César Eliud Lozada - cp's OEIS frontend

A371654 Numbers that are not multiples of 10 and that yield a prime if their digits are arranged in ascending order.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 32, 37, 47, 59, 67, 71, 73, 74, 76, 79, 89, 91, 92, 95, 97, 98, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 157, 167, 172, 173, 175, 176, 179, 193, 194, 197, 199, 203, 209, 217, 223, 227, 229, 232, 233, 239, 257

Offset: 1

César Eliud Lozada, Apr 01 2024

If N is a term then all numbers with the same digits as N are terms too.

			91 is a term because arranging its digits in ascending order yields 19, which is a prime.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A007500, A095179, A371653.

Select[Range[500],  Mod[#, 10] != 0 && PrimeQ[FromDigits[Sort[IntegerDigits[#]]]] &]

from sympy import isprime
def ok(n): return n%10 and isprime(int("".join(sorted(str(n)))))
print([k for k in range(260) if ok(k)]) # Michael S. Branicky, Apr 01 2024

from itertools import count, islice, combinations_with_replacement
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A371654_gen(): # generator of terms
    for l in count(1):
        xlist = []
        for p in combinations_with_replacement('0123456789',l):
            if isprime(int(''.join(p))):
                xlist.extend(int(''.join(d)) for d in multiset_permutations(p) if d[0] != '0' and d[-1] != '0')
        yield from sorted(xlist)
A371654_list = list(islice(A371654_gen(),20)) # Chai Wah Wu, Apr 10 2024

A371653 Numbers k such that the number formed by putting the digits of k in descending order is prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 14, 16, 17, 31, 34, 35, 37, 38, 41, 43, 53, 61, 71, 73, 79, 83, 97, 112, 113, 118, 119, 121, 124, 125, 128, 131, 133, 134, 136, 142, 143, 145, 146, 149, 152, 154, 157, 163, 164, 166, 167, 175, 176, 179, 181, 182, 188, 191, 194, 197, 199

Offset: 1

César Eliud Lozada, Apr 01 2024

Numbers k such that A004186(k) is prime. - Robert Israel, Apr 01 2024

If N is a term then all numbers with the same digits as N are terms too.

			142 is a term because its digits in decreasing order form 421 and this is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A004186, A007500, A095179, A371654.

dd:= proc(n) local L,i;
   L:= sort(convert(n,base,10));
   add(L[i]*10^(i-1),i=1..nops(L))
end proc:
select(isprime @ dd, [$1..1000]); # Robert Israel, Apr 01 2024

Select[Range[500], PrimeQ[FromDigits[ReverseSort[IntegerDigits[#]]]] &]

from sympy import isprime
def ok(n): return isprime(int("".join(sorted(str(n), reverse=True))))
print([k for k in range(200) if ok(k)]) # Michael S. Branicky, Apr 01 2024

from itertools import count, islice, combinations_with_replacement
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A371653_gen(): # generator of terms
    for l in count(1):
        xlist = []
        for p in combinations_with_replacement('987654321',l):
            if isprime(int(''.join(p))):
                xlist.extend(int(''.join(d)) for d in multiset_permutations(p))
        yield from sorted(xlist)
A371653_list = list(islice(A371653_gen(),30)) # Chai Wah Wu, Apr 10 2024

A371031 Number of distinct integers resulting from adding an n-digit non-multiple of 10 and its reverse.

Original entry on oeis.org

9, 17, 170, 323, 3230, 6137, 61370, 116603, 1166030, 2215457

Offset: 1

César Eliud Lozada, Mar 08 2024

			For n=2 there are 81 2-digit numbers not ending with 0: {11, 12, 13, ..., 99}. There are 17 distinct results when adding each of these to their reversal: {22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 165, 176, 187, 198}. Therefore a(2) = 17.

Cf. A056964, A067251, A358985.

A371031[n_] :=
Module[{nn, ss},
  nn = Select[Range[If[n == 1, 1, 10^(n - 1) + 1], 10^n - 1], Mod[#, 10] > 0 &];
  ss = Map[# + FromDigits[Reverse[IntegerDigits[#]]] &, nn];
  Return[CountDistinct[ss]]
];
Map[A371031[#]&, Range[7]]

For n > 1, empirically a(n+1) = 10 a(n) if n even, 19 a(n) / 10 if n odd, and thus a(n+2) = 19 a(n). - Michael S. Branicky, Mar 31 2024

a(9)-a(10) from Michael S. Branicky, Mar 30 2024

A348410 Number of nonnegative integer solutions to n = Sum_{i=1..n} (a_i + b_i), with b_i even.

