A018851 -id:A018851 - cp's OEIS frontend

A018796 Smallest square that begins with n.

0, 1, 25, 36, 4, 529, 64, 729, 81, 9, 100, 1156, 121, 1369, 144, 1521, 16, 1764, 1849, 196, 2025, 2116, 225, 2304, 2401, 25, 2601, 2704, 289, 2916, 3025, 3136, 324, 3364, 3481, 35344, 36, 3721, 3844, 3969, 400, 41209, 4225, 4356, 441, 45369, 4624, 4761, 484, 49

Offset: 0

David W. Wilson

If 4*(n+1) < 10^d then a(n) < (n+1)*10^d. - Robert Israel, Aug 01 2014

			Among the first 100001 terms, the largest is a(99999) = 99999515529 = 316227^2. - _Zak Seidov_, May 22 2016

Zak Seidov, Table of n, a(n) for n = 0..100000 (first 10000 terms from Chai Wah Wu)

Cf. A018851, A077502.

a:= proc(n) local k,d,x;
  if issqr(n) then return n
  else for d from 1 do
    for k from 0 to 10^d-1 do
    x:= 10^d*n+k;
    if issqr(x) then return x fi
    od
    od
  fi
end proc:
seq(a(n),n=1..100); # Robert Israel, Jul 31 2014

Table[With[{d = IntegerDigits@ n}, k = 1; While[Or[IntegerLength[k^2] < Length@ d, Take[IntegerDigits[k^2], Length@ d] != d], k++]; k^2], {n, 49}] (* Michael De Vlieger, May 23 2016 *)

\\Set precision high enough (for the cases where n+1 is a square)!
a(n) = {my(v=vector(2));if(issquare(n),return(n), v=[sqrt(n),sqrt(n+1-(10^-((#digits(n)+7))))]; while(ceil(v[1])>floor(v[2]),v*=sqrt(10)));ceil(v[1])^2
} \\ David A. Corneth, May 22 2016

n = 1
while n < 100:
    for k in range(10**3):
        if str(k**2).startswith(str(n)):
            print(k**2,end=', ')
            break
    n += 1 # Derek Orr, Jul 31 2014

from gmpy2 import isqrt
def A018796(n):
    if n == 0:
        return 0
    else:
        d, nd = 1, n
        while True:
            x = (isqrt(nd-1)+1)**2
            if x < nd+d:
                return int(x)
            d *= 10
            nd *= 10 # Chai Wah Wu, May 23 2016

a(0)=0 prepended by Zak Seidov, May 22 2016

A018852 a(n)^3 is smallest cube beginning with n.

Original entry on oeis.org

0, 1, 3, 7, 16, 8, 4, 9, 2, 21, 10, 48, 5, 11, 52, 25, 55, 12, 57, 27, 59, 6, 61, 62, 29, 63, 64, 3, 66, 31, 67, 68, 32, 15, 7, 33, 154, 72, 73, 34, 16, 161, 35, 76, 164, 77, 36, 78, 169, 17, 37, 8, 174, 81, 38, 82, 178, 83, 18, 39, 182, 85, 184, 86, 4, 87, 188, 189, 19, 191, 89, 193, 9

Offset: 0

David W. Wilson

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 1..1000 from T. D. Noe)

Cf. A018797.

Cf. A018851 (k=2), this sequence (k=3), A018853 (k=4), A018872 (k=5), A018874 (k=6), A018876 (k=7), A018878 (k=8), A018880 (k=9), A018882 (k=10).

f:= proc(n) local d,m;
  for d from 0 do
    m:= ceil((10^d*n)^(1/3));
    if m^3 < 10^d*(n+1) then return m fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Jul 26 2015

a(n)=k=1;while(k,d=digits(k^3);D=digits(n);if(#D<=#d,c=1;for(i=1,#D,if(D[i]!=d[i],c=0;break));if(c,return(k)));k++)
vector(100,n,a(n)) \\ Derek Orr, Jul 26 2015

for n in range(1,10**3):
    for k in range(10**3):
        if str(k**3).startswith(str(n)):
            print(k,end=', ')
            break
    n += 1 # Derek Orr, Aug 03 2014

a(n) >= n^(1/3), for all n > 0, with equality when n is a cube. - Derek Orr, Jul 26 2015

0 prepended by Seiichi Manyama, Jan 30 2017

A265432 a(n) = smallest k with concat(1,k) and concat(n,k) both square numbers.

