A346678 -id:A346678 - cp's OEIS frontend

A067251 Numbers with no trailing zeros in decimal representation.

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, 101, 102, 103, 104

Offset: 1

Reinhard Zumkeller, Mar 10 2002

Or, decimated numbers: every 10th number has been omitted from the natural numbers. - Cino Hilliard, Feb 21 2005. For example, The 10th number starting with 1 is 10 and is missing from the table because it was decimated.
The word "decimated" can be interpreted in several ways and should be used with caution. - N. J. A. Sloane, Feb 21 2005
Not the same as A052382, as 101 is included.
Numbers in here but not in A043095 are 81, 91, 92, 93, 94,... for example. - R. J. Mathar, Sep 30 2008
The integers 100*a(n) are precisely the numbers whose square ends with exactly 4 identical digits while the integers 10*a(n) form just a subsequence of the numbers whose square ends with exactly 2 identical digits (A346678). - Bernard Schott, Oct 04 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,1,-1).

Complement of A008592.
Cf. A076641 (reversed).
Cf. A004086, A043095, A052382, A168184.
Cf. A039685 (a subsequence), A346678, A346940, A346942.

a067251 n = a067251_list !! (n-1)
a067251_list = filter ((> 0) . flip mod 10) [0..]
-- Reinhard Zumkeller, Jul 11 2015, Dec 29 2011

S := seq(n + floor((n-1)/9), n=1..100); # Bernard Schott, Oct 04 2021

DeleteCases[Range[110],?(Divisible[#,10]&)] (* _Harvey P. Dale, May 16 2016 *)

f(n) = for(x=1,n,if(x%10,print1(x","))) \\ Cino Hilliard, Feb 21 2005

Vec(x*(x+1)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1)/((x-1)^2*(x^2+x+1)*(x^6+x^3+1)) + O(x^100)) \\ Colin Barker, Sep 28 2015

def a(n): return n + (n-1)//9
print([a(n) for n in range(1, 95)]) # Michael S. Branicky, Oct 04 2021

a(n) = n + floor((n-1)/9).
a(n) mod 10 > 0 for all n.
A004086(A004086(a(n))) = a(n).
A168184(a(n)) = 1. - Reinhard Zumkeller, Nov 30 2009
From Colin Barker, Sep 28 2015: (Start)
a(n) = a(n-1) + a(n-9) - a(n-10) for n>10.
G.f.: x*(x+1)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1) / ((x-1)^2*(x^2+x+1)*(x^6+x^3+1)). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (1/20 + 1/sqrt(5) - sqrt(1+2/sqrt(5))/5) * Pi. - Amiram Eldar, May 11 2025

Edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of R. J. Mathar

Typos corrected in a comment line by Reinhard Zumkeller, Apr 04 2010

A346774 Numbers whose square starts and ends with exactly 2 identical digits.

Original entry on oeis.org

88, 150, 210, 212, 338, 340, 470, 580, 670, 880, 940, 1050, 1060, 1062, 1070, 1080, 1088, 1090, 1488, 1510, 1512, 1820, 1830, 1838, 1840, 2110, 2112, 2120, 2350, 2360, 2362, 2570, 2580, 2588, 2780, 2790, 2970, 3150, 3160, 3320, 3330, 3350, 3360, 3362, 3370, 3380, 3388, 3390, 3410

Offset: 1

Bernard Schott, Aug 03 2021

The terminal digits are 00 or 44.

			150 is a term because 150^2 = 22500.
212 is a term because 212^2 = 44944 (smallest square with 2 times two 4's).
2788 is not a term because 2788^2 = 7772944.

Subsequence of A346678.

Cf. A123912, A186438, A186439.

Select[Range[32, 3500], (d = IntegerDigits[#^2])[[1]] == d[[2]] != d[[3]] && d[[-1]] == d[[-2]] != d[[-3]] &] (* Amiram Eldar, Aug 03 2021 *)

def ok(n):
    s = str(n*n)
    if len(s) < 4: return False # there are no ok squares with < 4 digits
    return s[0] == s[1] != s[2] and s[-1] == s[-2] != s[-3]
print(list(filter(ok, range(3411)))) # Michael S. Branicky, Aug 03 2021

A186439 Numbers whose squares end in three identical digits.

