A179239 -id:A179239 - cp's OEIS frontend

A276802 Non-repdigit numbers n such that A045876(n) ends with n.

554, 3328, 55553, 77764, 222221, 444442, 666663, 888865, 888884, 5555552, 6666595, 9999840, 33332680, 55555526, 66666557, 99998670, 333332176, 333333312, 555555551, 666665752, 666666624, 999997536, 999999936, 9999976480, 9999997844, 9999999668, 9999999923, 11111111110

Offset: 1

Altug Alkan, Sep 17 2016

			554 is a term because 455+545+554 = 1554 that ends with 554.
2338 is the least term having its digits. For all permutations p of digits of n, in this case 2338, (without leading zeros if any), A045876(n) = A045876(p). A045876(2338) = 53328. It contains the digits of 2338 and ends with its digits permuted. 2338 has 4 digits, as has 53328 mod 10^4 so 53328 mod 10^4 == 3328 is a term. - _David A. Corneth_, Oct 04 2016

David A. Corneth, Pari Program.

Cf. A045876, A139819 (non-repdigits), A179239.

A047726(n) = n=eval(Vec(Str(n))); (#n)!/prod(i=0, 9, sum(j=1, #n, n[j]==i)!);
A055642(n) = #Str(n);
A007953(n) = sumdigits(n);
A045876(n) = ((10^A055642(n)-1)/9)*(A047726(n)*A007953(n)/A055642(n));
isA010785(n) = {1==#Set(digits(n))}
is(n) = (A045876(n)-n) % 10^A055642(n) == 0 && !isA010785(n)

More terms from David A. Corneth, Oct 06 2016

A287478 Positive numbers m with the property that m is the least cyclic permutation of its digits with the same number of digits as m.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 40, 44, 45, 46, 47, 48, 49, 50, 55, 56, 57, 58, 59, 60, 66, 67, 68, 69, 70, 77, 78, 79, 80, 88, 89, 90, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120

Offset: 1

David A. Corneth, May 25 2017

First differs from A179239 at n = 83; a(83) = 120 and A179239(83) = 122. This sequence is a supersequence of A179239.
If there is a zero digit, then we do not consider the cyclic shift which begins with the zero digit.
The number 0 should also be in this list. The initial digit of any term must be its smallest nonzero digit. - M. F. Hasler, Oct 18 2019

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A179239, A287198.

Select[Range@ 120, Function[d, First@ Sort@ Map[FromDigits, DeleteCases[ NestList[RotateLeft, d, Length@ d - 1], ?(First@ # == 0 &)]] == #]@ IntegerDigits@ # &] (* _Michael De Vlieger, May 27 2017 *)

is(n) = my(d=digits(n), v=vector(#d), no=n, nn=n, l=List(n)); for(i=2,#d, no = nn\10; no = no+(nn-no*10)*10^(#d-1); if(#digits(no)==#d,listput(l, no)); nn=no); listsort(l); n==l[1]
is(n) = {my(d = digits(n), dd = concat(d, d)); for(i=2,#d, c=vector(#d, j, dd[i+j-1]); if(fromdigits(c) < n, if(c[1]!=0, return(0)))); 1}

is_A287478(n,D=digits(n))={!for(i=2,#D,((D[i]M. F. Hasler, Oct 18 2019

A328293 Composite numbers k such that k+A055012(k) is the cube of a prime.

Original entry on oeis.org

34, 12025, 12130, 22789, 102952, 103039, 205222, 226019, 300176, 492203, 492221, 570760, 1030144, 1224376, 1224466, 2570470, 2684090, 3307264, 3868067, 3868157, 4329380, 4656049, 4656427, 5176537, 6966262, 6966403, 6966421, 7186697, 7186787, 7187318, 7187516, 7644406, 11694973, 12007691, 12008315

Offset: 1

Will Gosnell and Robert Israel, Oct 11 2019

Computing the range of A055012(n) up to some upper limit using A179239 might help reduce the search space for finding terms. - David A. Corneth, Oct 11 2019

			a(3) = 12130 is included because 12130 is composite and 12130 + 1^3 + 2^3 + 1^3 + 3^3 + 0^3 = 12167 = 23^3 and 23 is prime.