Original entry on oeis.org

225, 6, 1025, 6, 225, 9937257544619140625, 80625, 225, 19025, 14797831640625, 5625, 89450791534674072265625, 96, 69, 44, 21, 1993672119140625, 2002541101386962890625, 225, 6, 8734765625, 99758030615478515625, 5625, 863225, 80625, 6, 40625, 225, 890625, 158764150390625

Offset: 0

David W. Wilson and Robert G. Wilson v, Dec 08 2015

k must be a positive integer (and of course cannot begin with 0). - N. J. A. Sloane, May 19 2016

Every term is a member of A272671, by definition. Certainly every term of A272671 which is a power of 100 times an earlier term of A272671 (such as 600, 2100, 4400) will not appear, by the "smallest k" condition. Does every other term of A272671 (that is, the terms of A272684) eventually appear? See A272685 and A273369 for the first appearance of these terms. - Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 23 2016

			a(0) = 225 because 1225 is a square as is (0)225. (In other words, 225 is the first term in A272672 that is itself a square). - _N. J. A. Sloane_, May 21 2016
a(2) = 1025 because concat(1,1025) = 11025 = 105^2 and concat(2,1025) = 21025 = 145^2.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000, May 25 2016 [First 10000 terms from David W. Wilson, Dec 08 2015]
N. J. A. Sloane, Table of n, a(n), sqrt(concat(n,a(n))), sqrt(concat(1,a(n))) (based on _David W. Wilson_ 10000-term b-file)
David W. Wilson, Table of n, a(n) for n = 0..10000, May 25 2016 [The graph of the first 10000 terms looks rather different from the graph of the first 20000 terms, so this file is worth preserving. - _N. J. A. Sloane_, May 25 2016]

Cf. A045855, A272671, A018796, A272684, A272685 and A273369 (smallest inverse).
For records see A272674, A272675.
For square roots referred to in definition see A272682, A272683.
A018851 is a simpler sequence in the same spirit.

<< Combinatorica`
A265432[0] = 225;
A265432[1] = 6;
A265432[n_] := Block[{x = {-1, 1, 0, 1}[[Mod[n, 4, 1]]], d = Infinity, l, i}, While[d > Sqrt[10.0^(x - 1)] (Sqrt[10.0 n + 1] - Sqrt[11.0]), x++; d = Infinity; l = Divisors[((n - 1) 10^x)/4]; i = BinarySearch[l, 0.5 Sqrt[(n + 1) 10.0^x - 1] - 0.5 Sqrt[2*10.0^x - 1]]; If[i <= Length@l, d = 2*l[[i + 1/2]]]]; (((n - 1) 10^x - d^2)/(2 d))^2 - 10^x] (* Davin Park, Apr 11 2017 *)

A018853 a(n)^4 is smallest fourth power beginning with n.

Original entry on oeis.org

0, 1, 4, 14, 8, 15, 5, 29, 3, 31, 10, 33, 6, 19, 11, 35, 2, 65, 37, 21, 12, 68, 69, 22, 7, 4, 72, 23, 13, 74, 132, 42, 134, 24, 43, 77, 138, 44, 14, 25, 8, 45, 144, 81, 46, 26, 147, 83, 47, 84, 15, 151, 85, 27, 86, 273, 154, 49, 276, 88, 157, 28, 5, 159, 283, 9, 286, 51, 91, 289, 29

Offset: 0

David W. Wilson

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A018798 (a(n)^4).

