a(n) = number of {x,y,z} in Z/nZ such that x+y+z = x*y*z is nonzero.

0, 1, 0, 1, 4, 2, 6, 6, 6, 21, 18, 7, 30, 30, 22, 28, 50, 37, 56, 63, 30, 86, 84, 42, 120, 145, 72, 87, 144, 124, 154, 140, 98, 245, 188, 123, 234, 274, 176, 274, 286, 169, 300, 253, 270, 410, 360, 196, 336, 607, 308, 437, 476, 388, 520, 378, 344, 709, 570, 429, 630, 758, 372, 600, 882, 517, 736, 751, 524, 968, 828, 546, 900, 1153, 810, 839, 720, 912, 1026, 1140, 738, 1413, 1134, 585, 1502, 1482, 930

1,5

Similar to A376183, but the sum equal to the product is restricted to be nonzero, as to exclude many rather trivial solutions.

Counting sets {x, y, z} means that the order of the three (not necessarily distinct) values is irrelevant. Equivalently, one could use representatives with, e.g., x <= y <= z.

For n=1, we can't have x+y+z = x*y*z nonzero because zero is the only element, so there are a(1) = 0 solutions.

For n=2, x+y+z = x*y*z != 0 implies x = y = z = 1 (in Z/2Z), so there is only one unique solution, and a(2) = 1.

For n=3, x+y+z = x*y*z != 0 is impossible: if x = y = z, the sum is zero (in Z/3Z), and 1 + 2 + z = 1*2*z <=> z = 0, so there is no solution, a(3) = 0.

For n=4, since 2*2 = 0 in Z/4Z, at most one among {x, y, z}, say z, can equal 2. In this case, x = -y = +-1, gives a solution {x, y, z} = {1, 2, 3}. One then checks that x = +-y = z = +-1 can't yield a solution, so a(4) = 1.

For n=5, we see that (x, y) = (2, 3) gives z = z, so we have a solution for any nonzero z, and exhaustive verification shows that these are no other solutions: a(5) = 4.

(PARI) apply( {A376240(n)=sum(x=1, n-1, sum(y=1, x, sum(z=1, y, (x+y+z-x*y*z)%n==0 && x*y*z%n)))}, [1..99])

Cf. A376183 (same without restriction to nonzero sum/product).

nonn,new

M. F. Hasler, Sep 24 2024

approved

Nonnegative integers N = x*y*z = x+y+z for some rational x, y, z.

0, 6, 7, 9, 13, 14, 15, 16, 19

1,2

Obviously all of x, y and z must be nonzero for all solutions N > 0. For any N = x*y*z = x+y+z, one gets -N from (-x, -y, -z), so considering only N >= 0 is not a restriction. Either none or exactly two among x, y and z must be negative.

For given N, the problem amounts to finding fractions x and y such that x*y^2 + x*(x - N)*y + N = 0, which in turn corresponds to finding rational points on the elliptic curve Y^2 = X^3 + N^2*(X+4)^2 (with X = -4*N/x and Y = 4*N*D/x^2, where D^2 is the discriminant of the previous quadratic in y).

It appears that (for N > 0) we have a rational solution iff this elliptic curve has nonzero rank. (Is there any counter-example?) If so, the sequence goes (0, 6, 7, 9, 13, 14, 15, 16, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 37, 38, 40, 43, 44, 45, 46, 48, 49, 52, 53, 55, 56, 58, 59, 60, 61, 62, 63, 64, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 86, 87, ...)

Allan MacLeod, <a href="https://doi.org/10.48550/arXiv.1610.03430">Elliptic Curves in Recreational Number Theory</a>, arXiv:1610.03430 [math.NT], Oct. 2016.

Victor Miller and others, in reply to Keith F. Lynch, <a href="https://mailman.xmission.com/hyperkitty/list/math-fun@mailman.xmission.com/message/T47KO4SAQEHPTUUX6I6GKZBFU6HS3JSP/">Re: Integer sums and products</a>, math-fun mailing list (available for subscribers), Sep. 2024.

The first few terms correspond to the following solutions (|x| <= |y| <= |z|):

N | x | y | z

-----+---------+---------+---------

0 | 0 | 0 | 0 (or any rational y = -z).

6 | 1 | 2 | 3 (and also {25/21, 54/35, 49/15}).

7 | 7/6 | 4/3 | 9/2

9 | 1/2 | 4 | 9/2

13 | 36/77 | 121/42 | 637/66

14 | 1/3 | 9 | 14/3

15 | 1/2 | 5/2 | 12

16 | -2/3 | -4/3 | 18

19 | 121/234 | 324/143 |3211/198

...

