keyword:new - cp's OEIS frontend

A386488 Record high points in A386487.

0, 1, 2, 3, 5, 9, 11, 25, 35, 64, 91, 108, 156, 194, 266, 471, 477, 538, 1308, 1485, 1945, 2479, 3256, 5328, 7247, 9008, 11566, 14859, 18714, 18815, 22031, 62149, 101919, 200924, 293629, 296260, 371947, 890616, 1177562, 2551932, 2827899, 3192227, 5538307, 6695946

Offset: 1

N. J. A. Sloane, Sep 01 2025

Needs a b-file!

Rémy Sigrist, PARI program

Cf. A386482, A387519-A387521.

r = -1; s = Import["https://oeis.org/A386482/b386482.txt","Data"][[All, -1]]; Reap[Do[If[# > r, r = #; Sow[#]] &[s[[n]] - n], {n, Length[s]}] ][[-1, 1]] (* Michael De Vlieger, Sep 01 2025 *)

PARI
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\\ See Links section.
```

More terms from Rémy Sigrist, Sep 02 2025

A386631 Values of u in the quartets (2, u, v, w) of type 3; i.e., values of u for solutions to 2(2 - u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

Original entry on oeis.org

5, 6, 7, 8, 8, 8, 9, 10, 10, 11, 11, 11, 12, 12, 12, 13, 14, 14, 14, 14, 14, 15, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18, 18, 19, 20, 20, 20, 20, 20, 20, 21, 22, 22, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 25, 26, 26, 26, 26, 26, 26, 26, 27, 27, 27, 28

Offset: 1

Clark Kimberling, Aug 22 2025

A 4-tuple (m, u, v, w) is a quartet of type 3 if m, u, v, w are distinct positive integers such that m < v and m*(m - u) = v*(v - w). Here, the values of u are arranged in nondecreasing order. When there is more than one solution for given m and u, the values of v are arranged in increasing order. Here, m = 2.

			First 20 quartets (2,u,v,w) of type 3:
   m    u    v    w
   2    5    6    7
   2    6    8    9
   2    7   10   11
   2    8    3    7
   2    8    4    7
   2    8   12   13
   2    9   14   15
   2   10    4    8
   2   10   16   17
   2   11    3    9
   2   11    6    9
   2   11   18   19
   2   12    4    9
   2   12    5    9
   2   12   20   21
   2   13   22   23
   2   14    3   11
   2   14    4   10
   2   14    6   10
   2   14    8   11
2(2-10) = 4(4-8), so (2, 10, 4, 8) is in the list.

Cf. A385182 (type 1), A386218 (type 2), A385476 (type 3, m=1), A387225, A387226.

ssolnsM[m_Integer?Positive, u_Integer?Positive] :=
  Module[{n = m  (m - u), nn, sgn, ds, tups}, If[n == 0, Return[{}]];
   sgn = Sign[n]; nn = Abs[n];
   ds = Divisors[nn];
   If[sgn > 0, ds = Select[ds, # < nn/# &]];
   tups = ({m, u, nn/#, nn/# - sgn  #} & /@ ds);
   Select[tups, #[[3]] > 1 && #[[4]] > 0 && #[[2]] =!= #[[4]] &&
   Length@DeleteDuplicates[#] == 4 &]];
(solns = Sort[Flatten[Map[solnsM[2, #] &, Range[2, 60]], 1]]) // ColumnForm
Map[#[[2]] &, solns] (*A386631*)
Map[#[[3]] &, solns] (*A387225*)
Map[#[[4]] &, solns] (*A387226*)
(* Peter J. C. Moses, Aug 22 2025 *)

A386655 E.g.f.: Sum_{n>=0} (2^n*x + LambertW(x))^n / n!.

Original entry on oeis.org

1, 3, 23, 708, 82677, 39043808, 75384175459, 594418947869568, 19030890530555146281, 2460681168464503636482816, 1280084112577610436036966382815, 2672769582069469500760580570122074560, 22366167041278673568399013569832022272725469

Offset: 0

Paul D. Hanna, Aug 23 2025

Conjecture: for n >= 6, a(n) (mod 6) equals [1, 0, 3, 0, 3, 2] repeating.
In general, the following sums are equal:
(C.1) Sum_{n>=0} (q^n + p)^n * r^n/n!,
(C.2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = 2 with p = LambertW(x)/x, r = x.

