A067251 -id:A067251 - cp's OEIS frontend

A355622 a(n) is the n-digit positive number with no trailing zeros and coprime to its digital reversal R(a(n)) at which abs(a(n)/R(a(n))-Pi) is minimized.

1, 92, 581, 5471, 52861, 998713, 7774742, 93630892, 422334431, 9190135292, 45425395441, 472539314051, 5784475521481, 49371008251751, 939253175379892, 9265811239939492, 52949745472445861, 952186420153090303, 9836241210282790313, 36386277546811128511, 442327789252803797041

Offset: 1

Stefano Spezia, Jul 10 2022

a(n) and R(a(n)) have the same number of digits.

Petr Beckmann wrote that the fraction 92/29, corresponding to the second term of the sequence, appeared as value of Pi in a document written in A.D. 718.

			n              fraction    approximated value
-   -------------------    ------------------
1                     1    1
2                 92/29    3.1724137931034...
3               581/185    3.1405405405405...
4             5471/1745    3.1352435530086...
5           52861/16825    3.1418127786033...
6         998713/317899    3.1416047235128...
7       7774742/2474777    3.1415929596889...
8     93630892/29803639    3.1415926088757...
9   422334431/134433224    3.1415926690860...
...

Petr Beckmann, A History of Pi, 3rd Ed., Boulder, Colorado: The Golem Press (1974): p. 27.

Bert Dobbelaere, Table of n, a(n) for n = 1..22

Cf. A000796, A002485, A004086, A067251.

Cf. A355623 (denominator or digital reversal).

nmax=9; a={}; For[n=1, n<=nmax, n++, minim=Infinity; For[k=10^(n-1), k<=10^n-1, k++, If[(dist=Abs[k/FromDigits[Reverse[IntegerDigits[k]]]-Pi]) < minim && Last[IntegerDigits[k]]!=0 && GCD[k,FromDigits[Reverse[IntegerDigits[k]]]]==1, minim=dist; kmin=k]]; AppendTo[a, kmin]]; a

a(10)-a(19) from Bert Dobbelaere, Jul 17 2022

a(20)-a(21) from Bert Dobbelaere, Sep 05 2022

A355623 a(n) is the n-digit positive number with no trailing zeros and coprime to its digital reversal R(a(n)) at which abs(R(a(n))/a(n)-Pi) is minimized.

Original entry on oeis.org

1, 29, 185, 1745, 16825, 317899, 2474777, 29803639, 134433224, 2925310919, 14459352454, 150413935274, 1841255744875, 15715280017394, 298973571352939, 2949399321185629, 16854427454794925, 303090351024681259, 3130972820121426389, 11582111864577268363, 140797308252987723244

Offset: 1

Stefano Spezia, Jul 10 2022

a(n) and R(a(n)) have the same number of digits.

Petr Beckmann wrote that the fraction 92/29, corresponding to the second term of the sequence, appeared as value of Pi in a document written in A.D. 718.

			n              fraction    approximated value
-   -------------------    ------------------
1                     1    1
2                 92/29    3.1724137931034...
3               581/185    3.1405405405405...
4             5471/1745    3.1352435530086...
5           52861/16825    3.1418127786033...
6         998713/317899    3.1416047235128...
7       7774742/2474777    3.1415929596889...
8     93630892/29803639    3.1415926088757...
9   422334431/134433224    3.1415926690860...
...

Petr Beckmann, A History of Pi, 3rd Ed., Boulder, Colorado: The Golem Press (1974): p. 27.

Bert Dobbelaere, Table of n, a(n) for n = 1..22

Cf. A000796, A002485, A004086, A067251.

