A018834 -id:A018834 - cp's OEIS frontend

A046831 Numbers k such that decimal expansion of k^2 contains k as a substring and k does not end in 0.

1, 5, 6, 25, 76, 376, 625, 3792, 9376, 14651, 90625, 109376, 495475, 505025, 890625, 971582, 1713526, 2890625, 4115964, 5133355, 6933808, 7109376, 10050125, 12890625, 48588526, 50050025, 66952741, 87109376, 88027284, 88819024

Offset: 1

David W. Wilson

Subsequence of A018834. - Chai Wah Wu, Apr 04 2023

Giovanni Resta, Table of n, a(n) for n = 1..64

Cf. A018834, A003226, A008851.

a046831 n = a046831_list !! (n-1)
a046831_list = filter ((> 0) . (`mod` 10)) a018834_list
-- Reinhard Zumkeller, Jul 27 2011

Reap[For[n = 1, n < 10^8, n++, If[Mod[n, 10] != 0, If[StringPosition[ToString[n^2], ToString[n]] != {}, Print[n]; Sow[n]]]]][[2, 1]] (* Jean-François Alcover, Apr 04 2013 *)

from itertools import count, islice
def A046831_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:n%10 and str(n) in str(n**2), count(max(startvalue,0)))
A046831_list = list(islice(A046831_gen(),20)) # Chai Wah Wu, Apr 04 2023

A046851 Numbers n such that n^2 can be obtained from n by inserting internal (but not necessarily contiguous) digits.

Original entry on oeis.org

0, 1, 10, 11, 95, 96, 100, 101, 105, 110, 125, 950, 960, 976, 995, 996, 1000, 1001, 1005, 1006, 1010, 1011, 1021, 1025, 1026, 1036, 1046, 1050, 1100, 1101, 1105, 1201, 1205, 1250, 1276, 1305, 1316, 1376, 1405, 9500, 9505, 9511, 9525, 9600, 9605, 9625

Offset: 1

David W. Wilson

Contains A038444.  In particular, the sequence is infinite. - Robert Israel, Oct 20 2016
If n is any positive term, then b_n(k) := n*10^k (k >= 0) is an infinite subsequence. - Rick L. Shepherd, Nov 01 2016
From Robert Israel's comment it follows that the subsequence of terms with no trailing zeros is also infinite (contains A000533). - Rick L. Shepherd, Nov 01 2016

			110^2 = 12100 (insert "2" and "0" into "1_1_0").

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

Cf. A045953, A008851, A018834, A038444, A086457 (subsequence).

import Data.List (isInfixOf)
a046851 n = a046851_list !! (n-1)
a046851_list = filter chi a008851_list where
   chi n = (x == y && xs `isSub` ys) where
      x:xs = show $ div n 10
      y:ys = show $ div (n^2) 10
   isSub [] ys       = True
   isSub _  []       = False
   isSub us'@(u:us) (v:vs)
         | u == v    = isSub us vs
         | otherwise = isSub us' vs
-- Reinhard Zumkeller, Jul 27 2011

IsSublist:= proc(a, b)
  local i,bp,j;
  bp:= b;
  for i from 1 to nops(a) do
    j:= ListTools:-Search(a[i],bp);
    if j = 0 then return false fi;
    bp:= bp[j+1..-1];
  od;
  true
end proc:
filter:= proc(n) local A,B;
  A:= convert(n,base,10);
  B:= convert(n^2,base,10);
  if not(A[1] = B[1] and A[-1] = B[-1]) then return false fi;
  if nops(A) <= 2 then return true fi;
  IsSublist(A[2..-2],B[2..-2])
end proc:
select(filter, [$0..10^4]); # Robert Israel, Oct 20 2016

id[n_]:=IntegerDigits[n];
insQ[n_]:=First[id[n]]==First[id[n^2]]&&Last[id[n]]==Last[id[n^2]];
sort[n_]:=Flatten/@Table[Position[id[n^2],id[n][[i]]],{i,1,Length[id[n]]}];
takeQ[n_]:=Module[{lst={First[sort[n][[1]]]}},
   Do[
    Do[
     If[Last[lst]Ivan N. Ianakiev, Oct 19 2016 *)

A029943 Substring of both its square and its cube.