			E.g.f.: A(x) = 1 + 3*x + 23*x^2/2! + 708*x^3/3! + 82677*x^4/4! + 39043808*x^5/5! + 75384175459*x^6/6! + 594418947869568*x^7/7! + ...
where A(x) = Sum_{n>=0} (2^n*x + LambertW(x))^n / n!.
RELATED SERIES.
LambertW(x) = x - 2*x^2/2! + 3^2*x^3/3! - 4^3*x^4/4! + 5^4*x^5/5! - 6^5*x^6/6! + 7^6*x^7/7! + ... + (-1)^(n-1) * n^(n-1)*x^n/n! + ...
where exp(LambertW(x)) = x/LambertW(x);
also, (x/LambertW(x))^y = Sum_{k>=0} y*(y - k)^(k-1) * x^k/k!.

Cf. A386656 (q=3), A386657 (q=4), A386658 (q=5), A386648.

nmax = 15; Join[{1}, Rest[CoefficientList[Series[Sum[(2^k*x + LambertW[x])^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!]] (* Vaclav Kotesovec, Aug 23 2025 *)

{a(n) = sum(k=0,n, binomial(n,k) * 2^(k*(k+1)) * (2^k - (n-k))^(n-k-1) )}
for(n=0, 12, print1(a(n), ", "))

{a(n) = my(A = sum(m=0, n, (2^m + lambertw(x +x^3*O(x^n))/x)^m *x^m/m! )+x*O(x^n)); n! * polcoeff(A, n)}
for(n=0, 12, print1(a(n), ", "))

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = Sum_{n>=0} (2^n*x + LambertW(x))^n / n!.
(2) A(x) = Sum_{n>=0} 2^(n^2) * exp( LambertW(x) * 2^n ) * x^n / n!.
(3) A(x) = Sum_{n>=0} 2^(n^2) * (x/LambertW(x))^(2^n) * x^n / n!.
(4) A(x) = Sum_{n>=0} 2^(n*(n+1)) * x^n/n! * Sum_{k>=0} (2^n - k)^(k-1) * x^k/k!.
a(n) = Sum_{k=0..n} binomial(n,k) * 2^(k*(k+1)) * (2^k - (n-k))^(n-k-1).
a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n*k) * (1 - (n-k)/2^k)^(n-k-1).
a(n) ~ 2^(n^2). - Vaclav Kotesovec, Aug 23 2025

A386656 E.g.f.: Sum_{n>=0} (3^n*x + LambertW(x))^n / n!.

Original entry on oeis.org

1, 4, 98, 21901, 45203076, 864855654349, 151334120052647134, 240066304912259832915171, 3437872829353908000927273009224, 443629285010311848968435132228644809721, 515464807017361539745514781011221080738833641050

Offset: 0

Paul D. Hanna, Aug 23 2025

Conjecture: for n >= 6, a(n) (mod 6) equals [4, 3, 2, 1, 0, 1] repeating.
In general, the following sums are equal:
(C.1) Sum_{n>=0} (q^n + p)^n * r^n/n!,
(C.2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = 3 with p = LambertW(x)/x, r = x.

			E.g.f.: A(x) = 1 + 4*x + 98*x^2/2! + 21901*x^3/3! + 45203076*x^4/4! + 864855654349*x^5/5! + 151334120052647134*x^6/6! + ...
where A(x) = Sum_{n>=0} (3^n*x + LambertW(x))^n / n!.
RELATED SERIES.
LambertW(x) = x - 2*x^2/2! + 3^2*x^3/3! - 4^3*x^4/4! + 5^4*x^5/5! - 6^5*x^6/6! + 7^6*x^7/7! + ... + (-1)^(n-1) * n^(n-1)*x^n/n! + ...
where exp(LambertW(x)) = x/LambertW(x);
also, (x/LambertW(x))^y = Sum_{k>=0} y*(y - k)^(k-1) * x^k/k!.

Cf. A386655 (q=2), A386657 (q=4), A386658 (q=5), A386648.

{a(n) = sum(k=0,n, binomial(n,k) * 3^(k*(k+1)) * (3^k - (n-k))^(n-k-1) )}
for(n=0, 12, print1(a(n), ", "))

{a(n) = my(A = sum(m=0, n, (3^m + lambertw(x +x^3*O(x^n))/x)^m *x^m/m! )+x*O(x^n)); n! * polcoeff(A, n)}
for(n=0, 12, print1(a(n), ", "))

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = Sum_{n>=0} (3^n*x + LambertW(x))^n / n!.
(2) A(x) = Sum_{n>=0} 3^(n^2) * exp( LambertW(x) * 3^n ) * x^n / n!.
(3) A(x) = Sum_{n>=0} 3^(n^2) * (x/LambertW(x))^(3^n) * x^n / n!.
(4) A(x) = Sum_{n>=0} 3^(n*(n+1)) * x^n/n! * Sum_{k>=0} (3^n - k)^(k-1) * x^k/k!.
a(n) = Sum_{k=0..n} binomial(n,k) * 3^(k*(k+1)) * (3^k - (n-k))^(n-k-1).
a(n) = Sum_{k=0..n} binomial(n,k) * 3^(n*k) * (1 - (n-k)/3^k)^(n-k-1).

A386657 E.g.f.: Sum_{n>=0} (4^n*x + LambertW(x))^n / n!.

Original entry on oeis.org