Original entry on oeis.org

1, 1, 5, 19, 85, 376, 1715, 7890, 36693, 171820, 809380, 3830619, 18201235, 86770516, 414836210, 1988138644, 9548771157, 45948159420, 221470766204, 1069091485500, 5167705849460, 25009724705460, 121171296320475, 587662804774890, 2852708925078675, 13859743127937876

Offset: 0

César Eliud Lozada, Oct 17 2021

Suppose n objects are to be distributed into 2n baskets, half of these white and half black. White baskets may contain 0 or any number of objects, while black baskets may contain 0 or an even number of objects. a(n) is the number of distinct possible distributions.

			Some examples (semicolon separates white basket from black baskets):
For n=1: {{1 ; 0}} - Total possible ways: 1.
For n=2: {{0, 0 ; 0, 2}, {0, 0 ; 2, 0}, {0, 2 ; 0, 0}, {1, 1 ; 0, 0}, {2, 0 ; 0, 0}} - Total possible ways: 5.

Alois P. Heinz, Table of n, a(n) for n = 0..1441

Cf. A033715, A033716, A034297, A063020, A088218, A234839, A294860, A348474, A351856, A351857, A352373.

b:= proc(n, t) option remember; `if`(t=0, 1-signum(n),
      add(b(n-j, t-1)*(1+iquo(j, 2)), j=0..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 17 2021

(* giveList=True produces the list of solutions *)
(* giveList=False gives the number of solutions *)
counter[objects_, giveList_: False] :=
  Module[{n = objects, nb, eq1, eqa, eqb, eqs, var, sol, var2, list},
   nb = n;
   eq1 = {Total[Map[a[#] + 2*b[#] &, Range[nb]]] - n == 0};
   eqa = {And @@ Map[0 <= a[#] <= n &, Range[nb]]};
   eqb = {And @@ Map[0 <= b[#] <= n &, Range[nb]]};
   eqs = {And @@ Join[eq1, eqa, eqb]};
   var = Flatten[Map[{a[#], b[#]} &, Range[nb]]];
   var = Join[Map[a[#] &, Range[nb]], Map[b[#] &, Range[nb]]];
   sol = Solve[eqs, var, Integers];
   var2 = Join[Map[a[#] &, Range[nb]], Map[2*b[#] &, Range[nb]]];
   list = Sort[Map[var2 /. # &, sol]];
   list = Map[StringReplace[ToString[#], {"," -> " ;"}, n] &, list];
   list = Map[StringReplace[#, {";" -> ","}, n - 1] &, list];
   Return[
    If[giveList, Print["Total: ", Length[list]]; list, Length[sol]]];
   ];
(* second program: *)
b[n_, t_] := b[n, t] = If[t == 0, 1 - Sign[n], Sum[b[n - j, t - 1]*(1 + Quotient[j, 2]), {j, 0, n}]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 16 2023, after Alois P. Heinz *)

Conjecture: D-finite with recurrence +7168*n*(2*n-1)*(n-1)*a(n) -64*(n-1)*(1759*n^2-5294*n+5112)*a(n-1) +12*(7561*n^3-75690*n^2+165271*n-101070)*a(n-2) +5*(110593*n^3-743946*n^2+1659971*n-1232778)*a(n-3) +2680*(4*n-15)*(2*n-7)*(4*n-13)*a(n-4)=0. - R. J. Mathar, Oct 19 2021
From Vaclav Kotesovec, Nov 01 2021: (Start)
Recurrence (of order 2): 16*(n-1)*n*(2*n - 1)*(51*n^2 - 162*n + 127)*a(n) = (n-1)*(5457*n^4 - 22791*n^3 + 32144*n^2 - 17536*n + 3072)*a(n-1) + 8*(2*n - 3)*(4*n - 7)*(4*n - 5)*(51*n^2 - 60*n + 16)*a(n-2).
a(n) ~ sqrt(3 + 5/sqrt(17)) * (107 + 51*sqrt(17))^n / (sqrt(Pi*n) * 2^(6*n+2)). (End)
From Peter Bala, Feb 21 2022: (Start)
a(n) = [x^n] ( (1 - x)*(1 - x^2) )^(-n). Cf. A234839.
a(n) = Sum_{k = 0..floor(n/2)} binomial(2*n-2*k-1,n-2*k)*binomial(n+k-1,k).
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 3*x^2 + 9*x^3 + 32*x^4 + 119*x^5 + ... is the g.f. of A063020.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k.
Conjecture: the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and positive integers n and k.
The o.g.f. A(x) is the diagonal of the bivariate rational function 1/(1 - t/((1-x)*(1-x^2))) and hence is an algebraic function over Q(x) by Stanley 1999, Theorem 6.33, p. 197.
Let F(x) = (1/x)*Series_Reversion( x*(1-x)*(1-x^2) ). Then A(x) = 1 + x*d/dx (log(F(x))). (End)
a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(2*n+k-1, k)*binomial(2*n-k-1, n-k). Cf. A352373. - Peter Bala, Jun 05 2024