Original entry on oeis.org

0, 1, 5, 6, 10, 25, 50, 60, 76, 100, 250, 376, 500, 600, 625, 760, 1000, 2500, 3760, 5000, 6000, 6250, 7600, 9376, 10000, 25000, 37600, 50000, 60000, 62500, 76000, 90625, 93760, 100000, 109376, 250000, 376000, 500000, 600000, 625000

Offset: 1

Patrick De Geest

Intersection of A018834 and A029942. - Reinhard Zumkeller, Feb 29 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Automorphic Number
Wikipedia, Automorphic number
Index entries for sequences related to automorphic numbers

import Data.List (isInfixOf)
a029943 n = a029943_list !! (n-1)
a029943_list = filter f [0..] where
   f x = show x `isInfixOf` show (x^2) && show x `isInfixOf` show (x^3)
-- Reinhard Zumkeller, Nov 26 2011

ssscQ[n_]:=Module[{idn=IntegerDigits[n],sq=IntegerDigits[n^2], cu=IntegerDigits[n^3],len=IntegerLength[n]},MemberQ[Partition[ sq,len,1], idn] &&MemberQ[Partition[cu,len,1],idn]]; Join[{0}, Select[Range[700000],ssscQ]] (* Harvey P. Dale, Apr 24 2011 *)

a(n) = A003226(m) * 10^k for appropriate m and k. [Reinhard Zumkeller, Nov 26 2011]

Offset corrected by Reinhard Zumkeller, Nov 26 2011

A089951 Numbers having the same leading decimal digits as their squares.

Original entry on oeis.org

0, 1, 10, 11, 12, 13, 14, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 895

Offset: 1

Reinhard Zumkeller, Jan 12 2004

A000030(a(n)) = A002993(a(n)) = A000030(A000290(a(n))).

			895*895 = 801025, therefore 895 is a term: a(55)=895.

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

Cf. A018834.
Cf. A144582. - Reinhard Zumkeller, Aug 17 2008
Cf. A000030, A002993, A000290, A256523 (subsequence).

a089951 n = a089951_list !! (n-1)
a089951_list = [x | x <- [0..], a000030 x == a000030 (x ^ 2)]
-- Reinhard Zumkeller, Apr 01 2015

F:= proc(d) $10^d .. floor(sqrt(2)*10^d), $ ceil(sqrt(80)*10^d) .. 9*10^d - 1, $ ceil(sqrt(90)*10^d) .. 10^(d+1)-1 end proc:
0, F(0), F(1), F(2), F(3); # Robert Israel, Mar 18 2015

d[n_] := IntegerDigits[n]; Select[Range[895],
First[d[#]] == First[d[#^2]] &] (* Jayanta Basu, Jun 03 2013 *)

a(n)={my(v = [1, sqrt(80), sqrt(90)], w=[(k)->10^k * ((sqrt(2) - 1))\1 + 1, (k)->9 * 10^k - ceil(sqrt(80) * 10^k), (k)->10 * 10^k - ceil(sqrt(90) * 10^k)],i = 1,k = 0); if(n==1, 0, n--; while(n>w[i](k), n-=w[i](k); i++; if(i == 4, i = 1; k++)); ceil(v[i]*10^k)+n-1)} \\ David A. Corneth, Feb 26 2015

isok(n) = (n == 0) || (digits(n)[1] == digits(n^2)[1]); \\ Michel Marcus, Mar 18 2015

A number n is in the sequence iff n = 0 or n/10^floor(log_10(n)) lies in one of the half-open intervals [1, sqrt(2)), [sqrt(80), 9) or [sqrt(90), 10). - David W. Wilson, May 29 2008

A045953 Numbers m such that m^2 can be obtained from m by inserting an internal block of (contiguous) digits.

Original entry on oeis.org

0, 1, 10, 11, 95, 96, 100, 101, 125, 976, 995, 996, 1000, 1001, 1025, 1376, 9625, 9976, 9995, 9996, 10000, 10001, 10025, 10376, 10625, 99376, 99625, 99976, 99995, 99996, 100000, 100001, 100025, 100376, 100625, 109376, 990625, 999376, 999625, 999976

Offset: 1

John "MazeMan" Knoderer (Webmaster(AT)Mazes.com)

All terms of this sequence appear in A086457. - Jeremy Gardiner, Jul 20 2003

It seems that for any nonnegative integer k the number of k-digit terms is 2k. - Ivan N. Ianakiev, Aug 17 2021

			95^2 = 9025 (insert '02' inside '95').