1, 5, 287, 274532, 4362420261, 1131407873777920, 4729288202285254702123, 317048074495318899943286044736, 340323907513179399929311813628104334217, 5846207259092593125133941613189798019292422881280

Offset: 0

Paul D. Hanna, Aug 23 2025

Conjecture: for n >= 6, a(n) (mod 6) equals [1, 2, 3, 2, 3, 4] repeating.
In general, the following sums are equal:
(C.1) Sum_{n>=0} (q^n + p)^n * r^n/n!,
(C.2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = 4 with p = LambertW(x)/x, r = x.

			E.g.f.: A(x) = 1 + 5*x + 287*x^2/2! + 274532*x^3/3! + 4362420261*x^4/4! + 1131407873777920*x^5/5! + 4729288202285254702123*x^6/6! + ...
where A(x) = Sum_{n>=0} (4^n*x + LambertW(x))^n / n!.
RELATED SERIES.
LambertW(x) = x - 2*x^2/2! + 3^2*x^3/3! - 4^3*x^4/4! + 5^4*x^5/5! - 6^5*x^6/6! + 7^6*x^7/7! + ... + (-1)^(n-1) * n^(n-1)*x^n/n! + ...
where exp(LambertW(x)) = x/LambertW(x);
also, (x/LambertW(x))^y = Sum_{k>=0} y*(y - k)^(k-1) * x^k/k!.

Cf. A386655 (q=2), A386656 (q=3), A386658 (q=5), A386648.

{a(n) = sum(k=0,n, binomial(n,k) * 4^(k*(k+1)) * (4^k - (n-k))^(n-k-1) )}
for(n=0, 12, print1(a(n), ", "))

{a(n) = my(A = sum(m=0, n, (4^m + lambertw(x +x^3*O(x^n))/x)^m *x^m/m! )+x*O(x^n)); n! * polcoeff(A, n)}
for(n=0, 12, print1(a(n), ", "))

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = Sum_{n>=0} (4^n*x + LambertW(x))^n / n!.
(2) A(x) = Sum_{n>=0} 4^(n^2) * exp( LambertW(x) * 4^n ) * x^n / n!.
(3) A(x) = Sum_{n>=0} 4^(n^2) * (x/LambertW(x))^(4^n) * x^n / n!.
(4) A(x) = Sum_{n>=0} 4^(n*(n+1)) * x^n/n! * Sum_{k>=0} (4^n - k)^(k-1) * x^k/k!.
a(n) = Sum_{k=0..n} binomial(n,k) * 4^(k*(k+1)) * (4^k - (n-k))^(n-k-1).
a(n) = Sum_{k=0..n} binomial(n,k) * 4^(n*k) * (1 - (n-k)/4^k)^(n-k-1).

A386658 E.g.f.: Sum_{n>=0} (5^n*x + LambertW(x))^n / n!.

Original entry on oeis.org

1, 6, 674, 2000229, 153566609748, 298500361403750381, 14557504055095871311168750, 17765160070810827062009088144577731, 542112188572462226990932242595876785196798632, 413592212104548192173492724488185195719396124921931347641

Offset: 0

Paul D. Hanna, Aug 23 2025

Conjecture: for n >= 6, a(n) (mod 6) equals [4, 3, 0, 3, 0, 5] repeating.
In general, the following sums are equal:
(C.1) Sum_{n>=0} (q^n + p)^n * r^n/n!,
(C.2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
here, q = 5 with p = LambertW(x)/x, r = x.

			E.g.f.: A(x) = 1 + 6*x + 674*x^2/2! + 2000229*x^3/3! + 153566609748*x^4/4! + 298500361403750381*x^5/5! + 14557504055095871311168750*x^6/6! + ...
where A(x) = Sum_{n>=0} (5^n*x + LambertW(x))^n / n!.
RELATED SERIES.
LambertW(x) = x - 2*x^2/2! + 3^2*x^3/3! - 4^3*x^4/4! + 5^4*x^5/5! - 6^5*x^6/6! + 7^6*x^7/7! + ... + (-1)^(n-1) * n^(n-1)*x^n/n! + ...
where exp(LambertW(x)) = x/LambertW(x);
also, (x/LambertW(x))^y = Sum_{k>=0} y*(y - k)^(k-1) * x^k/k!.

Cf. A386655 (q=2), A386656 (q=3), A386657 (q=4), A386648.

{a(n,q=5) = sum(k=0,n, binomial(n,k) * q^(k*(k+1)) * (q^k - (n-k))^(n-k-1) )}
for(n=0, 12, print1(a(n), ", "))

{a(n,q=5) = my(A = sum(m=0, n, (q^m + lambertw(x +x^3*O(x^n))/x)^m *x^m/m! )+x*O(x^n)); n! * polcoeff(A, n)}
for(n=0, 12, print1(a(n), ", "))

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = Sum_{n>=0} (5^n*x + LambertW(x))^n / n!.
(2) A(x) = Sum_{n>=0} 5^(n^2) * exp( LambertW(x) * 5^n ) * x^n / n!.
(3) A(x) = Sum_{n>=0} 5^(n^2) * (x/LambertW(x))^(5^n) * x^n / n!.
(4) A(x) = Sum_{n>=0} 5^(n*(n+1)) * x^n/n! * Sum_{k>=0} (5^n - k)^(k-1) * x^k/k!.
