A052210 -id:A052210 - cp's OEIS frontend

A052211 Numbers k such that k^4 starts with k itself (in base 10).

0, 1, 10, 100, 1000, 10000, 46416, 100000, 464159, 1000000, 2154435, 4641589, 10000000, 21544347, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 2154434690032, 4641588833613, 10000000000000, 21544346900319, 46415888336128

Offset: 1

Erich Friedman, Jan 29 2000

			46416^4 starts with 46416.

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A052210, A233451, A233452, A233453, A233454, A233455, A233456.

Join[{0}, Sort[ Table[ 10^i,{i,0,22} ]~Join~Select[ Table[ Ceiling[ 10.^(1/3)*10^i ],{i,0,22} ], Take[ IntegerDigits[ #^4 ],Length[ IntegerDigits[ # ] ] ]==IntegerDigits[ # ]& ]~Join~Select[ Table[ Ceiling[ 10.^(2/3)*10^i ],{i,0,22} ],Take[ IntegerDigits[ #^4 ],Length[ IntegerDigits[ # ] ] ]==IntegerDigits[ # ]& ] ] ]

r=41; print1(1, ", "); e=4; for(n=2, r, p=round((10^(1/(e-1)))^n); f=p^e; b=10^(#Str(f)-#Str(p)); if((f-lift(Mod(f, b)))/b==p, print1(p, ", "))); \\ Arkadiusz Wesolowski, Dec 10 2013

0 inserted by Juhani Heino, Aug 31 2015

A307588 Numbers k such that the digits of k^(1/3) begin with k.

Original entry on oeis.org

0, 1, 31, 999, 1000, 31622, 999999, 1000000, 31622776, 999999999, 1000000000, 31622776601, 999999999999, 1000000000000, 31622776601683, 999999999999999, 1000000000000000, 31622776601683792, 31622776601683793, 999999999999999999, 1000000000000000000

Offset: 1

Dmitry Kamenetsky, Apr 17 2019

Program is in A307371.

The subsequence {31, 31622, 31622776, 31622776601, 31622776601683, ...} looks like this subsequence of A052210 {32, 31623, 316228, 3162278, 31622777, ..., 316227766016838, ...}. - Bernard Schott, May 04 2019

			31622^(1/3) = 31.62251..., which begins with "31622", so 31622 is in the sequence.
The seeming pattern a(3k) = floor(10^(3k-3/2)), a(3k+1) = 10^(3k) - 1, a(3k+2) = 10^(3k), is broken at a(18) = a(19) - 1 = floor(10^(33/2)) - 1. - _Jon E. Schoenfield_, May 01 2019

Chai Wah Wu, Table of n, a(n) for n = 1..1172

Cf. A307371, A307600.

Cf. A052210 (analog for 3rd power instead of 1/3).

a(10)-a(21) from Jon E. Schoenfield, May 01 2019

A233451 Numbers k such that k^5 starts with k itself (in base 10).

Original entry on oeis.org

0, 1, 10, 18, 100, 178, 1000, 10000, 17783, 31623, 100000, 177828, 316228, 1000000, 10000000, 100000000, 1000000000, 5623413252, 10000000000, 100000000000, 177827941004, 316227766017, 1000000000000, 1778279410039, 10000000000000, 31622776601684

Offset: 1

Arkadiusz Wesolowski, Dec 10 2013

			18^5 = 1889568 begins with 18, so 18 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..1756

Cf. A052210, A052211, A233452, A233453, A233454, A233455, A233456.

R:= 0,1:
for d from 1 to 100 do
  k:= ceil(10^(d/4));
  if k^5 - 10^d * k < 10^d then
    R:= R, k
  fi
od:
R; # Robert Israel, Sep 11 2024

kmax=10^6; Select[Range[0,kmax],FromDigits[Drop[IntegerDigits[#^5],-(IntegerLength[#^5]-IntegerLength[#])]]==# &] (* Stefano Spezia, Aug 27 2023 *)

r=54; print1(1, ", "); e=5; for(n=2, r, p=round((10^(1/(e-1)))^n); f=p^e; b=10^(#Str(f)-#Str(p)); if((f-lift(Mod(f, b)))/b==p, print1(p, ", ")));

0 inserted by Juhani Heino, Aug 31 2015

A233452 Numbers k such that k^6 starts with k itself (in base 10).

Original entry on oeis.org

0, 1, 4, 10, 16, 40, 100, 631, 1000, 1585, 2512, 10000, 15849, 25119, 100000, 1000000, 10000000, 15848932, 100000000, 1000000000, 6309573445, 10000000000, 100000000000, 251188643151, 1000000000000, 3981071705535, 6309573444802, 10000000000000

Offset: 1

Arkadiusz Wesolowski, Dec 10 2013

			4^6 = 4096 begins with 4, so 4 is in the sequence.