Robert Israel, Table of n, a(n) for n = 1..2400

Cf. A030078, A055012, A179239.

filter:= proc(n) local x,t,F;
  if isprime(n) then return false fi;
  x:= n + add(t^3, t = convert(n,base,10));
  F:= ifactors(x)[2];
  nops(F)=1 and F[1][2]=3
end proc:
F:= proc(p,lastp) local n0;
  n0:= max(p^3 - 9^3*(1+ilog10(p^3)),lastp^3+1);
  select(filter, [$n0 .. p^3]);
end proc:
seq(op(F(ithprime(i),ithprime(i-1))),i=2..50);

(scan(a,b)=forcomposite(n=max(a,b-9^3*(logint(b,10)+1))+1,b, n+A055012(n)==b && printf(n","))); forprime(p=1+o=2,234, scan(o^3,p^3)) \\ M. F. Hasler, Oct 11 2019

A328515 Number of primes in permutations of digits per permutation class of the positive integers ordered by smallest member of this class excluding leading zeros.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 1, 0, 1, 2, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 0, 1, 3, 0, 1, 0, 0, 2, 2, 0, 0, 2, 2, 0, 2, 2, 0, 2, 1, 0, 3, 3, 0, 2, 0, 1, 2, 0, 0, 4, 0, 0, 3, 0, 0

Offset: 0

David A. Corneth, Oct 18 2019

Primitive sequence of A039999.

			A179239(67) = 103. Its permutations of digits without leading zeros are 103, 130, 301, 310. Of these, only 103 is prime which is one number. So a(67) = 1.

David A. Corneth, Table of n, a(n) for n = 0..9999

Cf. A039999, A166921, A179239.

A357096 Least number whose set of decimal digits coincides with the set of decimal digits of prime(n).

Original entry on oeis.org

2, 3, 5, 7, 1, 13, 17, 19, 23, 29, 13, 37, 14, 34, 47, 35, 59, 16, 67, 17, 37, 79, 38, 89, 79, 10, 103, 107, 109, 13, 127, 13, 137, 139, 149, 15, 157, 136, 167, 137, 179, 18, 19, 139, 179, 19, 12, 23, 27, 29, 23, 239, 124, 125, 257, 236, 269, 127, 27, 128, 238, 239

Offset: 1

Jean-Marc Rebert, Sep 12 2022

			prime(5) = 11 and 1 have the same set of digits {1}, and 1 is the smallest such number, hence a(5) = 1.

Cf. A179239, A179308.

f:= proc(p) local L,i;
  L:= sort(convert(convert(convert(p,base,10),set),list));
  if L[1] = 0 then L[1]:= L[2]; L[2]:= 0 fi;
  add(L[-i]*10^(i-1),i=1..nops(L)) end proc:
seq(f(ithprime(i)),i=1..100); # Robert Israel, Sep 12 2022

a[n_] := Module[{d = Union[IntegerDigits[Prime[n]]]}, If[d[[1]] == 0, d[[1;;2]] = d[[2;;1;;-1]]]; FromDigits[d]]; Array[a, 100] (* Amiram Eldar, Sep 13 2022 *)

a(n)=my(v=vecsort(digits(prime(n)),,8),w=v);if(v[1]==0,j=#v;w=if(j>2,v[3..j],[]);w=concat(Vecrev(v[1..2]),w));fromdigits(w)

from sympy import prime
def a(n):
    s = "".join(sorted(set(str(prime(n)))))
    return int(s) if "0" not in s else int(s[1] + "0" + s[2:])
print([a(n) for n in range(1, 63)]) # Michael S. Branicky, Sep 12 2022

A369203 a(n) is the first number that has exactly n anagrams that each have exactly n prime divisors, counted by multiplicity.

Original entry on oeis.org

2, 15, 117, 135, 1224, 10023, 10026, 50688, 104445, 100368, 1012257, 1002258, 1034568, 10027899, 10024569, 100002789, 100234566, 100236789, 1000024569, 1012566789, 10000224468, 10002367899, 10002345678, 100012344588, 100012234689, 100223456778, 1000012457889, 1002345566778

Offset: 1

Robert Israel, Jan 15 2024

a(n) is the first number that has n anagrams k such that A001222(k) = n.