More terms from Alois P. Heinz, Oct 17 2021

A336995 Numbers of the form x^3 + x^2y + xy^2 + y^3, where x and y are coprime positive integers.

Original entry on oeis.org

4, 15, 40, 65, 85, 156, 175, 203, 259, 272, 369, 400, 477, 580, 585, 671, 715, 803, 820, 888, 935, 1105, 1111, 1157, 1261, 1417, 1464, 1484, 1625, 1695, 1820, 1885, 2055, 2080, 2336, 2380, 2465, 2533, 2595, 2669, 2848, 2873, 2955, 3060, 3145, 3439, 3485, 3492

Offset: 1

César Eliud Lozada, Aug 10 2020

Equivalently, numbers of the form (x+y)*(x^2 + y^2) where x and y are coprime positive integers.

			For x=1, y=1, x^3+x^2*y+x*y^2+y^3 = 4, so 4 is in the sequence.
For x=1, y=2, x^3+x^2*y+x*y^2+y^3 = 15, so 15 is in the sequence.
For x=2, y=2, x^3+x^2*y+x*y^2+y^3 = 32, but GCD(x,y)<>1, so 32 is not in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Subsequence of A336607.

N:= 10000: # for terms <= N
S:= {}:
for x from 1 while (x+1)*(x^2+1) < N do
   V:= select(`<=`,map(y -> (x+y)*(x^2+y^2), select(y -> igcd(x,y)=1, {seq(i,i=1..min(x,(N-x^3)/x^2))})),N);
   S:= S union V;
od:
sort(convert(S,list)); # Robert Israel, Sep 21 2020

max = 5000; T0 = {}; xm = Ceiling[Sqrt[max]];
While[T = T0;
T0 = Table[x^3 + x^2 y + x y^2 + y^3, {x, 1, xm}, {y, x, xm}] //
     Flatten // Union // Select[#, # <= max &] &; T != T0, xm = 2 xm];
T (* T=A336607 *)
(* Now, exclude a(n) such that a(n)=k^3*a(m) for m=2 is an integer *)
T2 = T; n = 1;
While[n <= Length[T2],
  t1 = T2[[n]]; t2 = Last[T2]; max2 = 1 + (t2/t1)^(1/3);
  T2 = Complement[T2, Table[t1*k^3, {k, 2, max2}]];
  n++;
  ];
T2 (* T2=A336995 *)

upto(limit)={my(L=List(), b=sqrtnint(limit,3)); for(x=1, b, for(y=1, b, my(t=(x+y)*(x^2+y^2)); if(t<=limit && gcd(x,y)==1, listput(L,t)) )); Set(L)}
upto(4000) \\ Andrew Howroyd, Aug 10 2020

A336607 Numbers of the form x^3 + x^2y + xy^2 + y^3, where x and y are positive integers.

Original entry on oeis.org

4, 15, 32, 40, 65, 85, 108, 120, 156, 175, 203, 256, 259, 272, 320, 369, 400, 405, 477, 500, 520, 580, 585, 671, 680, 715, 803, 820, 864, 888, 935, 960, 1080, 1105, 1111, 1157, 1248, 1261, 1372, 1400, 1417, 1464, 1484, 1624, 1625, 1695, 1755, 1820, 1875, 1885

Offset: 1

César Eliud Lozada, Jul 27 2020

Numbers of the form (x+y)(x^2+y^2), where x and y are positive integers. - Chai Wah Wu, Aug 08 2020

No terms == 2 (mod 4). - Robert Israel, Sep 21 2020

			4=1^3+1^2*1+1*1^2+1^3, 15=1^3+1^2*2+1*2^2+2^3, 32=2^3+2^2*2+2*2^2+2^3, ...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A024614.