Cf. A355622 (numerator or digital reversal).

nmax=9; a={}; For[n=1, n<=nmax, n++, minim=Infinity; For[k=10^(n-1), k<=10^n-1, k++, If[(dist=Abs[FromDigits[Reverse[IntegerDigits[k]]]/k-Pi]) < minim && Last[IntegerDigits[k]]!=0 && GCD[k,FromDigits[Reverse[IntegerDigits[k]]]]==1, minim=dist; kmin=k]]; AppendTo[a, kmin]]; a

a(10)-a(19) from Bert Dobbelaere, Jul 17 2022

a(20)-a(21) from Bert Dobbelaere, Sep 05 2022

A209931 Numbers k such that smallest digit of all divisors of k is 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 111

Offset: 1

Jaroslav Krizek, Mar 20 2012

Also numbers k such that smallest digit of concatenation of all divisors of k (A037278 or A176558) is 1.
Sequence is not the same as A052382, first deviation is at a(173): A052382(173) = 212, a(173) = 213. [Corrected by Michael S. Branicky, Jul 01 2025.]
Sequence is not the same as A067251, first deviation is at a(91): A067251 (91) = 101, a(91) = 111.
Complement of A209932.

			Number 24 is in sequence because smallest digit of all divisors of 24 (1, 2, 4, 8, 3, 6, 12, 24) is 1.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A052382, A067251, A209929 (smallest digit of all divisors of n).

isA209931 := proc(n)
    digsdiv := {} ;
    for d in numtheory[divisors](n) do
        dgs := convert(convert(d,base,10),set) ;
        digsdiv := digsdiv union dgs ;
    end do:
    if 0 in digsdiv then
        false;
    else
        true ;
    end if;
end proc:
A209931 := proc(n)
    option remember;
    if n =1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA209931(a) then
                return a;
            end if;
        end do;
    end if;
end proc:
seq(A209931(n),n=1..120) ;# R. J. Mathar, Dec 28 2023

Select[Range[100], Min[IntegerDigits[Divisors[#]]] == 1 &] (* Paolo Xausa, Jul 03 2025 *)

from sympy import divisors
def ok(n): return all('0' not in str(d) for d in divisors(n, generator=True))
print([k for k in range(1, 112) if ok(k)]) # Michael S. Branicky, Jul 01 2025

A217562 Even numbers not divisible by 5.

Original entry on oeis.org

2, 4, 6, 8, 12, 14, 16, 18, 22, 24, 26, 28, 32, 34, 36, 38, 42, 44, 46, 48, 52, 54, 56, 58, 62, 64, 66, 68, 72, 74, 76, 78, 82, 84, 86, 88, 92, 94, 96, 98, 102, 104, 106, 108, 112, 114, 116, 118, 122, 124, 126, 128, 132

Offset: 1

Jeremy Gardiner, Oct 06 2012

Numbers ending with 2,4,6,8 in base 10.
No term is divisible by 10 therefore a subsequence of A067251 (Numbers with no trailing zeros in decimal representation).
Union of this sequence with A005408 (The odd numbers) gives A067251.
Union of this sequence with A045572 (Numbers that are odd but not divisible by 5) gives A047201.
The even numbers divisible by 5 are A008592 (Multiples of 10).

Jeremy Gardiner, Table of n, a(n) for n = 1..4000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A005408, A005843, A045572, A047201, A067251.

for n=1 to 199
if n mod 5 <> 0 and n mod 2 <> 1 then print str$(n)+", ";
next n
print

I:=[2, 4, 6, 8, 12]; [n le 5 select I[n] else Self(n-1) + Self(n-4) - Self(n-5): n in [1..60]]; // Vincenzo Librandi, Dec 28 2012

CoefficientList[Series[2*(1 + x + x^2 + x^3 + x^4)/((1 + x)*(1 + x^2)*(x - 1)^2), {x, 0, 100}], x] (* Vincenzo Librandi, Dec 28 2012 *)

A217562(n)=(n-1)*5\2+2 \\ M. F. Hasler, Oct 07 2012

def A217562(n): return (5*n-1>>1)&-2 # Chai Wah Wu, Apr 21 2025

a(n) = 2*A047201(n).
G.f.: 2*x*(1+x+x^2+x^3+x^4) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Oct 06 2012
a(n) = 2*(n+floor((n-1)/4)). - Aaron J Grech, Sep 28 2024
E.g.f.: (4 - cos(x) + (5*x - 3)*cosh(x) + sin(x) + (5*x - 2)*sinh(x))/2. - Stefano Spezia, Sep 28 2024

A122484 Numbers k not ending in zero such that the sum of digits of k is >= the sum of digits of k^4 (in base 10).