Does 9 divide a(n) for n > 6? - David A. Corneth, Jan 16 2024

			a(4) = 135 is a term because 135 has 4 anagrams having 4 prime divisors, counted by multiplicity: 135 = 3^3 * 5, 315 = 3^2 * 5 * 7, 351 = 3^3 * 13 and 513 == 3^3 * 19, and no number < 135 works.
a(6) != 2367 because 2367 has exactly 7 anagrams with each having exactly 6 prime divisors (namely 2673, 3276, 3726, 6237, 6372, 6732, 7236). - _David A. Corneth_, Jan 16 2024

David A. Corneth, Table of n, a(n) for n = 1..61
Robert Israel, Anagrams of a(n) with n prime divisors, for n = 1 to 18

Cf. A001222, A369184. All terms are in A179239.

f:= proc(n) # numbers k such that n has k anagrams with Omega = k
        local L, W,WS,V,d, w, x, i;
      L:= convert(n, base, 10); d:= nops(L);
      L:= select(t -> t[-1] <> 0, combinat:-permute(L));
      L:= map(t-> add(t[i]*10^(i-1), i=1..d), L);
      W:= map(t -> numtheory:-bigomega(t), L);
      WS:= convert(W,set);
      for x in WS do V[x]:= 0 od;
      for x in W do V[x]:= V[x]+1 od;
      select(x -> V[x] = x, WS);
end proc:
g:= proc(xin,d,n) # first anagrams with n digits starting xin, all other digits >= d
  option remember;
  local i;
  if 1 + ilog10(xin) = n then return xin fi;
  seq(procname(10*xin+i,i,n), i=d..9)
end proc:
h:= proc(n) # first anagrams with n digits
  local i,j;
  seq(seq(g(i*10^j,i,n),j=n-1..0,-1),i=1..9)
end proc:
V:= 'V': m:= 0:
for d from 1 to 9 do
  for x in h(d) do
    for y in f(x) do
      if not assigned(V[y]) then V[y]:= x: m:= max(m,y) fi
od od od:
seq(V[y],y=1..m);

from collections import Counter
from sympy import primeomega as W
from sympy.utilities.iterables import multiset_permutations as MP
from itertools import combinations_with_replacement, count, islice
def counteq(n):
    c = Counter(W(int("".join(p))) for p in MP(str(n)) if p[0]!='0')
    return [i for i in c if c[i] == i]
def agen(): # generator of terms
    adict, n = dict(), 1
    for d in count(len(str(2**n))):
        for f in "123456789":
            for r in combinations_with_replacement("0123456789", d-1):
                k = int(f+"".join(r))
                for v in counteq(k):
                    if v not in adict:
                        adict[v] = k
                while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 8))) # Michael S. Branicky, Jan 16 2024

More terms from David A. Corneth, Jan 16 2024

A373179 a(n) is the smallest n-digit integer whose digit permutations make the maximum possible number of n-digit primes.

Original entry on oeis.org

2, 13, 149, 1237, 13789, 123479, 1235789, 12345679, 102345679, 1123456789, 10123456789, 1011233456789, 1012334567789, 10123345677899

Offset: 1

Gonzalo Martínez, May 26 2024

A065851(n) is the maximum number of n-digit primes which can be made by permuting n digits.
a(n) = k is the smallest n-digit k for which A046810(k) = A065851(n).
a(n) has its relevant digits sorted and not beginning with 0 and may or may not be one of the primes (it is for n = 1 to 7, but not at n = 8).

			For n=3, A065851(3) = 4 primes are reached by permuting the digits of a(3) = 149, namely {149, 419, 491, 941}. (4 primes are also reached from 179 and 379, but they're bigger numbers.)

Kevin Ryde, C Code

Cf. A000040, A046810, A065851, A134596, A179239.

C
```
/* See links. */
```

from sympy import nextprime
from collections import Counter
def smallest(t):
    nz = "".join(sorted(c for c in t if c != "0"))
    s = "".join(t) if "0" not in t else nz[0]+"0"*t.count("0")+nz[1:]
    return int(s)
def a(n):
    c, p = Counter(), nextprime(10**(n-1))
    while p < 10**n:
        c["".join(sorted(str(p)))] += 1
        p = nextprime(p)
    m = min(c.most_common(1), key=lambda x:smallest(x[0]))
    return smallest(m[0])  # m[1] generates A065851
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, May 28 2024

a(9)-a(11) from Michael S. Branicky, May 27 2024

a(12)-a(14) from Kevin Ryde, Jul 16 2024

A383304 Nonnegative integers whose difference between the largest and smallest digits is equal to the arithmetic mean of its digits.

Original entry on oeis.org

0, 13, 26, 31, 39, 62, 93, 123, 132, 144, 213, 225, 231, 246, 252, 264, 267, 276, 288, 312, 321, 348, 369, 384, 396, 414, 426, 438, 441, 462, 483, 522, 624, 627, 639, 642, 672, 693, 726, 762, 828, 834, 843, 882, 936, 963, 1133, 1223, 1232, 1313, 1322, 1331, 1344, 1434, 1443

Offset: 1

Stefano Spezia, Apr 22 2025

			144 is a term since 4 - 1 = 3 = (1 + 4 + 4)/3.