All terms of A054000 (2*n^2-2: 0, 6, 16, 30, 48, 70, 96, 126, 160, 198, ...) are in the sequence, as product and sum of the triple (2*n^2, 1/n - 1, -1/n - 1).

(PARI) select( {is_A376243(n)=!n||ellrank(ellinit([0, 1, 0, 8, 16]*n^2))}, [0..30]) \\ Assuming there's a rational solution iff the elliptic curve has rank > 0. - M. F. Hasler, Sep 23 2024

Cf. A376241-A376242 for an enumeration of all possible solutions (not in the order of increasing N) using the Stern-Brocot sequence A002487.

A054000 (2*n^2-2) is a subsequence.

nonn,more,changed

M. F. Hasler, Sep 16 2024

approved

a(n) = least m >= 0 such that (x = f(A376241(n)), y = f(m), z = (x+y)/(xy-1)) yields an integer x+y+z = x*y*z, where f(m) = A002487(m)/A002487(m+1).

0, 1, 2, 1, 6, 2, 2, 14, 4, 30, 12, 35, 2, 4, 9, 20, 4, 8, 126, 56, 32, 152, 52, 254, 61, 84, 40, 16, 4, 510, 368, 320, 212, 48, 396, 72, 583, 1022, 792, 368, 98, 188, 340, 80, 583, 339, 140, 32, 233, 2046, 480, 384, 583, 2062, 852, 188, 328

1,3

A376241 uses the Stern-Brocot sequence s = A002487 to enumerate all (nonnegative) rational x = s(n)/s(n+1) and similarly y = s(m)/s(m+1), WLOG m <= n, which yield an integer x*y*z = x+y+z with (necessarily) z = (x+y)/(xy-1). The present sequence lists the m-values corresponding to the n-values listed in A376241.

The terms correspond to the following solutions, with y = A002487(m)/A002487(m+1):

m | x | y | z | xyz = x+y+z

-----+-----+-----+-----+------------

0 | 0 | 0 | 0 | 0

1 | 2 | 1 | 3 | 6

2 | 3/2 | 1/2 | -8 | -6

1 | 3 | 1 | 2 | 6

6 | 4/3 | 2/3 | -18 | -16

2 | 5/2 | 1/2 | 12 | 15

2 | 4 | 1/2 | 9/2 | 9

14 | 5/4 | 3/4 | -32 | -30

...| ... | ... | ... | ...

(PARI) A376242(n, k=A376241(n))={my(p, q=1, x=A002487(k)/A002487(k+1)); for(m=2, k, my(y=(p=q)/q=A002487(m)); x*y != 1 && denominator(x+y+(x+y)/(x*y-1))==1 && return(m-1))} \\ Short of a function A376241(n), one can simply provide a term k = A376241(n) as second argument and omit the first argument n.

Cf. A002487 (Stern-Brocot sequence), A376241 (corresponding n values), A376243 (set of absolute values of corresponding xyz = x+y+z).

nonn,changed

M. F. Hasler, Sep 16 2024

approved

Triangle read by rows: row n gives the arithmetic progression of n primes with minimal final term, cf. A005115.

2, 2, 3, 3, 5, 7, 5, 11, 17, 23, 5, 11, 17, 23, 29, 7, 37, 67, 97, 127, 157, 7, 157, 307, 457, 607, 757, 907, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089, 110437, 124297, 138157, 152017, 165877, 179737, 193597, 207457, 221317, 235177, 249037

1,1

The first 10 rows (i.e., 55 terms) are the same as for A133276 (where the common distance is minimal), but here T(11,1) = a(56) = 110437 while A133276(11,1) = 60858179. - M. F. Hasler, Jan 02 2020

OEIS wiki, <a href="/wiki/Primes_in_arithmetic_progression">Primes in arithmetic progression</a>.

<a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>

Triangle begins:

2;

2, 3;

3, 5, 7;

5, 11, 17, 23;

5, 11, 17, 23, 29;

7, 37, 67, 97, 127, 157;

7, 157, 307, 457, 607, 757, 907;

199, 409, 619, 829, 1039, 1249, 1459, 1669;

199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879;

199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089;

...

For common differences, see A093364. For initial terms, see A113827. For final terms, see A005115.

Differs from A133276 (from T(11,1) = a(56) on).

nonn,tabl

N. J. A. Sloane, Oct 17 2007

A-numbers in the Name and Crossrefs sections corrected by Bobby Jacobs, Dec 10 2016

Name edited by M. F. Hasler, Jan 02 2020

approved

Nonnegative integers N = x*y*z = x+y+z for some rational x, y, z.

0, 6, 7, 9, 13, 14, 15, 16, 19

1,2

Obviously all of x, y and z must be nonzero for all solutions N > 0. For any N = x*y*z = x+y+z, one gets -N from (-x, -y, -z), so considering only N >= 0 is not a restriction. Either none or exactly two among x, y and z must be negative.