Original entry on oeis.org

1, 7, 19, 67, 124499, 594959999, 1349969999, 57999659949, 84936699999, 498998999999

Offset: 1

Martin Raab, Sep 15 2006

I've also found 498998999999, 7494994999999, 34999974999999 and some larger numbers, but not all values in between have been checked.
One is likely to find an example of the form 5*10^j - m*10^floor(j/2) - 1 or 7.5*10^j - m*10^floor(j/2) - 1 for j > 12 within the first 10^(floor(j/2)-1) m's.
Is this sequence finite? - Charles R Greathouse IV, Jan 12 2012
This sequence is infinite: for N = 7.5*10^j - 40*10^floor(j/2) - 1 one has A007953(N) = 9j-2 and A007953(N^4) <= 9j-2 for all j > 16, with equality for all even j > 16. - M. F. Hasler, Jan 14 2012
a(11) > 10^12. - Delbert L. Johnson, May 01 2023

			67 is a term because 67 has a digital sum of 13 and 67^4 = 20151121 which also has a digital sum of 13.
594959999 has a digital sum of 68 and 594959999^4 has a digital sum of 67, i.e., less than 68.

Cf. A064210.

is_A122484(n)= n%10 && A007953(n) >= A007953(n^4) \\ M. F. Hasler, Jan 14 2012

A122484 = { k in A067251 | A007953(k) >= A007953(k^4) }. - M. F. Hasler, Jan 14 2012

a(8) and a(9) from Martin Raab, Oct 17 2008

a(10) from Delbert L. Johnson, May 01 2023

A178158 Numbers n that are divisible by every suffix of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 21, 22, 24, 25, 31, 32, 33, 35, 36, 41, 42, 44, 45, 48, 51, 52, 55, 61, 62, 63, 64, 65, 66, 71, 72, 75, 77, 81, 82, 84, 85, 88, 91, 92, 93, 95, 96, 99, 101, 102, 104, 105, 125, 201, 202, 204, 205, 208, 225, 301, 302, 303, 304, 305, 306, 312, 315, 325, 375, 401, 402, 404, 405, 408, 425, 501, 502, 504, 505, 525, 601, 602, 603, 604

Offset: 1

Michel Lagneau, Dec 17 2010

If n = y*10^d+z is in the sequence, where 1<=y<=9 and z < 10^d, then z | y*10^d. - Robert Israel, Oct 17 2018

			9375 is in the sequence because :
.     5 | 9375 ;
.    75 | 9375 ;
.   375 | 9375 ;
.  9375 | 9375 .

Robert Israel, Table of n, a(n) for n = 1..10000 (first 788 terms from Reinhard Zumkeller)
Math Overflow, The number of numbers no greater than n that are divisible by all their suffixes

Cf. A034709 .

Cf. A067251.

import Data.List (tails)
a178158 n = a178158_list !! (n-1)
a178158_list = filter (\suff -> all ((== 0) . (mod suff))
   (map read $ tail $ init $ tails $ show suff :: [Integer])) a067251_list
-- Reinhard Zumkeller, Mar 26 2012

with(numtheory):T:=array(1..5):for n from 1 to 10000 do:ind:=0:l:=length(n):n0:=n:s:=0:for m from 0 to l-1 do:q:=n0:u:=irem(q, 10):v:=iquo(q, 10):n0:=v : s:=s + u*10 ^m:if irem(n,10)<>0 and irem(n, s)=0 then ind:=ind+1:else fi:od:if ind=l then printf(`%d,`, n):else fi:od:
# Alternative:
filter:= proc(x)
  if x mod 10 = 0 then return false fi;
  andmap(t -> type(x/(x mod 10^t),integer), [$1..ilog10(x)])
end proc:
Res:= $1..9:
for d from 1 to 6 do
  for y from 1 to 9 do
    for z in sort(convert(select(`<`,numtheory:-divisors(y*10^d),10^d),list)) do
      if filter(y*10^d+z) then
         Res:= Res, y*10^d+z;
      fi
od od od:
Res; # Robert Israel, Oct 17 2018

