A108571 -id:A108571 - cp's OEIS frontend

A181392 Squares in A108571.

1, 435343551252544, 543345144355225, 21343774737727744, 525664562544416656, 555621544626466564, 645545246266556416, 656542564646552164, 666524445565146256, 2766717773326766736, 8823883385555888521

Offset: 1

Patrick Wieschollek, Oct 17 2010

The last term is 999999999786876856487355368576387875784644 = 999999999893438428238^2. - Giovanni Resta, May 19 2013

There are no squares with 43, 44, or 45 digits. Indeed, numbers of 45 digits have sum of digits 1^1+2^2+...+9^9 = 285, which mod 9 is equal to 6. It is easy to verify that no power can be equal to 6 mod 9, hence there are no squares, cubes, etc. of 45 digits. Similarly, the numbers of 44 and 43 digits can only be obtained by omitting the single 1 or the two 2's, so mod 9 they are equal to 5 and 2, respectively. Again, 2 and 5 are not squares or cubes mod 9, but they can be powers with exponents k = 5, 7, 11, 13, 17, 19, 23, 25,... (numbers not divisible by 2 or 3). Since 10^(44/k) is at most 6.3*10^8 (for k=5) excluding higher powers by generating them is not a tremendous computational effort, which can be further reduced noticing that certain candidates can be excluded based on their last digits. For example, 9993^5 mod 10000 is 3193, which contains a 1. So no number ending in 9993 can be the base for a 5th power of 44 digits (which should lack the 1). Since 4th powers are squares too, they can have at most 42 digits, and since 10^(42/4) is about 3.16*10^10, it is not difficult to ascertain that no 4th powers belong to A108571. - Giovanni Resta, Jul 26 2015

Patrick Wieschollek and Giovanni Resta, Table of n, a(n) for n = 1..1000 (first 160 terms from Patrick Wieschollek)

Cf. A108571, A225886.

Edited by N. J. A. Sloane, Oct 17 2010

A225886 Numbers whose cube is in A108571 (i.e., if it contains digit 'd' it contains d copies of it).

Original entry on oeis.org

1, 39943162355, 367743626989, 372479081599, 384263945797, 425427609997, 833230637077, 924004183216, 986073549499, 996147664414, 1139347038032, 1420866014167, 1531843001692, 1706360863129, 1861996227349, 1871975416279, 1899165154429, 2077205393866

Offset: 1

Giovanni Resta, May 19 2013

The last term in the sequence is 99997991522989, whose cube is 999939746899855868958748787736573465847669.

			39943162355 is in the sequence since its cube 63727566782531668573568877888875 belongs to A108571.

Cf. A108571, A181392.

A127007 a(n) = number of n-digit terms in A108571.

Original entry on oeis.org

1, 1, 4, 5, 16, 82, 169, 541, 2272, 17965, 44407, 201751, 801515, 4890886, 52218595, 165519640, 835947970, 4290442728, 24096524166, 179566203960, 2739764737710, 9938147178960, 60997160143920, 331360222255920, 2154105076695000

Offset: 1

Zak Seidov, Jan 02 2007

First 9 terms coincide with terms in A007837. Sum of all 45 terms gives the total number of terms in A108571: 66712890763701234740813164553708284.

Zak Seidov, Table of n, a(n) for n = 1..45.

Cf. A007837, A108571.

a[n_, w_:{0}] := If[n == 0, Total[w]!/Times @@ (w!), Sum[a[n-k, Append[w, k]], {k, 1 + Last@w, Min[9, n]}]]; Array[a, 45] (* Giovanni Resta, May 19 2013 *)

A356071 Numbers whose square is in A108571.

Original entry on oeis.org

1, 20864888, 23309765, 146095088, 725027284, 745400258, 803458304, 810273142, 816409484, 1663345356, 2970502211, 5035530306, 5705842932, 5801143392, 5969549784, 6143081912, 6582975516, 6801076116, 6828208884, 6830325216, 6911416408, 7248217206, 7248822984, 7317408465

Offset: 1

Jianing Song, Jul 25 2022

Square root of A181392. The last term is 999999999893438428238.