Reinhard Zumkeller, Table of n, a(n) for n = 1..100
J. Knoderer, What number comes next? (an interesting sequence)

Cf. A046851, A008851, A018834.

import Data.List (isPrefixOf, inits, isSuffixOf, tails)
a045953 n = a045953_list !! (n-1)
a045953_list = filter chi a008851_list where
   chi n = (x == y && xs `isSub'` ys) where
      x:xs = show $ div n 10
      y:ys = show $ div (n^2) 10
      isSub' us vs = any id $ zipWith (&&)
                              (map (`isPrefixOf` vs) $ inits us)
                              (map (`isSuffixOf` vs) $ tails us)
-- Reinhard Zumkeller, Jul 27 2011

A075904 Numbers k such that k^4 has k as a substring of its decimal expansion.

Original entry on oeis.org

0, 1, 5, 6, 10, 25, 50, 60, 76, 83, 92, 100, 107, 211, 217, 250, 352, 363, 376, 500, 556, 600, 625, 636, 760, 863, 909, 935, 1000, 1531, 1636, 2263, 2500, 2503, 3630, 3760, 4342, 5000, 5001, 6000, 6250, 7245, 7600, 8578, 9350, 9376, 10000, 25000, 28206, 32213

Offset: 1

Zak Seidov, Sep 27 2002

			6^4 = 129_6, 83^4 = 4745_83_21, 2503^4 = 39_2503_37770081.

Chai Wah Wu, Table of n, a(n) for n = 1..206

Cf. A018834 (squares), A029942 (cubes), A075905 (5th powers).

Select[Range[10000], StringPosition[ToString[ #^4], ToString[ # ]] != {} &] (* Tanya Khovanova, Oct 11 2007 *)
ssQ[n_]:=Module[{idn=IntegerDigits[n],idn4=IntegerDigits[n^4]}, MemberQ[ Partition[ idn4, Length[ idn],1], idn]]; Select[Range[10000],ssQ] (* Harvey P. Dale, Mar 13 2013 *)

A075904_list, m = [0], [24, -36, 14, -1, 0]
for n in range(1,10**9+1):
    for i in range(4):
        m[i+1] += m[i]
    if str(n) in str(m[-1]):
        A075904_list.append(n) # Chai Wah Wu, Nov 05 2014

More terms from Tanya Khovanova, Oct 11 2007

Added 0 to sequence by Chai Wah Wu, Nov 05 2014

A075905 Numbers k such that k^5 has k as a substring of its representation.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 18, 20, 24, 25, 30, 32, 40, 43, 45, 48, 49, 50, 51, 57, 60, 68, 70, 73, 75, 76, 80, 90, 93, 99, 100, 101, 125, 178, 192, 193, 195, 200, 205, 240, 249, 250, 251, 300, 307, 320, 375, 376, 400, 430, 432, 443, 480, 490, 499, 500, 501

Offset: 1

Zak Seidov, Sep 27 2002

			45^5 = 18_45_28125, 3637^5 = 6_3637_9975073041957, 3975^5 = 992_3975_07802734375.

Chai Wah Wu, Table of n, a(n) for n = 1..1091

Cf. A018834 (squares), A029942 (cubes), A075904 (4th powers).

Select[Range[0,600],SequenceCount[IntegerDigits[#^5],IntegerDigits[#]]>0&] (* Harvey P. Dale, Jul 06 2025 *)

A075905_list, m = [0], [120, -240, 150, -30, 1, 0]
for n in range(1,10**8+1):
    for i in range(5):
        m[i+1] += m[i]
    if str(n) in str(m[-1]):
        A075905_list.append(n) # Chai Wah Wu, Nov 05 2014

A239058 Numbers whose divisors all appear as a substring in their decimal expansion.

Original entry on oeis.org

1, 11, 13, 17, 19, 31, 41, 61, 71, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 241, 251, 271, 281, 311, 313, 317, 331, 401, 419, 421, 431, 461, 491, 521, 541, 571, 601, 613, 617, 619, 631, 641, 661, 691, 701, 719, 751, 761, 811, 821, 881, 911, 919, 941, 971

Offset: 1

M. F. Hasler, Mar 09 2014

A subsequence of A092911 (all divisors can be formed using the digits of the number) which is a subsequence of A011531 (numbers having the digit 1).
Are 1 and 125 the only nonprime terms in this sequence?
No: 17692313, 4482669527413081, 21465097175420089, and 567533481816008761 are members. - Charles R Greathouse IV, Mar 09 2014
See A239060 for the nonprime terms of this sequence, which include in particular the squares of terms of A115738 (unless such a square would not have a digit 1).

