A217657 -id:A217657 - cp's OEIS frontend

A292683 Numbers divisible by themselves with first digit removed (A217657), excluding multiples of 10.

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, 416, 425, 501

Offset: 1

M. F. Hasler, Oct 17 2017

Obviously, any term multiplied by 10 would again be a term, so we exclude trailing zeros.
This sequence cannot contain single-digit numbers (which would yield 0 with the initial digit removed), in contrast to A178158 (numbers divisible by every suffix of n) where the condition is vacuously satisfied for single-digit numbers.
416 is the first term in the present sequence which is not in A178158.
See A292684 and A292685 for the (number of) multiples of N = a(n) which have the same property and yield the same ratio N/A217657(N).

			12 is in the sequence because it is divisible by 2.
416 is in the sequence because it is divisible by 16, 416 = 4*4*25 + 16.

Harvey P. Dale, Table of n, a(n) for n = 1..712 (All terms up to 10^7.)

Cf. A217657, A178158, A034709.

Cf. A292684, A292685.

fQ[n_] := Mod[n, 10] > 0 && Mod[n, n - Quotient[n, 10^Floor@ Log10@ n] 10^Floor@ Log10@ n] == 0; Select[ Range[11, 501], fQ] (* Robert G. Wilson v, Oct 18 2017 *)
Select[Range[10,550],Mod[#,10]!=0&&Mod[#,FromDigits[Rest[IntegerDigits[#]]]]==0&] (* Harvey P. Dale, Sep 15 2024 *)

select( is(n)=n%10&&(m=n%10^logint(n,10))&&!(n%m), [0..500])

A292684 a(n) is the number of positive integers k not divisible by 10 such that f(kN) = f(N) for N = A292683(n) and f(x) = x / (x without its first digit: A217657(x)).

Original entry on oeis.org

9, 4, 1, 9, 9, 4, 3, 3, 3, 3, 1, 1, 9, 9, 9, 7, 4, 9, 9, 9, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 9, 9, 9, 9, 9, 1, 1, 1, 1, 1, 1, 9, 9, 9, 9, 3, 9, 9, 9, 9, 9, 7, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 9, 9, 9, 9, 9, 9, 9

Offset: 1

M. F. Hasler, Oct 17 2017

Sequence A292683 lists the numbers n which are divisible by A217657(n), i.e., by n with its first digit removed.
We exclude k with trailing 0's (just like in A292683) because if k*N has the property, then 10*k*N trivially also has the property.
Is there any number for which there are more than 9 possible k-values?
All of the k-values are listed in the table A292685.

			For A292683(1) = 11, we have k = 1, ..., 9 satisfying 11*k / A217657(11*k) = 11.
For A292683(2) = 12, we have k = 1, 2, 3, 4 satisfying 12*k / A217657(12*k) = 6.
For A292683(3) = 15, we have only k = 1 satisfying 15*k / A217657(15*k) = 3.
For A292683(4) = 21, we have k = 1, 2, 3, 4, 5, 15, 25, 35 and 45 satisfying 21*k / A217657(21*k) = 2.

Cf. A292683, A292685, A217657, A000030.

(A217657(n)=n%10^logint(n,10)); A292684(n,N=A292683(n),r=N/A217657(N),a=[1])={for(k=2,oo,k%10||next;k>10*a[#a]&&break;A217657(k*N)*r==k*N&&a=concat(a,k));#a} \\ Instead of the 1st arg. n, one can directly give N (= A292683(n) by default) as 2nd arg. One could store only the last 'a' (and increase a counter) instead of storing all 'a's.

A292685 Irregular table where row n lists the positive integers k not divisible by 10 such that f(kN) = f(N) for N = A292683(n) and f(x) = x / (x without its first digit: A217657(x)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 1, 1, 2, 3, 4, 5, 15, 25, 35, 45, 1, 2, 3, 4, 5, 15, 25, 35, 45, 1, 2, 5, 15, 1, 5, 15, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 1, 1, 2, 5, 15, 25, 75, 125, 175, 225, 1, 2, 5, 15, 25, 75, 125, 175, 225, 1, 2, 5, 15, 25, 75, 125, 175, 225

Offset: 1

M. F. Hasler, Oct 18 2017

The numbers N listed in A292683 are such that N is divisible by A217657(N) = N with its initial digit removed. Most of these numbers have several multiples k*N which again have this property, but furthermore, such that the ratio k*N / A217657(k*N) is always the same. The corresponding k-values are listed here.

It is not rare that there are 9 such k-values, although the set of these is usually different from { 1, ..., 9 }. Is there any N for which there more than 10 such k-values?

