A118938 -id:A118938 - 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

A118936 Sub-Kaprekar numbers: k such that k = |q - r| and k^2 = q*10^m + r, for some m >= 1, q >= 0, 0 <= r < 10^m, with k not a power of 10.

Original entry on oeis.org

11, 78, 101, 287, 364, 1001, 1078, 1096, 1287, 1364, 10001, 11096, 18183, 100001, 118183, 336634, 1000001, 1336634, 2727274, 10000001, 12727274, 19138757, 23529412, 25974026, 97744361, 100000001, 120879122, 123529412, 140017878

Offset: 1

Giovanni Resta, May 06 2006; corrected May 12 2006

Union of A118937 and A118938.

			287^2 = 82369 and |82 - 369| = 287, so 287 is a term.
1287^2 = 1656369 and |1656 - 369| = 1287, so 1287 is a term.

Cf. A006886, A118937, A118938.

f[n_] := !IntegerQ@Log[10,n] && Block[{p = 10^Range@Log[10,n^2]}, 0 == Times@@(n-Abs[Floor[n^2/p]-Mod[n^2,p]])]; Select[Range@400000,f]

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.

A259379 Numbers k of the form a - b + c, such that k^3 equals the decimal concatenation a//b//c and numbers k, b, and c have the same number of digits.

Original entry on oeis.org

155, 209, 274, 286, 287, 351, 364, 428, 573, 637, 715, 727, 846, 923, 1095, 1096, 2191, 8905, 18182, 18183, 81818, 81819, 326734, 336634, 663367, 673267, 2727273, 2727274, 4545454, 5454547, 7272727, 23529411, 23529412, 76470589

Offset: 1

Pieter Post, Jul 22 2015

This sequence is infinite because it has several infinite subsequences. For example:
  274, 326734, 332667334, 3..326..673..34 etc.;
  364, 336634, 333666334, 3..36..63..34 etc.;
  637, 663367, 666333667, 6..63..36..67 etc.;
  727, 673267, 667332667, 6..673..326..67 etc.
Note that: 274 + 727 = 364 + 637 = 1001 and 326734 + 673267 = 336634 + 663367 = 1000001.
Many numbers come in pairs, like: (286, 287), (1095, 1096), (18182, 18183) but also bigger number (140017877, 140017878) and (859982123, 859982124).
140017877 + 859982124 = 140017878 + 859982123 = 1000000001.

			155^3 = 3723875 and 155 = 3 - 723 + 875.
715^3 = 365525875 and 715 = 365 - 525 + 875.

Pieter Post, Table of n, a(n) for n = 1..189

Cf. A056733, A101311, A118937, A118938, A228381, A257796, A258482.

isok(n)=nb = #digits(n, 10); if (a = n^3 \ 10^(2*nb), c = n^3 % 10^nb; b = (n^3 - a*10^(2*nb))\10^nb; n^3 == (a-b+c)^3;); \\ Michel Marcus, Aug 05 2015

def modb(n,m):
    kk = 0
    l = 1
    while n > 0:
        na = n % m
        l += 1
        kk += ((-1)**l) * na
        n //= m
    return kk
for n in range (100, 10**9):
    ll = len(str(n))
    if modb(n**3, 10**ll) == n:
        print(n, end=', ') # corrected by David Radcliffe, May 09 2025

A260193 Numbers k of the form abs(a - b + c - d) such that k^4 equals the concatenation of a//b//c//d and numbers k,b,c,d have the same number of digits.

Original entry on oeis.org

198, 220, 221, 287, 352, 364, 484, 562, 627, 638, 672, 715, 716, 780, 793, 858, 901, 1095, 1233, 2328, 8905, 18183, 39753, 60248, 85207, 336734, 2727274, 5893504, 8620777, 17769557, 52818678, 70710735, 76470590, 82230444, 101318734, 101636206, 104263158, 105262158, 109891110, 109942690, 117883117, 119722383, 120826541

Offset: 1

Pieter Post, Jul 22 2015

Leading zeros in b, c, and d are allowed.
Many numbers come in pairs, like: (220, 221), (715, 716), (140017877, 140017878).
Some numbers are also member of A259379, for example: 287, 715, 1095 and also the pair (140017877, 140017878).

			198^4 = 1536953616 and 198 = abs (1 - 536 + 953 - 616 ).
8905^4 = 6288335365950625 and 8905 = abs (6288 - 3353 + 6595 - 0625 ).

Pieter Post, Table of n, a(n) for n = 1..239

Cf. A056733, A101311, A118937, A118938, A228381, A257796, A258482, A259379.

test[n_] := Block[{L=IntegerLength@ n, v}, v = IntegerDigits[ n^4, 10^L]; Length@ v == 4 && Abs@ Total[ {1, -1, 1, -1} v] == n]; Select[Range[10^5], test] (* Giovanni Resta, Aug 12 2015 *)

def modb(n, m):
    kk = 0
    l = 1
    while n > 0:
        na = n % m
        l += 1
        kk += ((-1)**l) * na
        n //= m
    return abs(kk)
for n in range (100, 10**9):
    ll = len(str(n))
    if modb(n**4, 10**ll) == n and n**4 >= 10**(ll*3):
         print (n, end=', ') # corrected by David Radcliffe, May 09 2025

cp's OEIS Frontend

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

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A118936 Sub-Kaprekar numbers: k such that k = |q - r| and k^2 = q*10^m + r, for some m >= 1, q >= 0, 0 <= r < 10^m, with k not a power of 10.

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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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A259379 Numbers k of the form a - b + c, such that k^3 equals the decimal concatenation a//b//c and numbers k, b, and c have the same number of digits.

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A260193 Numbers k of the form abs(a - b + c - d) such that k^4 equals the concatenation of a//b//c//d and numbers k,b,c,d have the same number of digits.

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Python