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.

A214129 Partitions of n into parts congruent to +-1, +-5 (mod 13).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 17, 19, 21, 24, 27, 31, 34, 38, 42, 47, 52, 58, 64, 71, 78, 87, 95, 105, 116, 128, 140, 154, 168, 185, 202, 221, 241, 264, 287, 314, 341, 373, 405, 441, 478, 520, 564, 612, 662, 719, 777, 842
Offset: 0

Views

Author

Michael Somos, Jul 04 2012

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + 4*x^10 + ...
G.f. = q^-1 + q^5 + q^11 + q^17 + q^23 + 2*q^29 + 2*q^35 + 2*q^41 + 3*q^47 + 3*q^53 + ...

References

Links

Crossrefs

Cf. A214130, A214131.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ q, q^13] QPochhammer[ q^5, q^13] QPochhammer[ q^8, q^13] QPochhammer[ q^12, q^13]), {q, 0, n}]
  • PARI
    {a(n) = if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 - [ 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1][k%13 + 1] * x^k, 1 + x * O(x^n)), n))}

Formula

Expansion of f(-x^13)^2 / (f(-x, -x^12) * f(-x^5, -x^8)) in powers of x where f() is Ramanujan's two-variable theta function.
Euler transform of period 13 sequence [ 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, ...].
G.f.: 1 / (Product_{k>0} (1 - x^(13*k - 1)) * (1 - x^(13*k - 5)) * (1 - x^(13*k - 8)) * (1 - x^(13*k - 12))).
a(n) = A214130(n) + A214131(n-1).

A214130 Partitions of n into parts congruent to +-2, +-3 (mod 13).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 7, 9, 9, 11, 12, 14, 15, 18, 19, 23, 24, 28, 30, 35, 37, 43, 46, 52, 56, 64, 68, 77, 84, 93, 101, 113, 121, 135, 146, 161, 174, 193, 207, 229, 247, 272, 292, 322, 346, 379, 408, 446, 479, 524, 562, 613, 659
Offset: 0

Views

Author

Michael Somos, Jul 04 2012

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			1 + x^2 + x^3 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + 2*x^9 + 3*x^10 + ...
q^-1 + q^11 + q^17 + q^23 + q^29 + 2*q^35 + q^41 + 2*q^47 + 2*q^53 + ...

Links

Crossrefs

Cf. A214129, A214131.

Programs

  • Maple
    with (numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(irem(d, 13) in [2, 3, 10, 11], d, 0),
          d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 23 2013
  • Mathematica
    a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ q^2, q^13] QPochhammer[ q^3, q^13] QPochhammer[ q^10, q^13] QPochhammer[ q^11, q^13]), {q, 0, n}]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[If[MemberQ[{2, 3, 10, 11}, Mod[d, 13]], d, 0], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 - [ 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0][k%13 + 1] * x^k, 1 + x * O(x^n)), n))}

Formula

Expansion of f(-x^13)^2 / (f(-x^2, -x^11) * f(-x^3, -x^10)) in powers of x where f() is Ramanujan's two-variable theta function.
Euler transform of period 13 sequence [ 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, ...].
G.f.: 1 / (Product_{k>0} (1 - x^(13*k - 2)) * (1 - x^(13*k - 3)) * (1 - x^(13*k - 10)) * (1 - x^(13*k - 11))).
A214129(n) = a(n) + A214131(n-1).

A233039 Larger member of primitive friendly pairs ordered by smallest maximal element.

Original entry on oeis.org

28, 200, 224, 234, 270, 496, 496, 819, 936, 1488, 1638, 3724, 6200, 6200, 6860, 6975, 8128, 8128, 8128, 10976, 13104, 18600, 21600, 24384, 24384, 24800, 27000, 27000, 29792, 40131, 40640, 43008, 50274, 54000, 54400, 58032, 87750, 93100, 154791, 160524
Offset: 1

Views

Author

Michel Marcus, Dec 03 2013

Keywords

Comments

Subsequence of A050973.
Friends m and n are primitive friendly if and only if they have no common prime factor of the same multiplicity (see A096366).
Perfect numbers greater than 6 (A000396) belong to this sequence as they form primitive friendly pairs (PFPs) with smaller perfect, so that the n-th perfect number will appear n-1 times in the sequence.
PFPs are quite useful to derive new greater amicable pairs from existing ones (see A230148).

Examples

			28 forms a friendly pair with the lesser integer 6, and this pair cannot be derived from a smaller pair, so it is primitive and 28 belongs to the sequence.
140 forms also a pair with 30, hence 140 belongs to A050973. But the pair (30, 140) can be derived from (6, 28) by multiplying both members by 5, so it is not primitive; hence 140 does not belong to the sequence.

Links

Crossrefs

Cf. A050973, A096366, A214131, A214133.

Programs

  • PARI
    vp(f) = {maxp = f[#f~, 1]; v = vector(primepi(maxp)); for (j=1, #f~, v[primepi(f[j, 1])] = f[j, 2];);v;}
ispfp(vpn, vpi) = {for (k=1, min(#vpn, #vpi), if (vpi[k] && (vpn[k] == vpi[k]), return (0));); return (1);}
lista(nn) = {for (n=2, nn, ab = sigma(n)/n; vpn = vp(factor(n)); for (i=2, n-1, if (sigma(i)/i == ab, if (ispfp(vpn, vp(factor(i))), print1(n, ", ")););););} \\ Michel Marcus, Dec 03 2013
Showing 1-3 of 3 results.

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