			The table starts as follows:
    n | N=A292683(n) | N/A217657(N) | A292685(n,k=1..A292684(n))
    1 |      11      |      11      | 1, 2, 3, 4, 5, 6, 7, 8, 9
    2 |      12      |       6      | 1, 2, 3, 4
    3 |      15      |       3      | 1
    4 |      21      |      21      | 1, 2, 3, 4, 5, 15, 25, 35, 45
    5 |      24      |       6      | 1, 2, 5, 15
      |    (...)     |    (...)     | (...)
   68 |     416      |      26      | 1, 2, 5, 15, 25, 75, 125, 175, 225
           (...)
For A292683(2) = 12, we have k = 1, 2, 3, 4 satisfying 12*k / A217657(12*k) = 6, e.g., 12*4 = 48, 48 / 8 = 6 (= 12 / 2).
There are other k such that 12*k is divisible by A217657(12*k), e.g., k = 6, 7, 8, 17, ... (=> 12*k = 72, 84, 96, 204: all divisible by their last digit), but which yield ratios (here 36, 21, 16, 51) different from 6.
For n = 4, we have, e.g., 21*15 = 315, 315 / 15 = 21 (= 21 / 1), or 21*45 = 945, 945 / 45 = 21. Here too, e.g., 21*24 = 504 is divisible by 04, but 504 / 4 = 126, not 21.

Cf. A292683, A292684 (gives the row lengths), A217657, A000030.

A292685_row(n, N=A292683(n), r=N/A217657(N), a=[1])={for(k=2, oo, if(k%10,A217657(k*N)*r==k*N&&a=concat(a,k), k<10*a[#a]||break)); a} \\ Instead of the 1st arg. n, one can directly give N (= A292683(n) by default) as 2nd arg. It is not checked whether N is in A292683 (else the resulting vector should be empty).

A000030 Initial digit of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

N. J. A. Sloane, Simon Plouffe

When n - a(n)*10^[log_10 n] >= 10^[(log_10 n) - 1], where [] denotes floor, or when n < 100 and 10|n, n is the concatenation of a(n) and A217657(n). - Reinhard Zumkeller, Oct 10 2012, improved by M. F. Hasler, Nov 17 2018, and corrected by Glen Whitney, Jul 01 2022

Equivalent definition: The initial a(0) = 0 is followed by each digit in S = {1,...,9} once. Thereafter, repeat 10 times each digit in S. Then, repeat 100 times each digit in S, etc.

			23 begins with a 2, so a(23) = 2.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David W. Wilson, Table of n, a(n) for n = 0..10000
A. Cobham, Uniform Tag Sequences, Mathematical Systems Theory, 6 (1972), 164-192.

Cf. A010879 (final digit of n).
Cf. A061681, A130571, A109453, A134010, A052038.
Cf. A002993, A089951, A002994, A143464, A098174, A098175, A072543, A072544, A073600, A073601, A037904. - Reinhard Zumkeller, Aug 17 2008

a000030 = until (< 10) (`div` 10) -- Reinhard Zumkeller, Feb 20 2012, Feb 11 2011

[Intseq(n)[#Intseq(n)]: n in [1..100]]; // Vincenzo Librandi, Nov 17 2018

A000030 := proc(n)
    if n = 0 then
        0;
    else
        convert(n,base,10) ;
        %[-1] ;
    end if;
end proc:
seq(A000030(n),n=0..200) ;# N. J. A. Sloane, Feb 10 2017

Join[{0},First[IntegerDigits[#]]&/@Range[90]] (* Harvey P. Dale, Mar 01 2011 *)
Table[Floor[n/10^(Floor[Log10[n]])], {n, 1, 50}] (* G. C. Greubel, May 16 2017 *)
Table[NumberDigit[n,IntegerLength[n]-1],{n,0,100}] (* Harvey P. Dale, Aug 29 2021 *)

a(n)=if(n<10,n,a(n\10)) \\ Mainly for illustration.

A000030(n)=n\10^logint(n+!n,10) \\ Twice as fast as a(n)=digits(n)[1]. Before digits() was added in PARI v.2.6.0 (2013), one could use, e.g., Vecsmall(Str(n))[1]-48. - M. F. Hasler, Nov 17 2018

def a(n): return int(str(n)[0])
print([a(n) for n in range(85)]) # Michael S. Branicky, Jul 01 2022

a(n) = [n / 10^([log_10(n)])] where [] denotes floor and log_10(n) is the logarithm is base 10. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001

a(n) = k for k*10^j <= n < (k+1)*10^j for some j. - M. F. Hasler, Mar 23 2015

A040997 Absolute value of first digit of n minus sum of other digits of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Offset: 1

Felice Russo

This is different from |A055017(n)| = |(x1 + x3 + ...) - (x2 + x4 + ...)|, where x1,...,xk are the digits of n. - M. F. Hasler, Nov 09 2019

			a(371) = |3-7-1| = 5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A037904, A040114, A040115, A040163, A055017.