Cf. A018851 (k=2), A018852 (k=3), this sequence (k=4), A018872 (k=5), A018874 (k=6), A018876 (k=7), A018878 (k=8), A018880 (k=9), A018882 (k=10).

v=[]; k=1; while(#v<100, d=digits(k^4); D=digits(#v+1); if(#D<=#d, c=1; for(i=1, #D, if(D[i]!=d[i], c=0; break)); if(c, v=concat(v, k); k=0)); k++); v \\ Derek Orr, Aug 12 2015

a(n^4) = n for n >= 0. - Seiichi Manyama, Jan 30 2017

Added initial 0. - Seiichi Manyama, Jan 30 2017

A018872 a(n)^5 is smallest fifth power beginning with n.

Original entry on oeis.org

0, 1, 3, 2, 21, 9, 23, 6, 61, 25, 4, 26, 66, 42, 17, 69, 7, 28, 18, 72, 29, 116, 47, 75, 3, 48, 192, 77, 31, 124, 79, 5, 2, 32, 51, 129, 205, 13, 52, 33, 21, 53, 212, 134, 85, 34, 136, 86, 137, 87, 55, 22, 35, 14, 352, 56, 224, 142, 226, 9, 36, 144, 91, 363, 23, 58, 146, 232, 147, 37

Offset: 0

David W. Wilson

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A018851 (k=2), A018852 (k=3), A018853 (k=4), this sequence (k=5), A018874 (k=6), A018876 (k=7), A018878 (k=8), A018880 (k=9), A018882 (k=10).

a(n) = my(k=0, ss=Str(n)); while(strsplit(Str(k^5), ss)[1] != "", k++); k; \\ Michel Marcus, Aug 19 2025

a(n^5) = n for n >= 0. - Seiichi Manyama, Jan 30 2017

Added initial 0. - Seiichi Manyama, Jan 30 2017

A018874 a(n)^6 is smallest sixth power beginning with n.

Original entry on oeis.org

0, 1, 8, 18, 4, 9, 2, 3, 21, 46, 10, 7, 33, 49, 23, 5, 16, 11, 35, 24, 77, 36, 53, 115, 17, 37, 8, 55, 81, 12, 26, 121, 83, 263, 18, 39, 124, 85, 27, 271, 4, 127, 59, 87, 188, 189, 6, 19, 13, 89, 131, 61, 132, 9, 42, 133, 62, 134, 197, 29, 92, 292, 63, 43, 2, 201, 137, 202, 64, 138

Offset: 0

David W. Wilson

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A018851 (k=2), A018852 (k=3), A018853 (k=4), A018872 (k=5), this sequence (k=6), A018876 (k=7), A018878 (k=8), A018880 (k=9), A018882 (k=10).

a(n^6) = n for n >= 0. - Seiichi Manyama, Jan 30 2017

Added initial 0. - Seiichi Manyama, Jan 30 2017

A018876 a(n)^7 is smallest seventh power beginning with n.

Original entry on oeis.org

0, 1, 3, 12, 9, 34, 13, 5, 7, 37, 10, 38, 2, 28, 76, 55, 4, 15, 21, 11, 8, 3, 58, 42, 22, 114, 16, 6, 116, 84, 117, 44, 85, 119, 23, 12, 167, 45, 234, 63, 88, 17, 33, 46, 89, 24, 173, 9, 174, 65, 47, 91, 34, 127, 66, 92, 128, 248, 48, 129, 67, 18, 13, 181, 35, 351, 131, 49, 183, 95, 255

Offset: 0

David W. Wilson

			a(2)=3, because 3^7=2187 is the first 7th power with the most significant digit 2;
a(16)=4, because 4^7=16384 is the first 7th power whose two most significant digits are 16.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A018851 (k=2), A018852 (k=3), A018853 (k=4), A018872 (k=5), A018874 (k=6), this sequence (k=7), A018878 (k=8), A018880 (k=9), A018882 (k=10).

a(n^7) = n for n >= 0. - Seiichi Manyama, Jan 30 2017

Added initial 0. - Seiichi Manyama, Jan 30 2017

A018878 a(n)^8 is smallest eighth power beginning with n.

