A257796 -id:A257796 - cp's OEIS frontend

A257588 If n = abcd... in decimal, a(n) = |a^2 - b^2 + c^2 - d^2 + ...|.

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 1, 0, 3, 8, 15, 24, 35, 48, 63, 80, 4, 3, 0, 5, 12, 21, 32, 45, 60, 77, 9, 8, 5, 0, 7, 16, 27, 40, 55, 72, 16, 15, 12, 7, 0, 9, 20, 33, 48, 65, 25, 24, 21, 16, 9, 0, 11, 24, 39, 56, 36, 35, 32, 27, 20, 11, 0, 13, 28, 45, 49

Offset: 0

N. J. A. Sloane, May 10 2015

a(n) = 0 iff n is in A352535. - Bernard Schott, Jul 08 2022

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A257587, A257796.

Cf. A225693, A003132, A352535.

a257588 = abs . f 1 where
   f _ 0 = 0
   f s x = s * d ^ 2 + f (negate s) x' where (x', d) = divMod x 10
-- Reinhard Zumkeller, May 10 2015

a:= n-> (l-> abs(add(l[i]^2*(-1)^i, i=1..nops(l))))(convert(n, base, 10)):
seq(a(n), n=0..70);  # Alois P. Heinz, Mar 24 2022

Array[Abs@ Total@ MapIndexed[(2 Boole@ EvenQ[First[#2]] - 1) (#1^2) &, IntegerDigits[#]] &, 70] (* Michael De Vlieger, Feb 27 2022 *)

a(n) = my(d=digits(n)); abs(sum(k=1, #d, (-1)^k*d[k]^2)); \\ Michel Marcus, Feb 27 2022

def A257588(n):
    return abs(sum((int(d)**2*(-1)**j for j,d in enumerate(str(n)))))
# Chai Wah Wu, May 10 2015

A257587 If n = abcd... in decimal, a(n) = a^2 - b^2 + c^2 - d^2 + ...

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 1, 0, -3, -8, -15, -24, -35, -48, -63, -80, 4, 3, 0, -5, -12, -21, -32, -45, -60, -77, 9, 8, 5, 0, -7, -16, -27, -40, -55, -72, 16, 15, 12, 7, 0, -9, -20, -33, -48, -65, 25, 24, 21, 16, 9, 0, -11, -24, -39, -56, 36, 35, 32

Offset: 0

N. J. A. Sloane, May 10 2015

Michel Marcus, Table of n, a(n) for n = 0..10000

First 100 terms coincide with those of A177894, but then they diverge.

Cf. A257588, A257796, A352535 (indices of zeros).

A257587[n_] := Total[-(-1)^Range[Max[IntegerLength[n], 1]]*IntegerDigits[n]^2];
Array[A257587, 100, 0] (* Paolo Xausa, Mar 11 2024 *)

a(n) = my(d=digits(n)); sum(k=1, #d, (-1)^(k+1)*d[k]^2); \\ Michel Marcus, Jul 12 2022

def a(n): return sum(int(d)**2*(-1)**i for i, d in enumerate(str(n)))
print([a(n) for n in range(63)]) # Michael S. Branicky, Jul 11 2022

a(A352535(n)) = 0. - Bernard Schott, Jul 12 2022

A258482 Positive numbers n with concatenations n=x//y such that n=x^2-y^2.

Original entry on oeis.org

100, 147, 10000, 13467, 1000000, 1010100, 1016127, 1034187, 1140399, 1190475, 1216512, 1300624, 1334667, 1416767, 1484847, 1530900, 100000000, 102341547, 102661652, 116604399, 133346667, 159809775, 10000000000, 10101010100, 13333466667, 14848484847

Offset: 1

Pieter Post, May 31 2015

The terms in this sequence have only an odd number of digits. If they would have an even number of digits both parts would have the same length. The maximum difference x^2 - y^2 would be (10^m-1)^2 - 1^2, which is (10^m-2)*10^m. But this is always less than (10^m-1)^2 + 1, so m never equals x^2 - y^2.
For example m=3: 999^2 - 1^2 < 999001.
The terms in this sequence all start with the digit '1'. Suppose they would start with the digit '2' (or more) the smallest possiblity of x^2 - y^2 would be (2*10^m)^2 - (10^m-1)^2 = 3*10^2*m + 2*10^m-1, but this is always more than 2*10^2*m + 10^3-1, so m never equals x^2 - y^2.
For example m=3: 2000^2 - 999^2 > 2000999.
This sequence has an infinite subsequence, since (10^m+(10^m+2)/3)*10^m+(2*10^m+1)/3 equals (10^m+(10^m+2)/3)^2 - ((2*10^m+1)/3)^2 for every positive m.
For example m=3: 1334667 = 1334^2 - 667^2.
This set is a subset of A113797.

			147 is a member, since 147 = 14^2 - 7^2.
1484847 is a member, since 1484847 = 1484^2- 847^2.
48 is a member of A113797 since 48 = |4^2 - 8^2|, but 48 is not equal to 4^2 - 8^2, so 48 is not a member of this sequence.

Giovanni Resta, Table of n, a(n) for n = 1..4397 (terms < 10^60)

Cf. A002654, A055616, A064942, A064943, A113797, A101311, A162700, A257796.

isok(n) = {d = digits(n); if (#d > 1, for (k=1, #d-1, vba = Vecrev(vector(k, i, d[i])); vbb = Vecrev(vector(#d-k, i, d[k+i])); da = sum(i=1, #vba, vba[i]*10^(i-1)); db = sum(i=1, #vbb, vbb[i]*10^(i-1)); if (da^2 - db^2 == n, return(1));););} \\ Michel Marcus, Jun 14 2015

for p in range(1, 7):
    for i in range(10**p, 10**(p + 1)):
        c = 10**(int((p - 1) / 2) + 1)
        a, b = i // c, i % c
        if i == a**2 - b**2:
            print(i, end=",")

n=x*10^d+y, where 10^(d-1)<=x<10^d and 0<=y<10^d and n=x^2-y^2.

More terms from Giovanni Resta, Jun 14 2015

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

A257588 If n = abcd... in decimal, a(n) = |a^2 - b^2 + c^2 - d^2 + ...|.

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A257587 If n = abcd... in decimal, a(n) = a^2 - b^2 + c^2 - d^2 + ...

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A258482 Positive numbers n with concatenations n=x//y such that n=x^2-y^2.

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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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