Cf. A000030, A007953, A217657.

a040997 n = abs $ a000030 n - a007953 (a217657 n) -- Reinhard Zumkeller, Oct 10 2012

apply( A040997(n)={abs(vecsum(n=digits(n))-n[1]*2)}, [1..199]) \\ M. F. Hasler, Nov 09 2019

If decimal expansion of n is x1 x2 ... xk then a(n) = |x1-x2-x3- ... -xk|.

a(n) = abs(A000030(n) - A007953(A217657(n))). - Reinhard Zumkeller, Oct 10 2012

Name edited and incorrect formula deleted by M. F. Hasler, Nov 09 2019

A226099 Positive integers that yield a prime when their most significant (i.e., leftmost) decimal digit is removed.

Original entry on oeis.org

12, 13, 15, 17, 22, 23, 25, 27, 32, 33, 35, 37, 42, 43, 45, 47, 52, 53, 55, 57, 62, 63, 65, 67, 72, 73, 75, 77, 82, 83, 85, 87, 92, 93, 95, 97, 102, 103, 105, 107, 111, 113, 117, 119, 123, 129, 131, 137, 141, 143, 147, 153, 159, 161, 167, 171, 173, 179, 183, 189, 197, 202, 203, 205, 207, 211, 213, 217

Offset: 1

Jonathan Vos Post, May 26 2013

Terms < 110 are the same as in A260181, numbers whose last digit is prime. - M. F. Hasler, Dec 20 2019

These are numbers with decimal expansion of the form k = xp where p is a prime and x is a single digit. Whether or not the number k itself is a prime is irrelevant. - N. J. A. Sloane, Dec 21 2019

			a(1) = 12 because when its most significant (or leftmost) digit (1) is removed, the remaining number 2 is prime, and it is the least such number.
102, 103, 105 and 107 are in the sequence because if the first digit is dropped, what is left is a 1-digit prime with a leading digit '0'.

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

Cf. A000040, A217657 (n without initial digit), A000030 (initial digit of n), A260181 (last digit is prime), A202262 (substrings are composite).

[k:k in [1..220]| IsPrime( k-Reverse(Intseq(k))[1]*10^(#Intseq(k)-1 ))]; // Marius A. Burtea, Dec 21 2019

Select[Range@ 300, PrimeQ@ FromDigits@ Rest@ IntegerDigits@ # &] (* Giovanni Resta, Dec 20 2019 *)

select( is(n)=isprime(n%10^logint(n+!n,10)), [0..222]) \\ M. F. Hasler, Dec 20 2019

From M. F. Hasler, Dec 21 2019: (Start)
n in A226099 (this sequence) <=> A217657(n) in A000040 (prime).
a(n) = a(n-4) + 10 for 4 < n < 41, i.e., 20 < a(n) < 110; a(n) = a(n-25) for 61 < n < 287, i.e., 200 < a(n) < 1100, etc. (End)

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

A344749 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, 11, 19, 42, 48, 55, 64, 66, 93, 97, 111, 248, 397, 464, 555, 666, 793, 842, 919, 1111, 1397, 1793, 1919, 5555, 6248, 6464, 6666, 6842, 11111, 26842, 31793, 46464, 55555, 66666, 71397, 86248, 91919, 111111, 191919, 426842, 486248

Offset: 1

Rémy Sigrist, May 28 2021

Positive terms are zeroless (A052382) and uniquely determined by their final digit (A010879) and the number of digits in their decimal expansion (A055642).

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

			- 7^1 = 7 mod 10,
- 7^2 = 9 mod 10,
- 7^3 = 3 mod 10,
- 7^4 = 1 mod 10,
- so 1397 belongs to the sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..2251

Cf. A010879, A052382, A055642, A217657, A344555, A344748, 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..7])[1..50])

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)]
      aklst.append(int("".join(map(str, d))))
    alst.extend(sorted(aklst))
  return alst
print(auptod(6)) # Michael S. Branicky, May 29 2021

A366198 Any a(n) replacing the first digit of a(n+1) forms a palindrome. This is the lexicographically earliest sequence of distinct nonnegative integers with this property.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 11, 21, 12, 31, 13, 41, 14, 51, 15, 61, 16, 71, 17, 81, 18, 91, 29, 22, 32, 23, 42, 24, 52, 25, 62, 26, 72, 27, 82, 28, 92, 39, 33, 43, 34, 53, 35, 63, 36, 73, 37, 83, 38, 93, 49, 44, 54, 45, 64, 46, 74, 47, 84, 48, 94, 59, 55, 65, 56, 75, 57, 85, 58, 95

Offset: 1

Eric Angelini, Oct 03 2023

No integer > 1 ending in zero will appear in the sequence.