Original entry on oeis.org

38, 100, 200, 300, 400, 462, 500, 538, 600, 700, 800, 900, 962, 1000, 1038, 1100, 1200, 1300, 1400, 1462, 1500, 1538, 1600, 1700, 1800, 1900, 1962, 2000, 2038, 2100, 2200, 2300, 2400, 2462, 2500, 2538, 2600, 2700, 2800, 2900, 2962, 3000, 3038, 3100, 3200, 3300, 3400, 3462

Offset: 1

Michel Lagneau, Feb 21 2011

The three ending digits of a(n)^2 are 000 or 444.

n is in the sequence iff either n == 0 mod 100 or n == (+/-)38 mod 500. - Robert Israel, Jul 03 2014

			462 is in the sequence because 462^2 = 213444.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Cf. A016742 (even squares), A186438.
Cf. A039685, A047336.
Cf. A346678.

with(numtheory):T:=array(1..10):for p from 1 to 10000 do:n:=p^2:l:=length(n):n0:=n:for
  m from 1 to l do:q:=n0:u:=irem(q,10):v:=iquo(q,10):n0:=v :T[m]:=u:od:if T[1]=T[2]
  and T[1]=T[3] then printf(`%d, `,p):else fi:od:
# second Maple program:
a:= proc(n) local m, r;
      r:= 1+ irem(n-1, 7, 'm');
      [38, 100, 200, 300, 400, 462, 500][r] +500*m
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 24 2011

Select[Range[11,10000],Mod[PowerMod[#,2,1000],111]==0&] (* Zak Seidov, Feb 23 2011 *)

for(n=11,10000,if((n^2%1000)%111==0,print1(n", "))) \\ Zak Seidov, Feb 23 2011

Vec(2*x*(19*x^2 +12*x +19)*(x^4 +x^3 +x^2 +x +1)/((x -1)^2*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)) + O(x^100)) \\ Colin Barker, Jul 03 2014

def ok(n): s = str(n*n); return len(s) > 2 and s[-1] == s[-2] == s[-3]
print(list(filter(ok, range(3463)))) # Michael S. Branicky, Jul 29 2021

a(A047336(n)) = A039685(n). - Bruno Berselli, Feb 22 2011
a(n) = a(n-7) + 500 for n > 7. - Zak Seidov and Bruno Berselli, Feb 23 2011
G.f.: 2*x*(19*x^2 +12*x +19)*(x^4 +x^3 +x^2 +x +1) / ((x -1)^2*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)). - Colin Barker, Jul 03 2014

A346812 Positive numbers whose square starts with exactly 2 identical digits.

Original entry on oeis.org

15, 21, 34, 47, 58, 67, 88, 94, 105, 106, 107, 108, 109, 150, 151, 182, 183, 184, 210, 211, 212, 235, 236, 257, 258, 278, 279, 297, 315, 316, 332, 333, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 470, 471, 473, 474, 475, 476, 477, 478, 479, 575, 576, 577, 578, 579, 580, 581

Offset: 1

Bernard Schott, Aug 05 2021

If m is a term, then 10*m is another term.

			34 is a term because 34^2 = 1156.
149 is not a term because 149^2 = 22201.

Subsequence of A123912.
A346774 is a subsequence.
Cf. A186438, A186439, A346678 (similar, with "ends").

Select[Range[10, 600], (d = IntegerDigits[#^2])[[1]] == d[[2]] != d[[3]] &] (* Amiram Eldar, Aug 05 2021 *)

isok(m) = my(d=digits(m^2)); (#d > 2) && (d[2] == d[1]) && (d[3] != d[2]); \\ Michel Marcus, Aug 05 2021

def ok(n): s = str(n*n); return len(s) > 2 and s[0] == s[1] != s[2]
print(list(filter(ok, range(582)))) # Michael S. Branicky, Aug 05 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

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A067251 Numbers with no trailing zeros in decimal representation.

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A346774 Numbers whose square starts and ends with exactly 2 identical digits.

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A186439 Numbers whose squares end in three identical digits.

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A346812 Positive numbers whose square starts with exactly 2 identical digits.

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

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