cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A008837 a(n) = p*(p-1)/2 for p = prime(n).

Original entry on oeis.org

1, 3, 10, 21, 55, 78, 136, 171, 253, 406, 465, 666, 820, 903, 1081, 1378, 1711, 1830, 2211, 2485, 2628, 3081, 3403, 3916, 4656, 5050, 5253, 5671, 5886, 6328, 8001, 8515, 9316, 9591, 11026, 11325, 12246, 13203, 13861, 14878, 15931, 16290, 18145, 18528, 19306
Offset: 1

Views

Author

N. J. A. Sloane, Ken Levasseur

Keywords

Comments

Whereas A034953 is the sequence of triangular numbers with prime indices, this is the sequence of triangular numbers with numbers one less than primes for indices. - Alonso del Arte, Aug 17 2014
From Jianing Song, Apr 13 2019: (Start)
a(n) is both the number of quadratic residues and the number of nonresidues modulo prime(n)^2 that are coprime to prime(n).
For k coprime to prime(n), k^a(n) == +-1 (mod prime(n)^2). (End)

Links

Crossrefs

Half the terms of A036689.
Cf. A000217 (triangular numbers), A112456 (least triangular number divisible by n-th prime). - Klaus Brockhaus, Nov 18 2008
Cf. A000010, A000040, A072230, A034953, A271780.
Column 1 of A257253. (Row 1 of A257254).

Programs

Formula

a(n) = binomial(prime(n), 2) = A000217(A000040(n)-1). - Enrique Pérez Herrero, Dec 10 2011
a(n) = (1/2)*A072230(A000040(n)). - L. Edson Jeffery, Apr 07 2012
a(n) = (phi(prime(n))^2 + phi(prime(n)))/2, where phi(n) is Euler's totient function, A000010. - Alonso del Arte, Aug 22 2014
a(n) = A036689(n)/2. - Antti Karttunen, May 01 2015
Product_{n>=2} (1 - 1/a(n)) = A271780. - Amiram Eldar, Nov 22 2022

Extensions

Offset changed from 2 to 1 by Harry J. Smith, Jul 25 2009

A225503 Least triangular number t such that t = prime(n)*triangular(m) for some m>0, or 0 if no such t exists.

Original entry on oeis.org

6, 3, 15, 21, 66, 78, 1326, 190, 1035, 435, 465, 17205, 861, 903, 9870, 5565, 1567335, 16836, 20100, 2556, 2628, 49770, 55278, 4005, 42195, 413595, 47895, 10100265, 5995, 1437360, 32131, 8646, 1352190, 19559385, 54397665, 1642578, 12246, 52975, 501501, 134940, 336324807802305
Offset: 1

Views

Author

Alex Ratushnyak, May 09 2013

Keywords

Comments

Conjecture: a(n) > 0.
a(n) = (x^2-1)/8 where x is the least odd solution > 1 of the Pell-like equation x^2 - prime(n)*y^2 = 1 - prime(n). - Robert Israel, Jan 08 2015

Examples

			See A225502.

Links

Crossrefs

Cf. A000217, A112456, A225502.