For n >= 10 the concatenation of a(n) and A217657(a(n+1)) is a palindrome.

			a(9)  =  8 replacing the first digit of a(10) =  9 forms   8, a palindrome;
a(10) =  9 replacing the first digit of a(11) = 19 forms  99, a palindrome;
a(11) = 19 replacing the first digit of a(12) = 11 forms 191, a palindrome;
a(12) = 11 replacing the first digit of a(13) = 21 forms 111, a palindrome;
a(13) = 21 replacing the first digit of a(14) = 12 forms 212, a palindrome; etc.

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

Cf. A082216, A217657, A318486.

terms=75; b[0]=0;
b[n_]:=b[n]=(k=1; While[MemberQ[Array[b,n-1],k]||!PalindromeQ[FromDigits@Flatten@ReplacePart[IntegerDigits@k,1-> IntegerDigits@b[n-1]]],k++]; k); t=0;While[Length[a=Join[Range[0,9],Flatten@Table[FromDigits@Flatten@Insert[#,Table[9,i],-2]&/@(IntegerDigits/@Array[b,9^2,10]),{i,0,t++}]]]Giorgos Kalogeropoulos, Oct 04 2023 *)

from itertools import count, islice
def ispal(n): return (s:=str(n))==s[::-1]
def agen(): # generator of terms
    an, seen = 0, set()
    while True:
        yield an; seen.add(an); s = str(an)
        an = next(k for k in count(0) if k not in seen and ispal(s+str(k)[1:]))
print(list(islice(agen(), 80))) # Michael S. Branicky, Oct 04 2023

# faster version suitable for generating b-file
from sympy import isprime
from itertools import count, islice, product
def pals(digs):
    yield from digs
    for d in count(2):
        for p in product(digs, repeat=d//2):
            left = "".join(p)
            for mid in [[""], digs][d%2]:
                yield left + mid + left[::-1]
def folds(s): # generator of suffixes of palindromes starting with s
    for i in range((len(s)+1)//2, len(s)+1):
        for mid in [True, False]:
            t = s[:i] + (s[:i-1][::-1] if mid else s[:i][::-1])
            if t.startswith(s):
                yield t[len(s):]
    yield from ("".join(p)+s[::-1] for p in pals("0123456789"))
def agen():
    s, seen = "0", {"0"}
    while True:
        yield int(s)
        found = False
        for end in folds(s):
            for start in "123456789":
                t = start + end
                if t not in seen:
                    found = True; break
            if found: break
        s, seen = t, seen | {t}
print(list(islice(agen(), 60))) # Michael S. Branicky, Oct 04 2023

For n >= 92, a(n) = 10*a(n-81) + 90 - 9*(a(n-81) mod 10). - David A. Corneth, Oct 04 2023

A274580 Digital difference of n: the most significant decimal digit of n minus the sum of the other digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Felix Fröhlich, Jun 29 2016

If A007953(n) is seen as giving the resulting sum when inserting "+" signs between the digits of n, then a(n) gives the resulting difference when inserting "-" signs between the digits of n.
a(n) = 0 if and only if n is in A193772.
Signed version of A040997.
First differs from A225693 at n = 101.
abs(a(n)) first differs from abs(A055017(n)) at n = 102.

			a(13) = 1 - 3 = -2.
a(74) = 7 - 4 = 3.
a(211) = 2 - 1 - 1 = 0.

Cf. A007953, A007954.

Table[Fold[#1 - #2 &, IntegerDigits@ n], {n, 76}] (* Michael De Vlieger, Jun 30 2016 *)
Table[-Differences[(Total/@TakeDrop[IntegerDigits[n],1])],{n,100}]//Flatten (* Harvey P. Dale, Dec 27 2022 *)

diffdigits(n) = my(d=digits(n), dd=d[1]); for(k=2, #d, dd=dd-d[k]); dd
a(n) = diffdigits(n)

a(n) = A000030(n) - A007953(A217657(n)).

cp's OEIS Frontend

A292683 Numbers divisible by themselves with first digit removed (A217657), excluding multiples of 10.

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A292684 a(n) is the number of positive integers k not divisible by 10 such that f(kN) = f(N) for N = A292683(n) and f(x) = x / (x without its first digit: A217657(x)).

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A292685 Irregular table where row n lists the positive integers k not divisible by 10 such that f(kN) = f(N) for N = A292683(n) and f(x) = x / (x without its first digit: A217657(x)).

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A000030 Initial digit of n.

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PARI

PARI

Python

Formula

A040997 Absolute value of first digit of n minus sum of other digits of n.

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Haskell

PARI

Formula

Extensions

A226099 Positive integers that yield a prime when their most significant (i.e., leftmost) decimal digit is removed.

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Magma

Mathematica

PARI

Formula

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

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