There are no perfect powers beyond 1 in this sequence; see the comment in A181392.

			23309765 is a term since 23309765^2 = 543345144355225 has 1 digit '1', 2 digits '2', 3 digits '3', 4 digits '4' and 5 digits '5'.

Michael De Vlieger, Table of n, a(n) for n = 1..1000, computed from b-file at A181392.

Cf. A108571, A181392, A225886, A356072.

Map[Sqrt, Import["https://oeis.org/A181392/b181392.txt", "Data"][[1 ;; 24, -1]] ] (* Michael De Vlieger, Jul 25 2022, computed from b-file at A181392 *)

A356072 Cubes in A108571.

Original entry on oeis.org

1, 63727566782531668573568877888875, 49731947369965677347595574679599669, 51677995937349669479966437557564799, 56739945449541793697765966999776573, 76997569379457651594939677645469973, 578489779534895499899488878979775533, 788899738636778969999897877368221696

Offset: 1

Jianing Song, Jul 25 2022

Cube of A225886. The last term is 999939746899855868958748787736573465847669.

			63727566782531668573568877888875 is a term since it is the cube of 39943162355, and it has 1 digit '1', 2 digits '2', 3 digits '3', 5 digits '5', 6 digits '6', 7 digits '7' and 8 digits '8'.

Cf. A108571, A356071, A181392, A225886.

A125573 a(n) = number of n-digit terms with different sets of decimal digits in A108571.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 17, 18, 19, 21, 21, 22, 23, 23, 23, 23, 22, 21, 21, 19, 18, 17, 15, 13, 12, 10, 9, 8, 6, 5, 4, 3, 2, 2, 1, 1, 1

Offset: 1

Zak Seidov, Jan 02 2007

Cf. A007837, A034139, A108571, A127007.

a(n) = A034139(n). - R. J. Mathar, Mar 17 2025

A333369 Positive integers in which any odd digit, if present, occurs an odd number of times, and any even digit, if present, occurs an even number of times.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 17, 19, 22, 31, 35, 37, 39, 44, 51, 53, 57, 59, 66, 71, 73, 75, 79, 88, 91, 93, 95, 97, 100, 111, 122, 135, 137, 139, 144, 153, 157, 159, 166, 173, 175, 179, 188, 193, 195, 197, 212, 221, 223, 225, 227, 229, 232, 252, 272, 292, 300, 315, 317, 319, 322

Offset: 1

Bernard Schott, Mar 17 2020

Inspired by the 520th problem of Project Euler (see link) where such a number is called a "simber".
This sequence has little mathematical interest. The name "simber", which might be interpreted as "silly number", is deprecated. - N. J. A. Sloane, Aug 04 2022
The number of terms with respectively 1, 2, 3, ... digits is 5, 24, 130, ...

			656 is a 3-digit term because it has one 5 and two 6's.
447977 is a 6-digit term because it has one 9, two 4's and three 7's.

Michel Marcus, Table of n, a(n) for n = 1..10000
Project Euler, Problem 520: Simbers.

Cf. A108571 (finite subsequence), A353007.

seqQ[n_] := AllTrue[Tally @ IntegerDigits[n], EvenQ[Plus @@ #] &]; Select[Range[300], seqQ] (* Amiram Eldar, Mar 17 2020 *)

isok(m) = my(d=digits(m), s=Set(d)); for (i=1, #s, if (#select(x->(x==s[i]), d) % 2 != (s[i] % 2), return (0))); return (1); \\ Michel Marcus, Mar 17 2020

def ok(n): s = str(n); return all(s.count(d)%2 == int(d)%2 for d in set(s))
print([k for k in range(323) if ok(k)]) # Michael S. Branicky, Apr 15 2022

A139337 Replace each digit with its number of occurrences in decimal representation of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 22, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 22, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 22, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 22, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 22, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 22, 11, 11, 11, 11, 11, 11

Offset: 0

Reinhard Zumkeller, Apr 14 2008

A055642(a(n)) = A055642(n) for n < (10^10-1)/9; fixed points: a(A108571(n)) = A108571(n).

			a(373) = 212, since, reading the digits of 373 from left to right, 3 appeared twice, 7 once, 3 twice.