a(n) = Sum_{k=0..n} binomial(n,k) * 5^(k*(k+1)) * (5^k - (n-k))^(n-k-1).
a(n) = Sum_{k=0..n} binomial(n,k) * 5^(n*k) * (1 - (n-k)/5^k)^(n-k-1).

A386901 Integers y such that there exist two integers 0

Original entry on oeis.org

80850, 158340, 161070, 161700, 232050, 242550, 316680, 322140, 323400, 404250, 464100, 474810, 475020, 483210, 485100, 485940, 565950, 633360, 641550, 644280, 646800, 662340, 696150, 727650, 791700, 805350, 808500, 963270, 966420, 967890, 970200, 971880

Offset: 1

S. I. Dimitrov, Aug 07 2025

The numbers x, y and z form a psi-amicable triple.

			158340 is in the sequence since psi(150150) = psi(158340) = psi(175350) = 483840 = 150150 + 158340 + 175350. Other examples: (232050, 232050, 261660), (7091700, 7098630, 7098630).

S. I. Dimitrov, On psi-amicable numbers and their generalizations, arXiv:2508.02318 [math.NT], 2025.

Cf. A001615, A125491, A323329, A385852, A386933.

A386933 Integers z such that there exist two integers 0

Original entry on oeis.org

81900, 161700, 163800, 175350, 245700, 261660, 323400, 327600, 350700, 409500, 485100, 490770, 491400, 499380, 523320, 526050, 526260, 573300, 646800, 647010, 655200, 671370, 701400, 702450, 737100, 784980, 808500, 819000, 876750, 970200, 971880, 981540, 982800, 990150, 998760

Offset: 1

S. I. Dimitrov, Aug 09 2025

The numbers x, y and z form a psi-amicable triple.

			163800 is in the sequence since psi(158340) = psi(161700) = psi(163800) = 564480 = 158340 + 161700 + 163800. Other examples: (322140, 322140, 323400), (14127960, 14224980, 14224980).

S. I. Dimitrov, On psi-amicable numbers and their generalizations, arXiv:2508.02318 [math.NT], 2025.

Cf. A001615, A125492, A323330, A385852, A386901.

A386943 Ordered hypotenuses of nonprimitive Pythagorean triples of the form (u^2 - v^2, 2uv, u^2 + v^2), where u and v are positive integers.

Original entry on oeis.org

10, 20, 26, 34, 40, 45, 50, 52, 58, 68, 74, 80, 82, 90, 100, 104, 106, 116, 117, 122, 125, 130, 130, 136, 146, 148, 153, 160, 164, 170, 170, 178, 180, 194, 200, 202, 208, 212, 218, 225, 226, 232, 234, 244, 245, 250, 250, 260, 260, 261, 272, 274, 290, 290, 292, 296

Offset: 1

Felix Huber, Aug 24 2025

In the form (u^2 - v^2, 2*u*v, u^2 + v^2), u^2 + v^2 is the hypotenuse, max(u^2 - v^2, 2*u*v) is the long leg and min(u^2 - v^2, 2*u*v) is the short leg.
A101930(n) gives the total number of Pythagorean triples <= 10^n.
               number of terms <= h     total number of
       h       in this sequence         hypotenuses <= h      percentage
      10                 1                    2                 50.0 %
     100                15                   52                 28.8 %
    1000               209                  881                 23.7 %
   10000              2249                12471                 18.0 %
  100000             23086               161436                 14.3 %

			The nonprimitive Pythagorean triple (6, 8, 10) is of the form (u^2 - v^2, 2*u*v, u^2 + v^2): From u = 3 and v = 1 follows u^2 - v^2 = 8 (long leg), 2*u*v = 6 (short leg), u^2 - v^2 = 10 (hypotenuse). Therefore, 10 is a term.

Felix Huber, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple

Subsequence of A009000.

Cf. A020882, A101930, A366428, A380072, A386307, A386944, A386945.

A386943:=proc(N) # To get all hypotenuses <= N
    local i,l,u,v;
    l:=[];