Original entry on oeis.org

0, 1, 2, 5, 9, 7, 3, 23, 13, 42, 10, 18, 58, 78, 14, 25, 6, 34, 81, 61, 26, 11, 35, 47, 63, 2, 113, 85, 27, 86, 115, 65, 87, 49, 277, 37, 66, 21, 158, 5, 67, 283, 12, 9, 286, 51, 287, 91, 289, 122, 29, 69, 123, 39, 22, 93, 221, 7, 222, 125, 94, 223, 53, 126, 71, 3, 95, 127, 226, 17

Offset: 0

David W. Wilson

a(2)=2, because 2^8=256 is the first 8th power with the most significant decimal digit equal to 2;

a(16)=6, because 6^8=1679616 is the first 8th power with two most significant decimal digits equal to 16.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A018851 (k=2), A018852 (k=3), A018853 (k=4), A018872 (k=5), A018874 (k=6), A018876 (k=7), this sequence (k=8), A018880 (k=9), A018882 (k=10).

a(n^8) = n for n >= 0. - Seiichi Manyama, Jan 30 2017

Added initial 0. - Seiichi Manyama, Jan 30 2017

A018880 a(n)^9 is smallest ninth power beginning with n.

Original entry on oeis.org

0, 1, 4, 9, 7, 2, 16, 21, 59, 46, 6, 17, 22, 8, 29, 63, 38, 137, 23, 3, 14, 109, 141, 11, 184, 86, 4, 52, 87, 188, 113, 68, 19, 191, 148, 32, 149, 193, 9, 54, 7, 117, 91, 152, 118, 71, 33, 92, 154, 332, 43, 2, 93, 201, 26, 121, 261, 94, 73, 122, 34, 44, 264, 57, 123, 343, 74, 344, 16

Offset: 0

David W. Wilson

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A018851 (k=2), A018852 (k=3), A018853 (k=4), A018872 (k=5), A018874 (k=6), A018876 (k=7), A018878 (k=8), this sequence (k=9), A018882 (k=10).

a(n^9) = n for n >= 0. - Seiichi Manyama, Jan 30 2017

Added initial 0. - Seiichi Manyama, Jan 30 2017

A018882 a(n)^10 is smallest tenth power beginning with n.

Original entry on oeis.org

0, 1, 7, 9, 23, 3, 6, 49, 31, 5, 2, 32, 81, 13, 26, 33, 21, 42, 67, 85, 17, 43, 86, 109, 69, 11, 22, 35, 7, 28, 56, 89, 71, 113, 9, 18, 36, 72, 91, 182, 115, 23, 29, 58, 116, 232, 185, 147, 37, 74, 148, 59, 47, 94, 375, 188, 75, 15, 189, 3, 6, 12, 38, 24, 48, 152, 96, 121, 192, 305

Offset: 0

David W. Wilson

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A018851 (k=2), A018852 (k=3), A018853 (k=4), A018872 (k=5), A018874 (k=6), A018876 (k=7), A018878 (k=8), A018880 (k=9), this sequence (k=10).

a(n^10) = n for n >= 0. - Seiichi Manyama, Jan 30 2017

Added initial 0. - Seiichi Manyama, Jan 30 2017

cp's OEIS Frontend

A018796 Smallest square that begins with n.

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A018852 a(n)^3 is smallest cube beginning with n.

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A265432 a(n) = smallest k with concat(1,k) and concat(n,k) both square numbers.

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A018853 a(n)^4 is smallest fourth power beginning with n.

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A018872 a(n)^5 is smallest fifth power beginning with n.

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A018874 a(n)^6 is smallest sixth power beginning with n.

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A018876 a(n)^7 is smallest seventh power beginning with n.

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A018878 a(n)^8 is smallest eighth power beginning with n.

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