A075772 -id:A075772 - cp's OEIS frontend

A106265 Numbers a > 0 such that the Diophantine equation a + b^2 = c^3 has integer solutions b and c.

1, 2, 4, 7, 8, 11, 13, 15, 18, 19, 20, 23, 25, 26, 27, 28, 35, 39, 40, 44, 45, 47, 48, 49, 53, 54, 55, 56, 60, 61, 63, 64, 67, 71, 72, 74, 76, 79, 81, 83, 87, 89, 95, 100, 104, 106, 107, 109, 112, 116, 118, 121, 124, 125, 126, 127, 128, 135, 139, 143, 146, 147, 148, 150, 151, 152, 153

Offset: 1

Zak Seidov, Apr 28 2005

nonn

A given a(n) can have multiple solutions with distinct (b,c), e.g., a=4 with b=2, c=2 (4 + 2^2 = 2^3) or with b=11, c=5 (4 + 11^2 = 5^3). (See also A181138.) Sequences A106266 and A106267 list the minimal values. - M. F. Hasler, Oct 04 2013
The cubes A000578 = (1, 8, 27, 64, ...) form a subsequence of this sequence, corresponding to b=0, a=c^3. If b=0 is excluded, these terms are not present, except for a few exceptions, a = 216, 343, 12167, ... (6^3 + 28^2 = 10^3, 7^3 + 13^2 = 8^3, 23^3 + 588^2 = 71^3, ...), cf. A038597 for the possible b-values. - M. F. Hasler, Oct 05 2013
This is the complement of A081121. The values do indeed correspond to solutions listed in Gebel's file. - M. F. Hasler, Oct 05 2013
B-file corrected following a remark by Alois P. Heinz, May 24 2019. A double-check would be appreciated in view of two values that were missing, for unknown reasons, in the earlier version of the b-file. - M. F. Hasler, Aug 10 2024

			a = 1,2,4,7,8,11,13,15,18,19,20,23,25,26,27,28,35,39,40,44,45,47,48,49,53, ...
b = 0,5,2,1,0, 4,70, 7, 3,18,14, 2,10, 1, 0, 6,36, 5,52, 9,96,13,4,524,26, ...
c = 1,3,2,2,2, 3,17, 4, 3, 7, 6, 3, 5, 3, 3, 4,11, 4,14, 5,21, 6, 4,65, 9, ...
Here are the values grouped together:
{{1, 0, 1}, {2, 5, 3}, {4, 2, 2}, {7, 1, 2}, {8, 0, 2}, {11, 4, 3}, {13, 70, 17}, {15, 7, 4}, {18, 3, 3}, {19, 18, 7}, {20, 14, 6}, {23, 2, 3}, {25, 10, 5}, {26, 1, 3}, {27, 0, 3}, {28, 6, 4}, {35, 36, 11}, {39, 5, 4}, {40, 52, 14}, {44, 9, 5}, {45, 96, 21}, {47, 13, 6}, {48, 4, 4}, {49, 524, 65}, {53, 26, 9}, {54, 17, 7}, {55, 3, 4}, {56, 76, 18}, {60, 2, 4}, {61, 8, 5}, {63, 1, 4}, {64, 0, 4}, {67, 110, 23}, {71, 21, 8}, ... }
a(2243) = 10000 = 25^3 - 75^2. - _M. F. Hasler_, Oct 05 2013, index corrected Aug 10 2024
a(136) = 366 = 11815^3 - 1284253^2 (has c/a(n) ~ 32.3); a(939) = 3607 = 244772^3 - 121099571^2 (has c/a(n) ~ 67.9); a(1090) = 4265 = 84521^3 - 24572364^2 (has c/a(n) ~ 19.8). - _M. F. Hasler_, Aug 10 2024

M. F. Hasler, Table of n, a(n) for n = 1..2500 (corrected and extended Aug 10 2024)
J. Gebel, Integer points on Mordell curves, negative k values [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Cf. A106266, A106267 for respective minimal values of b and c.
Cf. A023055: (Apparent) differences between adjacent perfect powers (integers of form a^b, a >= 1, b >= 2); A076438: n which appear to have a unique representation as the difference of two perfect powers; that is, there is only one solution to Pillai's equation a^x - b^y = n, with a>0, b>0, x>1, y>1; A076440: n which appear to have a unique representation as the difference of two perfect powers and one of those powers is odd; that is, there is only one solution to Pillai's equation a^x - b^y = n, with a>0, b>0, x>1, y>1 and that solution has odd x or odd y (or both odd); A075772: Difference between n-th perfect power and the closest perfect power, etc.
Cf. A023055, A075772, A076438, A076440.
Cf. A054504, A081121, A081120; A179386 - A179388.

f[n_] := Block[{k = Floor[n^(1/3) + 1]}, While[k < 10^6 && !IntegerQ[ Sqrt[k^3 - n]], k++ ]; If[k == 10^6, 0, k]]; Select[ Range[ 154], f[ # ] != 0 &] (* Robert G. Wilson v, Apr 28 2005 *)

select( {is_A106265(a, L=99)=for(c=sqrtnint(a, 3), (a+9)*L, issquare(c^3-a, &b) && return(c))}, [1..199]) \\ The function is_A106265 returns 0 if n isn't a term, or else the c-value (A106267) which can't be zero if n is a term. The L-value can be used to increase the search limit but so far no instance is known that requires L>68. - M. F. Hasler, Aug 10 2024

a(n) = A106267(n)^3 - A106266(n)^2.