Cf. A037904, A054054, A054055, A179239, A371383, A371384, A383305.

Subsequence of A061383.

Select[Range[0,1500], Max[d=IntegerDigits[#]]-Min[d]==Mean[d] &]

def ok(n): return sum(d:=list(map(int, str(n)))) == (max(d) - min(d))*len(d)
print([k for k in range(1500) if ok(k)]) # Michael S. Branicky, Apr 23 2025

A276739 Least k such that A045876(k) is divisible by 10^n.

Original entry on oeis.org

1, 19, 10699, 102589, 10000112389

Offset: 0

Altug Alkan, Sep 16 2016

Corresponding values of A045876(k) are 1, 110, 3333300, 333333000, ...
Sequence is infinite.
a(5) > 10^18. - Giovanni Resta, Sep 27 2016
Subsequence of A179239. - David A. Corneth, Oct 01 2016

			a(1) = 19 because 19+91 = 110 is divisible by 10.

Cf. A045876.

Table[k = 1; While[! Divisible[Total[FromDigits /@ Permutations@ IntegerDigits@ k], 10^n], k++]; k, {n, 0, 3}] (* Michael De Vlieger, Sep 16 2016 *)

A047726(n) = n=digits(n); (#n)!/prod(i=0, 9, sum(j=1, #n, n[j]==i)!);
A055642(n) = #Str(n);
A007953(n) = sumdigits(n);
A045876(n) = ((10^A055642(n)-1)/9)*(A047726(n)*A007953(n)/A055642(n));
a(n) = {my(k = 1); while (A045876(k) % (10^n), k++); k;}

a(4) from Giovanni Resta, Sep 27 2016

A276758 Numbers n such that A045876(n) = A045876(n+1).

Original entry on oeis.org

10, 1010, 1100, 1119, 1339, 1519, 3139, 5119, 8899, 27799, 46699, 48499, 50559, 55059, 64699, 72799, 84499, 100110, 101010, 101100, 110010, 110100, 111000, 111229, 112129, 117799, 121129, 136699, 147499, 163699, 168199, 171799, 174499, 177199, 186199

Offset: 1

Altug Alkan, Sep 17 2016

A138147 is a subsequence. Therefore, the sequence is infinite. - David A. Corneth, Sep 17 2016
Suppose a term is of the form SDN, where S is a sequence of digits without leading zeros, D is a digit less than 9 and N is a sequence of digits 9 (possibly 0 nines; terms from A002283) and SDN is a concatenation of S, D and N. Let S' be a permutation of digits of S without leading zeros. Then S'DN is also in the sequence. To search terms one may choose S from A179239. - David A. Corneth, Sep 18 2016
Since (n + 8*k) = (n - k + 1)*(n - k) has solutions that are n = k + 3*sqrt(k) and n = k - 3*sqrt(k), for square values of k there are infinitely many terms such that: 1119, 1111119999, 111111111999999999, ...

			1339 is a term because A045876(1339) = A045876(1340).
See 2nd comment. As 27799 is in the sequence, we can see S = 27, D = 7 and N = 99. Now all permutations S' (distinct) of S without leading zeros give terms. They are 72, giving term 72799. - _David A. Corneth_, Sep 18 2016

Cf. A045876, A179239.

A047726(n) = n=digits(n); (#n)!/prod(i=0, 9, sum(j=1, #n, n[j]==i)!);
A007953(n) = sumdigits(n);
lista(nn) = for(n=1, nn, if(A047726(n)*A007953(n) == A047726(n+1)*A007953(n+1), print1(n, ", ")))

cp's OEIS Frontend

A276802 Non-repdigit numbers n such that A045876(n) ends with n.

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A287478 Positive numbers m with the property that m is the least cyclic permutation of its digits with the same number of digits as m.

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A328293 Composite numbers k such that k+A055012(k) is the cube of a prime.

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A328515 Number of primes in permutations of digits per permutation class of the positive integers ordered by smallest member of this class excluding leading zeros.

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A357096 Least number whose set of decimal digits coincides with the set of decimal digits of prime(n).

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A369203 a(n) is the first number that has exactly n anagrams that each have exactly n prime divisors, counted by multiplicity.

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A373179 a(n) is the smallest n-digit integer whose digit permutations make the maximum possible number of n-digit primes.

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C

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A383304 Nonnegative integers whose difference between the largest and smallest digits is equal to the arithmetic mean of its digits.

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