are all distinct primes (where asc(p) and desc(p) are the digits of p in ascending and descending order, respectively) and p is the minimum of all permutations of its digits with this property.

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