R. Zumkeller, Table of n, a(n) for n = 0..25000

import Data.List (group, sort); import Data.Maybe (mapMaybe)
a139337 n = read $ concatMap show $ mapMaybe (flip lookup ls) ds :: Int
   where ls = zip (map head zss) (map length zss)
         zss = group $ sort ds
         ds = map (read . return) $ show n :: [Int]
-- Reinhard Zumkeller, Mar 14 2014

a[n_] := IntegerDigits[n] /. Thread[{1, 2, 3, 4, 5, 6, 7, 8, 9, 0} -> DigitCount[n]] // FromDigits; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 28 2013 *)

A078348 Primes p such that every decimal digit d in p appears exactly d times.

Original entry on oeis.org

3313, 3331, 32233, 32323, 33223, 123323, 132233, 223133, 223313, 223331, 231323, 233231, 312233, 321323, 323123, 3344443, 3434443, 3443443, 4434343, 4443433, 14334443, 14443343, 14443433, 31434443, 31443443, 33434441, 33555553

Offset: 1

Carlos Rivera, Nov 22 2002

The largest term is the prime 99999999988888888777777766666655555444223343.

			In the prime 3313 the digit "1" appears exactly one time and the digit "3" appears exactly three times.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Primes in A108571.

ddp[x_]:=Select[FromDigits/@Permutations[Flatten[PadRight[{},#,#]&/@x]], PrimeQ]; Take[Flatten[ddp/@Subsets[Range[5]]]//Sort,40] (* Harvey P. Dale, May 13 2020 *)

from sympy import isprime
from itertools import chain, combinations as C, count, islice
from sympy.utilities.iterables import multiset_permutations as mp
def powerset(s):
    return chain.from_iterable(C(s, r) for r in range(len(s)+1))
def agen():
    sumlst = [[] for i in range(46)]
    for s in powerset(range(1, 10)): sumlst[sum(s)].append(s)
    for numdigits in count(1):
        found = set()
        for t in sumlst[numdigits]:
            diglst = "".join(str(i)*i for i in t)
            for m in mp(diglst, numdigits):
                t = int("".join(m))
                if isprime(t): found.add(t)
        yield from sorted(found)
print(list(islice(agen(), 30))) # Michael S. Branicky, Aug 10 2022

A105776 Self-describing integers with the rule: if the digit d, part of the integer i, is odd then there are d odd digits in this integer; if the digit d is even there are d even digits.

Original entry on oeis.org

1, 22, 122, 212, 221, 333, 4444, 14444, 22333, 23233, 23323, 23332, 32233, 32323, 32332, 33223, 33232, 33322, 41444, 44144, 44414, 44441, 555555, 666666, 1666666, 2255555, 2525555, 2552555, 2555255, 2555525, 2555552, 3334444, 3343444, 3344344, 3344434, 3344443

Offset: 1

Eric Angelini, May 04 2005

The subsequence of A108571 consisting of all terms that have at most one even and at most one odd digit (which of course may appear several times; otherwise said, no two different odd and no two different even digits are allowed). - M. F. Hasler, Sep 22 2014

			Integer 122 has 1 odd digit (1) and 2 even digits (2 and 2).

Cf. A108571.

More terms from M. F. Hasler, Sep 22 2014

cp's OEIS Frontend

A181392 Squares in A108571.

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A225886 Numbers whose cube is in A108571 (i.e., if it contains digit 'd' it contains d copies of it).

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A127007 a(n) = number of n-digit terms in A108571.

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A356071 Numbers whose square is in A108571.

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A356072 Cubes in A108571.

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A125573 a(n) = number of n-digit terms with different sets of decimal digits in A108571.

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A333369 Positive integers in which any odd digit, if present, occurs an odd number of times, and any even digit, if present, occurs an even number of times.

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A139337 Replace each digit with its number of occurrences in decimal representation of n.

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A078348 Primes p such that every decimal digit d in p appears exactly d times.

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