			All primes having the digit 1 (A208270) are in this sequence, because {1, p} are the only divisors of a prime p.
The divisors of 125 are {1, 5, 25, 125}; it can be seen that all of them occur as a substring in 125, therefore 125 is in this sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to decimal expansion of n

Cf. A092911, A011531, A121041, A121022-A121040, A018834.

is(n,d=vecextract(divisors(n),"^-1"))={ setminus(select(x->x<10,d),Set(digits(n)))&&return;!for(L=2,#Str(d[#d]),setminus(select(x->x
<10^L&&x>=10^(L-1),d),Set(concat(vector(L,o,digits(n\10^(L-o),10^L)))))&&return)}

overlap(long,short)=my(D=10^#digits(short)); while(long>=short, if(long%D==short,return(1));long\=10); 0
is(n)=my(d=divisors(n)); forstep(i=#d-1,1,-1, if(!overlap(n,d[i]), return(0))); 1 \\ Charles R Greathouse IV, Mar 09 2014

A239060 Nonprime numbers whose divisors all appear as a substring in the number's decimal expansion.

Original entry on oeis.org

1, 125, 17692313

Offset: 1

M. F. Hasler, Mar 09 2014

This is the subsequence of A239058 without the primes having a digit 1, A208270. It is thus a subsequence of A092911 (all divisors can be formed using the digits of the number) which is a subsequence of A011531 (numbers having the digit 1).
The term a(3)=17692313=A239058(870356), as well as the numbers 4482669527413081, 21465097175420089, and 567533481816008761 which are also members, were found by Charles R Greathouse IV, Mar 09 2014
The square of any term of A115738 is a member of this sequence. The above larger examples are of that form.
a(4) > 10^12. - Giovanni Resta, Sep 08 2018

			The divisors of 17692313 are {1, 23, 769231, 17692313}; it can be seen that all of them occur as a substring in 17692313, therefore 17692313 is in this sequence.

Cf. A092911, A011531, A121041, A121022-A121040, A018834.

PARI
```
is(n)=!isprime(n)&&is_A239058(n)
```

overlap(long,short)=my(D=10^#digits(short)); while(long>=short, if(long%D==short,return(1));long\=10); 0
is(n)=my(d=divisors(n)); #d!=2 && !forstep(i=#d-1,1,-1, if(!overlap(n,d[i]), return(0))) \\ Charles R Greathouse IV, Mar 09 2014

A281822 Least number k such that (k+n)^2 contains k as a substring.

Original entry on oeis.org

1, 8, 6, 1, 4, 9, 62, 6, 1, 1, 1, 1, 1, 1, 2, 2, 2, 4, 1, 5, 1, 2, 9, 2, 4, 2, 92, 9, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 3, 1, 3, 3, 1, 2, 1, 4, 4, 2, 4, 4, 2, 4, 1, 5, 1, 1, 5, 2, 4, 2, 4, 2, 1, 5, 1, 6, 5, 2, 4, 8, 6

Offset: 0

Paolo P. Lava, Jan 31 2017

			a(1) = 8 because (8 + 1)^2 = 9^2 = 81 contains 8 as a substring and it is the least number with this property.

Paolo P. Lava, Table of n, a(n) for n = 0..10000

Cf. A018834, A281823.

with(numtheory): P:= proc(q) local a,b,d,j,k,n,ok;
for n from 0 to q do for k from 1 to q do a:=ilog10(k)+1; b:=(n+k)^2; d:=ilog10((k+n)^2)-ilog10(k)+1;
ok:=0; for j from 1 to d do if k=(b mod 10^a) then ok:=1; break; else b:=trunc(b/10); fi; od;
if ok=1 then print(k); break; fi; od; od; end: P(10^6);

Typo in definition corrected by Harvey P. Dale, Feb 27 2017.

Entries, Maple code and b-file corrected at the suggestion of Harvey P. Dale, Feb 28 2017.

cp's OEIS Frontend

A046831 Numbers k such that decimal expansion of k^2 contains k as a substring and k does not end in 0.

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A046851 Numbers n such that n^2 can be obtained from n by inserting internal (but not necessarily contiguous) digits.

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A029943 Substring of both its square and its cube.

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A089951 Numbers having the same leading decimal digits as their squares.

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A045953 Numbers m such that m^2 can be obtained from m by inserting an internal block of (contiguous) digits.

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A075904 Numbers k such that k^4 has k as a substring of its decimal expansion.

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A075905 Numbers k such that k^5 has k as a substring of its representation.

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