N:= 10000: # for terms <= N
S:= {}:
for x from 1 while (x+1)*(x^2+1) < N do
   V:= select(`<=`,map(y -> (x+y)*(x^2+y^2), {seq(i,i=1..min(x,(N-x^3)/x^2))}),N);
   S:= S union V;
od:
sort(convert(S,list)); # Robert Israel, Sep 21 2020

max = 5000; T0 = {}; xm = Ceiling[Sqrt[max]]; While[T = T0;
T0 = Table[x^3 + x^2 y + x y^2 + y^3, {x, 1, xm}, {y, x, xm}] //
     Flatten // Union // Select[#, # <= max &] &; T != T0, xm = 2 xm];
T

A331428 Divide each side of a triangle into 2*n-1 (n>=1) equal parts and trace the corresponding cevians, i.e., join every point, except for the first and last ones, with the opposite vertex. a(n) is the number of points at which three cevians meet.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 42, 0, 0, 0, 0, 12, 0, 0, 0, 48, 6, 0, 0, 0, 0, 6, 12, 0, 0, 0, 6, 0, 0, 12, 0, 0, 0, 24, 0, 0, 90, 0, 0, 0, 0, 0, 12, 12, 0, 0, 0, 0, 0, 0, 0, 66, 0, 0, 0, 24, 0, 0, 0, 0, 12, 0, 0, 0, 12, 0

Offset: 1

César Eliud Lozada, Jan 16 2020

nonn

A bisection of A331423.

Cf. A331423, A331425.

CevIntersections[n_] := Length[Solve[(n - i)*(n - j)*(n - k) - i*j*k == 0 && 0 < i < n &&  0 < j < n && 0 < k < n, {i, j, k}, Integers]];
Map[CevIntersections[#] &, Range[1,51,2]]

a(n) = A331423(2*n-1).

A331425 Divide each side of a triangle into 2*n (n>=1) equal parts and trace the corresponding cevians, i.e., join every point, except for the first and last ones, with the opposite vertex. a(n) is the number of points at which three cevians meet.

Original entry on oeis.org

1, 7, 13, 19, 25, 31, 37, 43, 49, 61, 61, 91, 73, 79, 91, 91, 97, 103, 109, 133, 133, 127, 133, 187, 145, 151, 157, 175, 169, 235, 181, 187, 205, 199, 229, 283, 217, 223, 235, 325, 241, 283, 253, 271, 331, 271, 277, 343, 289, 301, 301, 319, 313, 319, 349, 439

Offset: 1

César Eliud Lozada, Jan 16 2020

nonn

A bisection of A331423.

Cf. A331423, A331428.

CevIntersections[n_] := Length[Solve[(n - i)*(n - j)*(n - k) - i*j*k == 0 && 0 < i < n &&  0 < j < n && 0 < k < n, {i, j, k}, Integers]];
Map[CevIntersections[#] &, Range[2,50,2]]

a(n) = A331423(2*n).

A331423 Divide each side of a triangle into n>=1 equal parts and trace the corresponding cevians, i.e., join every point, except for the first and last ones, with the opposite vertex. a(n) is the number of points at which three cevians meet.

Original entry on oeis.org

0, 1, 0, 7, 0, 13, 0, 19, 0, 25, 0, 31, 0, 37, 6, 43, 0, 49, 0, 61, 0, 61, 0, 91, 0, 73, 0, 79, 0, 91, 0, 91, 0, 97, 12, 103, 0, 109, 0, 133, 0, 133, 0, 127, 42, 133, 0, 187, 0, 145, 0, 151, 0, 157, 12, 175, 0, 169, 0, 235, 0, 181, 48, 187, 6, 205, 0, 199, 0

Offset: 1

César Eliud Lozada, Jan 16 2020

Denote the cevians by a0, a1,...,an, b0, b1,...,bn, c0, c1,...,cn. For any given n, the indices (i,j,k) of (ai, bj, ck) meeting at a point are the integer solutions of:
n^3 - (i + j + k)*n^2 + (j*k + k*i + i*j)*n - 2*i*j*k = 0,  with 0 < i, j, k < n
or, equivalently and shorter,
(n-i)*(n-j)*(n-k) - i*j*k = 0, with 0 < i, j, k < n.
From N. J. A. Sloane, Feb 14 2020: (Start)
Stated another way, a(n) = number of triples (i,j,k) in [1,n-1] X [1,n-1] X [1,n-1] such that (i/(n-i))*(j/(n-j))*(k/(n-k)) = 1.
This is the quantity N3 mentioned in A091908.
Indices of zeros are precisely all odd numbers except those listed in A332378.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..1045 (first 200 terms from Robert Israel)
Peter Kagey, An illustration of A331423(4) = 7.
Hugo Pfoertner, Visualization of diagonal intersections in an equilateral triangle.
Jim Propp and Adam Propp-Gubin, Counting Triangles in Triangles, arXiv:2409.17117 [math.CO], 2024. See p. 9.