For given N, the problem amounts to finding fractions x and y such that x*y^2 + x*(x - N)*y + N = 0, which in turn corresponds to finding rational points on the elliptic curve Y^2 = X^3 + N^2*(X+4)^2 (with X = -4*N/x and Y = 4*N*D/x^2, where D^2 is the discriminant of the previous quadratic in y).

Allan MacLeod, <a href="https://doi.org/10.48550/arXiv.1610.03430">Elliptic Curves in Recreational Number Theory</a>, arXiv:1610.03430 [math.NT], Oct. 2016.

Victor Miller and others, in reply to Keith F. Lynch, <a href="https://mailman.xmission.com/hyperkitty/list/math-fun@mailman.xmission.com/message/T47KO4SAQEHPTUUX6I6GKZBFU6HS3JSP/">Re: Integer sums and products</a>, math-fun mailing list (available for subscribers), Sep. 2024.

The first few terms correspond to the following solutions (|x| <= |y| <= |z|):

N | x | y | z

-----+---------+---------+---------

0 | 0 | 0 | 0 (or any rational y = -z).

6 | 1 | 2 | 3 (and also {25/21, 54/35, 49/15}).

7 | 7/6 | 4/3 | 9/2

9 | 1/2 | 4 | 9/2

13 | 36/77 | 121/42 | 637/66

14 | 1/3 | 9 | 14/3

15 | 1/2 | 5/2 | 12

16 | -2/3 | -4/3 | 18

19 | 121/234 | 324/143 |3211/198

...

All terms of A054000 (2*n^2-2: 0, 6, 16, 30, 48, 70, 96, 126, 160, 198, ...) are in the sequence, as product and sum of the triple (2*n^2, 1/n - 1, -1/n - 1).

Cf. A376241-A376242 for an enumeration of all possible solutions (not in the order of increasing N) using the Stern-Brocot sequence A002487.

A054000 (2*n^2-2) is a subsequence.

nonn,more,changed

M. F. Hasler, Sep 16 2024

approved

Least positive integer k such that n^2 + k is a cube.

1, 7, 4, 18, 11, 2, 28, 15, 61, 44, 25, 4, 72, 47, 20, 118, 87, 54, 19, 151, 112, 71, 28, 200, 153, 104, 53, 271, 216, 159, 100, 39, 307, 242, 175, 106, 35, 359, 284, 207, 128, 47, 433, 348, 261, 172, 81, 535, 440, 343, 244, 143, 40, 566, 459, 350, 239, 126, 11

0,2

a(n) = A070923(n) if n is not cube. Zak Seidov, Mar 26 2013

See A229618 for the range of this sequence. A179386 gives the range of b(n) = min{ a(m); m >= n }. The indices of jumps in this sequence are given in A179388 = { n | a(m)>a(n) for all m > n } = { 0, 5, 11, 181, 207, 225, 500, 524, 1586, ... }. - M. F. Hasler, Sep 26 2013

Bruno Berselli, <a href="/A181138/b181138.txt">Table of n, a(n) for n = 0..1000</a> (Corrected Jan 19 2019)

a(n) << n^(4/3). - Charles R Greathouse IV, Mar 26 2013

a(11) = 4 because 11^2 + k is never a cube for k < 4, but 11^2 + 4 = 5^3. - Bruno Berselli, Jan 29 2013

Table[(1 + Floor[n^(2/3)])^3 - n^2, {n, 100}] (* Zak Seidov, Mar 26 2013 *)

(Magma)

S:=[];

k:=1;

for n in [0..60] do

while not IsPower(n^2+k, 3) do

k:=k+1;

end while;

Append(~S, k);

k:=1;

end for;

S; // Bruno Berselli, Jan 29 2013

(PARI) A181138(n)=(sqrtnint(n^2, 3)+1)^3-n^2 \\ Charles R Greathouse IV, Mar 26 2013

(PARI) A181138(n)=(n^(2/3)\(1-.1^precision(1.))+1)^3-n^2 \\ In the current stable version 2.5.5, the function sqrtnint is not yet introduced, it exists only in development versions from PARI V.2.6 on. Unfortunately, in the standard precision, floor(125^(2/3))=24. - M. F. Hasler, Sep 26 2013

Cf. A070923, A077116.

nonn,easy

Jason Earls, Oct 06 2010

Extended to a(0)=1 by M. F. Hasler, Sep 26 2013

approved

Least c such that A106265(n) + b^2 = c^3 for some b (given in A106266).