More terms from Robert G. Wilson v, Apr 28 2005

Definition corrected, solutions with b=0 added by M. F. Hasler, Sep 30 2013

A076433 Perfect powers for which the two closest perfect powers are greater.

Original entry on oeis.org

25, 121, 2187, 6431296, 6434856, 6956883693, 27027009001, 34359738368, 42618264157, 312078649600, 312079600999, 328080365089, 11305780833649, 11305786504384, 19287643015432, 62854896459664, 79723523012809

Offset: 1

Neil Fernandez, Oct 10 2002

nonn

			The two closest perfect powers to 25 are 27 (difference = 2) and 32 (difference = 7). The third closest is 16 (difference = 9). Both 27 and 32 are greater than 25, so 25 is in the list.

Chai Wah Wu, Table of n, a(n) for n = 1..22

Cf. A001597, A053289, A075772, A076431, A076432.

from itertools import count, islice
from sympy import mobius, integer_nthroot
def A076433_gen(): # generator of terms
    def f(x): return int(x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    a = bisection(f)
    b = bisection(lambda x:f(x)+1,a,a)
    c = bisection(lambda x:f(x)+2,b,b)
    for i in count(3):
        d = bisection(lambda x:f(x)+i,c,c)
        if b-a > d-b:
            yield b
        a,b,c=b,c,d
A076433_list = list(islice(A076433_gen(),5)) # Chai Wah Wu, Sep 09 2024

More terms from Jud McCranie and Robert G. Wilson v, Oct 11 2002

a(6)-a(17) from Donovan Johnson, Sep 03 2008

A076431 Perfect powers for which the two closest perfect powers are smaller.

Original entry on oeis.org

9, 16, 36, 128, 144, 256, 361, 529, 1024, 1369, 1764, 2209, 2809, 3136, 3481, 5041, 5929, 6889, 8281, 9409, 10816, 12321, 13924, 16900, 17689, 19881, 22201, 24649, 27225, 29929, 33124, 36100, 39601, 43264, 51076, 55225, 59536, 64009, 69169

Offset: 1

Neil Fernandez, Oct 10 2002

nonn

			The two closest perfect powers to 9 are 8 (difference = 1) and 4 (difference =5). The third closest is 16 (difference = 7). Both 8 and 4 are smaller than 9, so 9 is in the list.

Cf. A001597, A053289, A075772, A076432, A076433.

A076432 Perfect powers for which the three closest perfect powers are smaller.

Original entry on oeis.org

36, 144, 2209, 6436369, 312079766881, 328081510656, 11305787558464, 62854912315881, 79723540870416, 4550858480922601, 11435943732416784, 3109406220195722500, 6258210474706101136, 7596357574791306304, 21016258678615763761, 32645304184825666489

Offset: 1

Neil Fernandez, Oct 10 2002

nonn

			The three closest perfect powers to 36 are 32 (difference = 4), 27 (difference = 9) and 25 (difference = 11). The fourth closest is 49 (difference = 13). 32, 27 and 25 are smaller than 36, so 36 is in the sequence.

Cf. A001597, A053289, A075772, A076431, A076433.

from itertools import count, islice
from sympy import mobius, integer_nthroot
def A076432_gen(): # generator of terms
    def f(x): return int(x+sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    a = bisection(f)
    b = bisection(lambda x:f(x)+1,a,a)
    c = bisection(lambda x:f(x)+2,b,b)
    d = bisection(lambda x:f(x)+3,c,c)
    for i in count(4):
        e = bisection(lambda x:f(x)+i,d,d)
        if d-a < e-d:
            yield d
        a,b,c,d=b,c,d,e
A076432_list = list(islice(A076432_gen(),5)) # Chai Wah Wu, Sep 09 2024

More terms from Jud McCranie and Robert G. Wilson v, Oct 11 2002
a(5)-a(10) from Donovan Johnson, Sep 03 2008
a(11)-a(16) from Donovan Johnson, Aug 01 2013

A075773 Let {b(n)} be the sequence of perfect powers (A001597); then a(n) = max { b(n)-b(n-1), b(n+1)-b(n) }.

Original entry on oeis.org

3, 4, 4, 7, 9, 9, 5, 5, 13, 15, 17, 19, 21, 21, 4, 16, 25, 27, 27, 20, 18, 18, 33, 35, 35, 19, 39, 41, 43, 43, 28, 47, 49, 51, 53, 55, 57, 59, 61, 61, 39, 65, 67, 69, 71, 71, 38, 75, 77, 79, 81, 81, 47, 85, 87, 89, 89, 68, 71, 71, 12, 95, 97, 99, 101, 103, 103

Offset: 1

Neil Fernandez, Oct 09 2002

nonn

			The perfect powers are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, etc. The 7th is 27. This is 2 larger than the 6th (25) and 5 smaller than the 8th (32). So a(7)=5.

Cf. A001597, A053289, A075772.

Missing 4 inserted and more terms from Sean A. Irvine, Mar 06 2025

cp's OEIS Frontend

A106265 Numbers a > 0 such that the Diophantine equation a + b^2 = c^3 has integer solutions b and c.

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A076433 Perfect powers for which the two closest perfect powers are greater.

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A076431 Perfect powers for which the two closest perfect powers are smaller.

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A076432 Perfect powers for which the three closest perfect powers are smaller.

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A075773 Let {b(n)} be the sequence of perfect powers (A001597); then a(n) = max { b(n)-b(n-1), b(n+1)-b(n) }.

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