A118936 -id:A118936 - cp's OEIS frontend

A228381 Unabridged sub-Kaprekar numbers (A118936, but allowing powers of ten).

10, 11, 78, 100, 101, 287, 364, 1000, 1001, 1078, 1096, 1287, 1364, 10000, 10001, 11096, 18183, 100000, 100001, 118183, 336634, 1000000, 1000001, 1336634, 2727274, 10000000, 10000001, 12727274, 19138757, 23529412, 25974026, 97744361, 100000000, 100000001, 120879122

Offset: 1

Hans Havermann, Aug 21 2013

Square roots of A228103.

Excluding powers-of-ten and powers-of-ten-plus-one, what remains may be arranged into pairs (x,y), y>x, where y-x is a power of ten. The x terms correspond to A118938.

			10^2 = (10-0)^2.
11^2 = (12-1)^2.
78^2 = (6-084)^2.

Hans Havermann, Table of n, a(n) for n = 1..1016

Cf. A045913, A118936, A118938, A228103.

k=3; While[k<10^8, k++; s=k^2; d=IntegerDigits[s]; l=Length[d]; Do[a=FromDigits[Take[d, {1, i}]]; b=FromDigits[Take[d, {i+1, l}]]; If[k==Abs[a-b], Print[k]], {i, l-1}]]

lista(nn) = my(d, s, t=1, v=List([])); while(t(x>1&&x<=nn), v)); \\ Jinyuan Wang, Jan 02 2025

A118938 Sub-Kaprekar numbers (2): n such that n=r-q and n^2=q*10^m+r, for some m>=1, q>=0, 0<=r<10^m, with n not a power of 10.

Original entry on oeis.org

78, 287, 364, 1096, 18183, 336634, 2727274, 19138757, 23529412, 25974026, 97744361, 120879122, 140017878, 165991904, 237762239, 288553552, 307692308, 333666334, 405436669, 428571430, 440553516, 447710186, 454545455, 473684212

Offset: 1

Giovanni Resta, May 06 2006

			287^2 = 82369 and 369-82 = 287.
A larger example is 1980198021^2 = 3921184202372316441, and 2372316441-392118420 = 1980198021.

Hans Havermann, Table of n, a(n) for n = 1..1000
Hans Havermann, A large number of terms pairing them with the their A118937 partners.

Cf. A006886, A118936, A118937, A228381.

A228103 Numbers k whose base-10 digits can be split into two parts, q and r, with k = (q-r)^2.

Original entry on oeis.org

100, 121, 6084, 10000, 10201, 82369, 132496, 1000000, 1002001, 1162084, 1201216, 1656369, 1860496, 100000000, 100020001, 123121216, 330621489, 10000000000, 10000200001, 13967221489, 113322449956, 1000000000000, 1000002000001, 1786590449956, 7438023471076

Offset: 1

Hans Havermann, Aug 10 2013

q*10^m+r = (q-r)^2 = A228381^2; q,m>0; 0<=r<10^m. - Hans Havermann, Aug 21 2013

			100 = (10-0)^2.
121 = (12-1)^2.
6084 = (6-084)^2.

Hans Havermann, Table of n, a(n) for n = 1..1016
Daniel Joyce, Robert Israel and Lars Blomberg, Can more of these terms be found? [broken link?]
J. B. Wood, Quick Solution to Puzzle?

Cf. A102766, A118936, A228381.

k=3; While[k<10^8, k++; s=k^2; d=IntegerDigits[s]; l=Length[d]; Do[a=FromDigits[Take[d,{1,i}]]; b=FromDigits[Take[d,{i+1,l}]]; If[k==Abs[a-b], w=ToString[s]; Print[StringTake[w,{1,i}], "'", StringTake[w,{i+1,l}]]], {i,l-1}]] (* Hans Havermann, Aug 10 2013, Aug 20 2013 *)

A118937 Sub-Kaprekar numbers (1): n such that n=q-r and n^2=q*10^m+r, for some m>=1, q>=0, 0<=r<10^m, with n not a power of 10.

Original entry on oeis.org

11, 101, 1001, 1078, 1287, 1364, 10001, 11096, 100001, 118183, 1000001, 1336634, 10000001, 12727274, 100000001, 123529412, 1000000001, 1019138757, 1025974026, 1097744361, 1120879122, 1140017878, 1165991904, 1237762239, 1288553552

Offset: 1

Giovanni Resta, May 06 2006

			1287^2 = 1656369 and 1656-369 = 1287.
A larger example: 1594563333^2 = 2542632222948068889 and
2542632222-948068889=1594563333.

Cf. A006886, A118936, A118938.

cp's OEIS Frontend

A228381 Unabridged sub-Kaprekar numbers (A118936, but allowing powers of ten).

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A118938 Sub-Kaprekar numbers (2): n such that n=r-q and n^2=q*10^m+r, for some m>=1, q>=0, 0<=r<10^m, with n not a power of 10.

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A228103 Numbers k whose base-10 digits can be split into two parts, q and r, with k = (q-r)^2.

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A118937 Sub-Kaprekar numbers (1): n such that n=q-r and n^2=q*10^m+r, for some m>=1, q>=0, 0<=r<10^m, with n not a power of 10.

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