Cf. A052210, A052211, A233451, A233453, A233454, A233455, A233456.

kmax=10^6; Select[Range[0,kmax],FromDigits[Drop[IntegerDigits[#^6],-(IntegerLength[#^6]-IntegerLength[#])]]==# &] (* Stefano Spezia, Aug 27 2023 *)

r=65; print1(1, ", "); e=6; for(n=2, r, p=round((10^(1/(e-1)))^n); f=p^e; b=10^(#Str(f)-#Str(p)); if((f-lift(Mod(f, b)))/b==p, print1(p, ", ")));

0 inserted by Juhani Heino, Aug 31 2015

A233453 Numbers k such that k^7 starts with k itself (in base 10).

Original entry on oeis.org

0, 1, 10, 100, 1000, 6813, 10000, 14678, 46416, 100000, 146780, 464159, 1000000, 4641589, 10000000, 21544347, 68129207, 100000000, 1000000000, 10000000000, 100000000000, 316227766017, 681292069058, 1000000000000, 2154434690032, 10000000000000, 21544346900319

Offset: 1

Arkadiusz Wesolowski, Dec 10 2013

			6813^7 = 681347587591081074493576917 begins with 6813, so 6813 is in the sequence.

Cf. A052210, A052211, A233451, A233452, A233454, A233455, A233456.

Join[{0},Select[Range[0,5*10^6],IntegerDigits[#]==Take[IntegerDigits[ #^7],IntegerLength[ #]]&]] (* The program generates the first 14 terms of the sequence. To generate more, increase the Range constant. *) (* Harvey P. Dale, Mar 31 2022 *)

r=80; print1(1, ", "); e=7; for(n=2, r, p=round((10^(1/(e-1)))^n); f=p^e; b=10^(#Str(f)-#Str(p)); if((f-lift(Mod(f, b)))/b==p, print1(p, ", ")));

0 inserted by Juhani Heino, Aug 31 2015

A233454 Numbers k such that k^8 starts with k itself (in base 10).

Original entry on oeis.org

0, 1, 2, 10, 14, 72, 100, 139, 518, 1000, 10000, 13895, 19307, 26827, 37276, 100000, 1000000, 10000000, 13894955, 26826958, 100000000, 193069773, 517947468, 1000000000, 1930697729, 10000000000, 100000000000, 268269579528, 1000000000000, 3727593720315

Offset: 1

Arkadiusz Wesolowski, Dec 10 2013

			2^8 = 256 begins with 2, so 2 is in the sequence.

Cf. A052210, A052211, A233451, A233452, A233453, A233455, A233456.

Join[{0}, Select[Range@ 1000000, Take[IntegerDigits[#^8], IntegerLength@ #] == IntegerDigits@ # &]] (* Michael De Vlieger, Aug 31 2015 *)

r=88; print1(1, ", "); e=8; for(n=2, r, p=round((10^(1/(e-1)))^n); f=p^e; b=10^(#Str(f)-#Str(p)); if((f-lift(Mod(f, b)))/b==p, print1(p, ", ")));

0 inserted by Juhani Heino, Aug 31 2015

A233455 Numbers k such that k^9 starts with k itself (in base 10).

Original entry on oeis.org

0, 1, 10, 75, 100, 750, 1000, 4217, 7499, 10000, 100000, 177828, 1000000, 10000000, 74989421, 100000000, 1000000000, 5623413252, 10000000000, 100000000000, 177827941004, 1000000000000, 1778279410039, 10000000000000, 56234132519035, 100000000000000

Offset: 1

Arkadiusz Wesolowski, Dec 10 2013

			750^9 = 75084686279296875000000000 begins with 750, so 750 is in the sequence.

G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 7499

Cf. A052210, A052211, A233451, A233452, A233453, A233454, A233456.

Join[{0}, Select[Range@ 1000000, Take[IntegerDigits[#^9], IntegerLength@ #] == IntegerDigits@ # &]] (* Michael De Vlieger, Aug 31 2015 *)

r=112; print1(1, ", "); e=9; for(n=2, r, p=round((10^(1/(e-1)))^n); f=p^e; b=10^(#Str(f)-#Str(p)); if((f-lift(Mod(f, b)))/b==p, print1(p, ", ")));

0 inserted by Juhani Heino, Aug 31 2015

A233456 Numbers k such that k^10 starts with k itself (in base 10).

Original entry on oeis.org

0, 1, 6, 10, 13, 36, 60, 100, 1000, 10000, 100000, 129155, 278256, 1000000, 10000000, 21544347, 100000000, 1000000000, 10000000000, 59948425032, 100000000000, 599484250319, 1000000000000, 10000000000000, 100000000000000, 464158883361278, 1000000000000000

Offset: 1

Arkadiusz Wesolowski, Dec 10 2013

			6^10 = 60466176 begins with 6, so 6 is in the sequence.