Programs

  • C
    #include 
#define TOP 300
typedef unsigned long long U64;
U64 isTriangular(U64 a) {
    U64 sr = 1ULL<<32, s, b, t;
    if (a < (sr/2)*(sr+1))  sr>>=1;
    while (a < sr*(sr+1)/2)  sr>>=1;
    for (b = sr>>1; b; b>>=1) {
        s = sr+b;
        if (s&1) t = s*((s+1)/2);
        else     t = (s/2)*(s+1);
        if (t >= s && a >= t)  sr = s;
    }
    return (sr*(sr+1)/2 == a);
}
int main() {
  U64 i, j, k, m, tm, p, pp = 1, primes[TOP];
  for (primes[0]=2, i = 3; pp < TOP; i+=2) {
    for (p = 1; p < pp; ++p) if (i%primes[p]==0) break;
    if (p==pp) {
        primes[pp++] = i;
        for (j=p=primes[pp-2], m=tm=1; ; j=k, m++, tm+=m) {
           if ((k = p*tm) < j) k=0;
           if (isTriangular(k)) break;
        }
        printf("%llu, ", k);
    }
  }
  return 0;
}
  • Maple
    F:= proc(n) local p, S,x,y, z, cands, s;
      p:= ithprime(n);
      S:= {isolve(x^2 - p*y^2 = 1-p)};
      for z from 0 do
        cands:= select(s -> (subs(s,x) > 1 and subs(s,x)::odd), simplify(eval(S,_Z1=z)));
        if cands <> {} then
           x:= min(map(subs,cands, x));
           return((x^2-1)/8)
        fi
      od;
end proc:
map(F, [$1..100]); # Robert Israel, Jan 08 2015
  • Mathematica
    a[n_] := Module[{p, x0, sol, x, y}, p = Prime[n]; x0 = Which[n == 1, 7, n == 2, 5, True, sol = Table[Solve[x > 1 && y > 1 && x^2 - p y^2 == 1 - p, {x, y}, Integers] /. C[1] -> c, {c, 0, 1}] // Simplify; Select[x /. Flatten[sol, 1], OddQ] // Min]; (x0^2 - 1)/8];
Array[a, 171] (* Jean-François Alcover, Apr 02 2019, after Robert Israel *)
  • PARI
    a(n) = {p = prime(n); k = 1; while (! ((t=k*(k+1)/2) && ((t % p) == 0) && ispolygonal(t/p, 3)), k++); t;} \\ Michel Marcus, Jan 08 2015

Extensions

a(171) from Giovanni Resta, Jun 19 2013

A225502 Least m > 0 such that prime(n)*triangular(m) is a triangular number, or 0 if no such m exists.

Original entry on oeis.org

2, 1, 2, 2, 3, 3, 12, 4, 9, 5, 5, 30, 6, 6, 20, 14, 230, 23, 24, 8, 8, 35, 36, 9, 29, 90, 30, 434, 10, 159, 22, 11, 140, 530, 854, 147, 12, 25, 77, 39, 1938509, 13, 41, 69, 182, 70, 14, 104, 105, 60, 30, 15, 15, 47, 240, 65274, 6314, 16, 17009, 33, 50, 68, 17, 264, 371
Offset: 1

Views

Author

Alex Ratushnyak, May 09 2013

Keywords

Comments

Conjecture: a(n) > 0.

Examples

			n    prime(n)    m     tri(m)   prime(n)*tri(m)
1      2         2       3              6
2      3         1       1              3
3      5         2       3             15
4      7         2       3             21
5     11         3       6             66
6     13         3       6             78
7     17        12      78           1326
8     19         4      10            190

Crossrefs

Cf. A000217, A112456, A225503.

Programs

  • C
    #include 
#define TOP 300
typedef unsigned long long U64;
U64 isTriangular(U64 a) {
    U64 sr = 1ULL<<32, s, b, t;
    if (a < (sr/2)*(sr+1))  sr>>=1;
    while (a < sr*(sr+1)/2)  sr>>=1;
    for (b = sr>>1; b; b>>=1) {
        s = sr+b;
        if (s&1) t = s*((s+1)/2);
        else     t = (s/2)*(s+1);
        if (t >= s && a >= t)  sr = s;
    }
    return (sr*(sr+1)/2 == a);
}
int main() {
  U64 i, j, k, m, tm, p, pp = 1, primes[TOP];
  for (primes[0]=2, i = 3; pp < TOP; i+=2) {
    for (p = 1; p < pp; ++p) if (i%primes[p]==0) break;
    if (p==pp) {
        primes[pp++] = i;
        for (j=p=primes[pp-2], m=tm=1; ; j=k, m++, tm+=m) {
           if ((k = p*tm) < j) { m=0; break; }
           if (isTriangular(k)) break;
        }
        printf("%llu, ", m);
    }
  }
  return 0;
}
  • Mathematica
    lm[n_]:=Module[{m=1,p=Prime[n]},While[!OddQ[Sqrt[8(p (m(m+1))/2)+1]], m++];m]; Array[lm,68] (* Harvey P. Dale, Mar 16 2018 *)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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