A255430 Natural numbers n, not multiples of 10, such that n and n^2 lack the digit 1 in their decimal expansions.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 22, 23, 24, 25, 26, 27, 28, 45, 47, 48, 52, 53, 55, 57, 58, 62, 63, 64, 65, 66, 67, 68, 73, 74, 75, 76, 77, 78, 82, 83, 84, 85, 86, 87, 88, 92, 93, 94, 95, 97, 98, 202, 205, 206, 207, 208, 222, 223, 225, 232, 233, 234, 235, 236, 238, 242, 243, 244, 245, 252, 253, 255

Offset: 1

Zak Seidov, Feb 23 2015

No trailing zeros allowed.

Cf. A000027, A052383, A067251, A255398.

A342950 7-smooth numbers not divisible by 10: positive numbers whose prime divisors are all <= 7 but do not contain both 2 and 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 15, 16, 18, 21, 24, 25, 27, 28, 32, 35, 36, 42, 45, 48, 49, 54, 56, 63, 64, 72, 75, 81, 84, 96, 98, 105, 108, 112, 125, 126, 128, 135, 144, 147, 162, 168, 175, 189, 192, 196, 216, 224, 225, 243, 245, 252, 256, 288, 294, 315, 324

Offset: 1

David A. Corneth, Mar 30 2021

nonn

			12 is in the sequence as all of its prime divisors are <= 7 and 12 is not divisible by 10.

David A. Corneth, Table of n, a(n) for n = 1..10195

Union of A108319 and A108347.

Intersection of A002473 and A067251.

Select[Range@500,Max[First/@FactorInteger@#]<=7&&Mod[#,10]!=0&] (* Giorgos Kalogeropoulos, Mar 30 2021 *)

is(n) = if(n%10 == 0, return(0)); forprime(p = 2, 7, n/=p^valuation(n, p)); n==1

A342950_list, n = [], 1
while n < 10**9:
    if n % 10:
        m = n
        for p in (2,3,5,7):
            q, r = divmod(m,p)
            while r == 0:
                m = q
                q, r = divmod(m,p)
        if m == 1:
            A342950_list.append(n)
    n += 1 # Chai Wah Wu, Mar 31 2021

from sympy import integer_log
def A342950(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n+x
        for i in range(integer_log(x,7)[0]+1):
            for j in range(integer_log(m:=x//7**i,3)[0]+1):
                c -= (k:=m//3**j).bit_length()+integer_log(k,5)[0]
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 17 2024

# faster for initial segment of sequence
import heapq
from itertools import islice
def A342950gen(): # generator of terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], [2, 3, 5, 7]
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield v
            oldv = v
            for p in psmooth_primes:
                if not (p==2 and v%5==0) and not (p==5 and v&1==0):
                    heapq.heappush(h, v*p)
print(list(islice(A342950gen(), 65))) # Michael S. Branicky, Sep 17 2024

Sum_{n>=1} 1/a(n) = 63/16. - Amiram Eldar, Apr 01 2021

A344748 Numbers m with decimal expansion (d_k, ..., d_1) such that d_i = m * i mod 10 for i = 1..k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 21, 26, 42, 47, 63, 68, 84, 89, 147, 284, 321, 468, 505, 642, 789, 826, 963, 2468, 2963, 4321, 4826, 6284, 6789, 8147, 8642, 50505, 52963, 54321, 56789, 58147, 208642, 258147, 406284, 456789, 604826, 654321, 802468, 852963

Offset: 1

Rémy Sigrist, May 28 2021

Positive terms have no trailing zero in decimal representation (A067251), and are uniquely determined by their final digit d (A010879) and the number of digits, say k, in their decimal expansion (A055642); d*k cannot be a multiple of 10.

If m belongs to the sequence, then A217657(m) also belongs to the sequence.