Cf. A091908, A332378. Bisections are A331425, A331428.

Ceva:= proc(n) local a, i, j, k; a:=0;
for i from 1 to n-1 do
for j from 1 to n-1 do
for k from 1 to n-1 do
if i*j*k/((n-i)*(n-j)*(n-k)) = 1 then a:=a+1; fi;
od: od: od: a; end;
t1:=[seq(Ceva(n), n=1..80)];  # N. J. A. Sloane, Feb 14 2020

CevIntersections[n_] := Length[Solve[(n - i)*(n - j)*(n - k) - i*j*k == 0 && 0 < i < n &&  0 < j < n && 0 < k < n, {i, j, k}, Integers]];
Map[CevIntersections[#] &, Range[50]]

A331423(n) = sum(i=1, n-1, sum(j=1, n-1, sum(k=1, n-1, (1==(i*j*k)/((n-i)*(n-j)*(n-k)))))); \\ (After the Maple program) - Antti Karttunen, Dec 12 2021

A307417 Numbers that can be expressed in a base in such a way that the sum of cubes of their digits in this base equals the original number.

Original entry on oeis.org

8, 9, 16, 17, 27, 28, 29, 35, 43, 54, 55, 62, 64, 65, 72, 91, 92, 99, 118, 125, 126, 127, 128, 133, 134, 152, 153, 189, 190, 216, 217, 224, 243, 244, 250, 251, 280, 307, 341, 342, 343, 344, 351, 370, 371, 407, 432, 433, 468, 469, 512, 513, 514, 520, 539, 559

Offset: 1

César Eliud Lozada, Apr 07 2019

There are infinitely many such numbers (proof in the second Johnson link).

			a(1) = 8 = [2, 0] (base 4) =  2^3 + 0^3
a(2) = 9 = [2, 1] (base 4) =  2^3 + 1^3
a(3) = 16 = [2, 2] (base 7) =  2^3 + 2^3
a(4) = 17 = [1, 2, 2] (base 3) =  1^3 + 2^3 + 2^3

César Eliud Lozada, Table of n, a(n) for n = 1..114
Allan Wm. Johnson Jr., Crux Mathematicorum, Vol. 5, No. 1, January 1979, problem 407, 16.
Allan Wm. Johnson Jr., Crux Mathematicorum, Vol. 5, No. 9, November 1979, solution to problem 407, 273-277.

Cf. A005188, A023052, A046197.

sqn:= []; lis:=[];
for n to 1000 do
  b := 2;
  while b < n do #needs to be adjusted
    q := convert(n, base, b);
    s := convert(map(proc (X) options operator, arrow; X^3 end proc, q), `+`);
    if evalb(s = n) then
      sqn := [op(sqn), n];
      lis := [op(lis), [n, b, ListTools[Reverse](q)]];
      break
    end if;
    b := b+1
  end do
end do;
lis := lis; #list of decompositions [number, base, conversion]
sqn := sqn; #sequence

cp's OEIS Frontend

User: César Eliud Lozada

A371654 Numbers that are not multiples of 10 and that yield a prime if their digits are arranged in ascending order.

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A371653 Numbers k such that the number formed by putting the digits of k in descending order is prime.

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A371031 Number of distinct integers resulting from adding an n-digit non-multiple of 10 and its reverse.

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A348410 Number of nonnegative integer solutions to n = Sum_{i=1..n} (a_i + b_i), with b_i even.

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A336995 Numbers of the form x^3 + x^2*y + x*y^2 + y^3, where x and y are coprime positive integers.

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A336607 Numbers of the form x^3 + x^2*y + x*y^2 + y^3, where x and y are positive integers.

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A331428 Divide each side of a triangle into 2*n-1 (n>=1) equal parts and trace the corresponding cevians, i.e., join every point, except for the first and last ones, with the opposite vertex. a(n) is the number of points at which three cevians meet.

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A336995 Numbers of the form x^3 + x^2y + xy^2 + y^3, where x and y are coprime positive integers.

A336607 Numbers of the form x^3 + x^2y + xy^2 + y^3, where x and y are positive integers.