    for u from 2 to floor(sqrt(N-1)) do
        for v to min(u-1,floor(sqrt(N-u^2))) do
            if gcd(u,v)>1 or is(u-v,even) then
                l:=[op(l),[u^2+v^2,max(2*u*v,u^2-v^2),min(2*u*v,u^2-v^2)]]
            fi
        od
    od;
    l:=sort(l);
    return seq(l[i,1],i=1..nops(l));
end proc;
A386943(296);

a(n) = sqrt(A386944(n)^2 + A386945(n)^2).

{A009000(n)} = {a(n)} union {A020882(n)} union {A386307(n)}.

A386944 Long legs of Pythagorean triples of the form (u^2 - v^2, 2uv, u^2 + v^2), ordered by increasing hypotenuse (A386943).

Original entry on oeis.org

8, 16, 24, 30, 32, 36, 48, 48, 42, 60, 70, 64, 80, 72, 96, 96, 90, 84, 108, 120, 100, 112, 126, 120, 110, 140, 135, 128, 160, 154, 168, 160, 144, 144, 192, 198, 192, 180, 182, 216, 224, 168, 216, 240, 196, 200, 234, 224, 252, 189, 240, 210, 286, 288, 220, 280, 280

Offset: 1

Felix Huber, Aug 24 2025

In the form (u^2 - v^2, 2*u*v, u^2 + v^2), u^2 + v^2 is the hypotenuse, max(u^2 - v^2, 2*u*v) is the long leg and min(u^2 - v^2, 2*u*v) is the short leg.

			The nonprimitive Pythagorean triple (6, 8, 10) is of the form (u^2 - v^2, 2*u*v, u^2 + v^2): From u = 3 and v = 1 follows u^2 - v^2 = 8 (long leg), 2*u*v = 6 (short leg), u^2 - v^2 = 10 (hypotenuse). Therefore, 8 is a term.

Felix Huber, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple

Subsequence of A046084.

Cf. A046087, A366675, A380073, A386308, A386943, A386945.

A386944:=proc(N) # To get all hypotenuses <= N
    local i,l,u,v;
    l:=[];
    for u from 2 to floor(sqrt(N-1)) do
        for v to min(u-1,floor(sqrt(N-u^2))) do
            if gcd(u,v)>1 or is(u-v,even) then
                l:=[op(l),[u^2+v^2,max(2*u*v,u^2-v^2),min(2*u*v,u^2-v^2)]]
            fi
        od
    od;
    l:=sort(l);
    return seq(l[i,2],i=1..nops(l));
end proc;
A386944(296);

a(n) = sqrt(A386943(n)^2 - A386945(n)^2).

{A046084(n)} = {a(n)} union {A046087(n)} union {A386308(n)}.

cp's OEIS Frontend

A386488 Record high points in A386487.

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A386631 Values of u in the quartets (2, u, v, w) of type 3; i.e., values of u for solutions to 2(2 - u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

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A386655 E.g.f.: Sum_{n>=0} (2^n*x + LambertW(x))^n / n!.

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A386656 E.g.f.: Sum_{n>=0} (3^n*x + LambertW(x))^n / n!.

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PARI

PARI

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A386657 E.g.f.: Sum_{n>=0} (4^n*x + LambertW(x))^n / n!.

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PARI

PARI

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A386658 E.g.f.: Sum_{n>=0} (5^n*x + LambertW(x))^n / n!.

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PARI

PARI

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A386901 Integers y such that there exist two integers 0

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A386933 Integers z such that there exist two integers 0

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A386943 Ordered hypotenuses of nonprimitive Pythagorean triples of the form (u^2 - v^2, 2uv, u^2 + v^2), where u and v are positive integers.

A386944 Long legs of Pythagorean triples of the form (u^2 - v^2, 2uv, u^2 + v^2), ordered by increasing hypotenuse (A386943).