Cf. A052210, A052211, A233451, A233452, A233453, A233454, A233455.

kmax=10^6; Select[Range[0,kmax],FromDigits[Drop[IntegerDigits[#^10],-(IntegerLength[#^10]-IntegerLength[#])]]==# &] (* Stefano Spezia, Aug 27 2023 *)

r=135; print1(1, ", "); e=10; for(n=2, r, p=round((10^(1/(e-1)))^n); f=p^e; b=10^(#Str(f)-#Str(p)); if((f-lift(Mod(f, b)))/b==p, print1(p, ", ")));

0 inserted by Juhani Heino, Aug 31 2015

A261751 Numbers n with property that binary expansion of n^3 begins with the binary expansion of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 16, 23, 32, 64, 91, 128, 256, 512, 1024, 2048, 4096, 5793, 8192, 16384, 32768, 46341, 65536, 92682, 131072, 185364, 262144, 370728, 524288, 1048576, 2097152, 2965821, 4194304, 5931642, 8388608, 16777216, 33554432, 47453133, 67108864, 94906266

Offset: 1

Dhilan Lahoti, Aug 30 2015

2^k is always a term in this sequence.

It appears that all solutions are either a power of 2 or approximately sqrt(2) * a power of 2. - Andrew Howroyd, Dec 24 2019

			23 is a term of this sequence because its cube written in base 2 (10111110000111) starts with its representation in base 2 (10111).

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Base 2 version of A052210.

Cf. A004539.

SetBeginSet[set1_, set2_] :=
  Do[For[i = 1, i <= Length[set1], i++,If[! set1[[i]] == set2[[i]], Return[False]]];Return[True], {1}];
For[k = 0; set = {}, k <= 100000, k++,If[SetBeginSet[IntegerDigits[k, 2], IntegerDigits[k^3, 2]],Print[k]]]

ok(n)={my(t=n^3); t == 0 || t>>(logint(t,2)-logint(n,2))==n} \\ Andrew Howroyd, Dec 23 2019

\\ for larger values
viable(b,k)={my(p=b^3, q=(b+2^k-1)^3, s=logint(q,2), t=s-logint(b,2)+k); (p>>s)==0 || ((p>>t)<=(b>>k) && (b>>k)<=(q>>t))}
upto(n)={
  local(L=List([0]));
  my(recurse(b,k)=; if(b <= n && viable(b,k), k--; if(k<0, listput(L, b), self()(b,k); self()(b+2^k,k))));
  for(k=0, logint(n,2), recurse(2^k, k));
  Vec(L);
} \\ Andrew Howroyd, Dec 24 2019

Terms a(31) and beyond from Andrew Howroyd, Dec 23 2019

A240925 Lexicographically earliest sequence of distinct terms such that, in base 10, the digits of the cubes of the terms correspond to the digits of the sequence itself.

Original entry on oeis.org

1, 32, 7, 6, 8, 3, 4, 321, 65, 12, 2, 76, 43, 30, 761, 61, 27, 46, 25, 17, 28, 84, 38, 9, 767, 950, 72, 7000, 440, 71, 10, 81, 22, 69, 811, 96, 83, 97, 33, 615, 62, 5, 49, 13, 21, 95, 259, 270, 45, 48, 727, 29, 451, 217, 66, 385, 73, 75000

Offset: 1

Paul Tek, Aug 03 2014

Leading zeros are ignored.

			+--------+---------------------------------------------+
|  a(n)  | 1,32, 7,6,8,3,4,321,  65, 12, 2,76, 43, ... |
+--------+---------------------------------------------+
| digits | 1 3 2 7 6 8 3 4 3 2 1 6 5 1 2 2 7 6 4 3 ... |
+--------+---------------------------------------------+
| a(n)^3 | 1,32768,    343,  216,  512,  27, 64, ...   |
+--------+---------------------------------------------+

Paul Tek, Table of n, a(n) for n = 1..4264
Paul Tek, Perl program for this sequence

Cf. A052210.

Perl
```
See Link section.
```

cp's OEIS Frontend

A052211 Numbers k such that k^4 starts with k itself (in base 10).

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A307588 Numbers k such that the digits of k^(1/3) begin with k.

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A233451 Numbers k such that k^5 starts with k itself (in base 10).

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A233452 Numbers k such that k^6 starts with k itself (in base 10).

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A233453 Numbers k such that k^7 starts with k itself (in base 10).

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A233454 Numbers k such that k^8 starts with k itself (in base 10).

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A233455 Numbers k such that k^9 starts with k itself (in base 10).

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A233456 Numbers k such that k^10 starts with k itself (in base 10).

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