			- 6 * 1 = 6 mod 10,
- 6 * 2 = 2 mod 10,
- 6 * 3 = 8 mod 10,
- 6 * 4 = 4 mod 10,
- so 4826 belongs to the sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..1826

Cf. A010879, A055642, A067251, A217657, A344749, A344822.

is(n) = { my (r=n); for (k=1, oo, if (r==0, return (1), (n*k)%10!=r%10, return (0), r\=10)) }

print (setbinop((d,k) -> sum(i=1, k, 10^(i-1) * ((d*i)%10)), [1..9], [0..6]))

def ok(m):
  d = str(m)
  return all(d[-i] == str((m*i)%10) for i in range(1, len(d)+1))
print(list(filter(ok, range(10**6)))) # Michael S. Branicky, May 29 2021

def auptod(maxdigits):
  alst = [0]
  for k in range(1, maxdigits+1):
    aklst = []
    for d1 in range(1, 10):
      d = [(d1*i)%10 for i in range(k, 0, -1)]
      if d[0] != 0: aklst.append(int("".join(map(str, d))))
    alst.extend(sorted(aklst))
  return alst
print(auptod(6)) # Michael S. Branicky, May 29 2021

A346942 Numbers whose square starts and ends with exactly 4 identical digits.

Original entry on oeis.org

235700, 258200, 333400, 471400, 577400, 666700, 816500, 881900, 942800, 1054200, 1054300, 1054400, 1054500, 1490700, 1490800, 1490900, 1825700, 1825800, 1825900, 2108100, 2108200, 2108300, 2357100, 2581900, 2788800, 2788900, 2981300, 2981400, 3162200, 3333200, 3333300

Offset: 1

Bernard Schott, Aug 08 2021

Terms are equal to 100 times the primitive terms of A346940, those that have no trailing zero in decimal representation, hence all terms end with exactly 00.

			258200 is a term because 258200^2 = 66667240000 starts with four 6's and ends with four 0's.
3334700 is not a term because 3334700^2 = 1111155560000 starts with five 1's (and ends with four 0's).

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

Numbers whose square '....' with exactly k identical digits:
  ---------------------------------------------------------------------------
  |    k  \'....'|    starts     |       ends         |   starts and ends   |
  ---------------------------------------------------------------------------
  |     k = 2    |    A346812    |      A346678       |       A346774       |
  |     k = 3    |    A346891    |      A039685       |       A346892       |
  |     k = 4    |    A346940    |    100*A067251     |    this sequence    |
  ---------------------------------------------------------------------------
Cf. A346926.

q[n_] := SameQ @@ (d = IntegerDigits[n^2])[[1 ;; 4]] && d[[5]] != d[[1]] && SameQ @@ d[[-4 ;; -1]] && d[[-5]] != d[[-1]]; Select[Range[10000, 3333300], q] (* Amiram Eldar, Aug 08 2021 *)

def ok(n):
  s = str(n*n)
  return len(s) > 4 and s[0] == s[1] == s[2] == s[3] != s[4] and s[-1] == s[-2] == s[-3] == s[-4] != s[-5]
print(list(filter(ok, range(3333333)))) # Michael S. Branicky, Aug 08 2021

A346942_list = [100*n for n in range(99,10**6) if n % 10 and (lambda x:x[0]==x[1]==x[2]==x[3]!=x[4])(str(n**2))] # Chai Wah Wu, Oct 02 2021

cp's OEIS Frontend

A355622 a(n) is the n-digit positive number with no trailing zeros and coprime to its digital reversal R(a(n)) at which abs(a(n)/R(a(n))-Pi) is minimized.

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A355623 a(n) is the n-digit positive number with no trailing zeros and coprime to its digital reversal R(a(n)) at which abs(R(a(n))/a(n)-Pi) is minimized.

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A209931 Numbers k such that smallest digit of all divisors of k is 1.

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A217562 Even numbers not divisible by 5.

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BASIC

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A122484 Numbers k not ending in zero such that the sum of digits of k is >= the sum of digits of k^4 (in base 10).

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PARI

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A178158 Numbers n that are divisible by every suffix of n.

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Haskell

Maple

A255430 Natural numbers n, not multiples of 10, such that n and n^2 lack the digit 1 in their decimal expansions.

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A342950 7-smooth numbers not divisible by 10: positive numbers whose prime divisors are all <= 7 but do not contain both 2 and 5.