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, 7, 4, 18, 4, 5, 4, 4, 23, 8, 6, 99, 5, 20, 13, 27, 7, 5, 6, 5, 9, 11, 143, 5, 8, 5, 7, 5, 5, 5, 15, 16, 12, 6, 47, 14, 195, 7, 197, 175, 8, 6, 9, 51, 10, 6, 59, 13, 7, 11, 6

1,2

M. F. Hasler, <a href="/A106267/b106267_4.txt">Table of n, a(n) for n = 1..2500</a> (replacing an incomplete version from 2013), Aug 10 2024

a(n) = (A106265(n) + A106266(n)^2)^(1/3). - M. F. Hasler, Oct 04 2013

(PARI) for(a=1, 999, (c=is_A106265(a))&& print1(c", ")) \\ _M. F. Hasler_, Oct 04 2013, edited Aug 10 2024: main program moved to A106265, as function is_A106265 which returns the (least) c-value a(n), always nonzero for terms of A106265, or 0 if the argument isn't a term of A106265.

(PARI) A106267(n)={for(c=sqrtn(a=A106265(n), 3)\1, 9e9, issquare(c^3-a)&return(c))} \\ - M. F. Hasler, Oct 05 2013

Cf. A106265 (main entry), A106266 (the b-values).

nonn

Zak Seidov, Apr 28 2005

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

All values recomputed from A106266 and A106265 by R. J. Mathar, Aug 24 2008

Initial term 1 and other solutions with b=0 added by M. F. Hasler, Oct 04 2013

approved

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

1,2

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

Needs b-file corrected.

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

M. F. Hasler, <a href="/A106265/b106265_4.txt">Table of n, a(n) for n = 1..2500</a> (corrected and extended Aug 10 2024)

J. Gebel, <a href="/A106265/a106265.txt">Integer points on Mordell curves</a>, negative k values [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

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

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

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 *)

(PARI) 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

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.

nonn

Zak Seidov, Apr 28 2005

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

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

approved

Least b such that A106265(n) + b^2 = c^3 for some c (given in A106267)

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, 17, 3, 76, 2, 8, 1, 0, 110, 21, 12, 985, 7, 89, 46, 140, 16, 6, 11, 5, 25, 35, 1710, 4, 20, 3, 15, 2, 1, 0, 57, 63, 40, 9, 322, 51, 2723, 14, 2765, 2315, 19, 8, 24, 364, 29

1,2

The values A106266(n) = 0 correspond to cubes A106265(n) = c^3. See the main entry A106265 for further comments and references.

M. F. Hasler, <a href="/A106266/b106266_1.txt">Table of n, a(n) for n = 1..2500</a> (corrected and extended Aug 10 2024)

a(n) = sqrt(A106267(n)^3-A106265(n)). - M. F. Hasler, Oct 05 2013

(PARI) [sqrtint(is_A106265(a)^3-a) | a<-A106265] \\ Assumes is_A106265(a) returns the c-value corresponding to a, and A106265 is a list or vector of values of that sequence. -M. F. Hasler, Aug 10 2024

Cf. A106265 (main entry), A106266 (the b-values).

nonn

Zak Seidov, Apr 28 2005

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

Edited by M. F. Hasler, Oct 04 2013

approved

Primes p such that the p-th digit in the decimal expansion of Pi is 1.

2, 41, 139, 149, 199, 239, 251, 397, 433, 439, 443, 491, 569, 599, 641, 647, 661, 787, 853, 883, 1031, 1087, 1097, 1153, 1187, 1319, 1423, 1613, 1619, 1637, 1657, 1667, 1697, 1759, 1789, 2081, 2129, 2143, 2221, 2239, 2459, 2633, 2689, 2741, 2753, 2777

1,1

The 2nd 1 in Pi is in the 2nd place of the digits 3,1,4,1,5,...

Harvey P. Dale, <a href="/A088565/b088565.txt">Table of n, a(n) for n = 1..1000</a>

The 1st digit 1 in Pi is in the 2nd place of the digits 3,1,4,1,5,9,..., and 2 is prime, whence a(1) = 2. [Corrected and edited by M. F. Hasler, Jul 29 2024]

Select[Flatten[Position[RealDigits[Pi, 10, 2800][[1]], 1]], PrimeQ] (* Harvey P. Dale, May 05 2019 *)

(PARI) pizeros(n, d) = { default(realprecision, 5000); p = Pi; v = Vec(Str(p)); for(x=1, n, if(v[x] == Str(d) && isprime(x-1), print1(x-1", ")) ) }

(PARI) A088565_upto(N=3456, d=1)={localprec(N+20); [p|p<-primes([1, #N=digits(Pi\10^-N)]), N[p]==d]} \\ M. F. Hasler, Jul 29 2024

Primes in A014976.

Cf. A088563, A088566 (the same for digits 0 and 2), A000796 (decimal digits of Pi).

nonn,base

Cino Hilliard, Nov 19 2003

approved