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 1-5 of 5 results.

A272169 Duplicate of A084911.

Original entry on oeis.org

7, 5, 2, 0, 1, 0, 7, 4, 2, 3
Offset: 0

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Author

Keywords

A272339 First differences of 1/p(n), reciprocal of the number p(n) of unrestricted partitions of n (negated numerator).

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 4, 7, 2, 1, 1, 3, 24, 34, 41, 5, 2, 8, 3, 137, 5, 35, 253, 64, 383, 239, 41, 177, 7, 1039, 619, 137, 26, 2167, 2573, 3094, 3660, 398, 94, 293, 115, 71, 917, 11914, 13959, 4106, 4799, 3217, 26252, 2791, 3247, 1262, 2302, 8032, 1329, 75547, 87331, 50533, 53, 134647
Offset: 0

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Author

Jean-François Alcover, Apr 26 2016

Keywords

Examples

			Fractions begin: 0, -1/2, -1/6, -2/15, -2/35, -4/77, -4/165, -7/330, ...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, p. 274.

Links

Crossrefs

Cf. A000041, A084911, A272340 (denominators).

Programs

  • Mathematica
    -(Table[1/PartitionsP[n], {n, 0, 60}] // Differences) // Numerator
  • PARI
    a(n) = -numerator(1/numbpart(n+1) - 1/numbpart(n)); \\ Michel Marcus, Nov 03 2020

Formula

a(n) / A272340(n) = 1/p(n+1) - 1/p(n).
Product_{p prime} (1 - Sum_{n>=1} (a(n)/A272340(n))/p^n) = A272169. - Amiram Eldar, Nov 03 2020

A379359 Numerators of the partial sums of the reciprocals of the number of abelian groups function (A000688).

Original entry on oeis.org

1, 2, 3, 7, 9, 11, 13, 41, 22, 25, 28, 59, 65, 71, 77, 391, 421, 218, 233, 481, 511, 541, 571, 581, 298, 313, 106, 217, 227, 237, 247, 1739, 1809, 1879, 1949, 3933, 4073, 4213, 4353, 13199, 13619, 14039, 14459, 14669, 14879, 15299, 15719, 15803, 16013, 16223, 16643
Offset: 1

Views

Author

Amiram Eldar, Dec 21 2024

Keywords

Examples

			Fractions begin with 1, 2, 3, 7/2, 9/2, 11/2, 13/2, 41/6, 22/3, 25/3, 28/3, 59/6, ...

References

  • Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions, North-Holland Publishing Company, Amsterdam, Netherlands, 1980. See pp. 13-16, Theorem 1.3.
  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003. See section 5.1, Abelian group enumeration constants, p. 274.

Links

Crossrefs

Cf. A000688, A063966, A084911, A370897, A379360 (denominators), A379361.

Programs

  • Mathematica
    Numerator[Accumulate[Table[1/FiniteAbelianGroupCount[n], {n, 1, 100}]]]
  • PARI
    f(n) = vecprod(apply(numbpart, factor(n)[, 2]));
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / f(k); print1(numerator(s), ", "))};

Formula

a(n) = numerator(Sum_{k=1..n} 1/A000688(k)).
a(n)/A379360(n) = D * n + O(sqrt(n/log(n))), where D = A084911.

A379361 Numerators of the partial alternating sums of the reciprocals of the number of abelian groups function (A000688).

Original entry on oeis.org

1, 0, 1, 1, 3, 1, 3, 7, 5, 2, 5, 7, 13, 7, 13, 59, 89, 37, 52, 89, 119, 89, 119, 109, 62, 47, 52, 89, 119, 89, 119, 803, 1013, 803, 1013, 1921, 2341, 1921, 2341, 2201, 2621, 2201, 2621, 2411, 2621, 2201, 2621, 2537, 2747, 2537, 2957, 2747, 3167, 1009, 1149, 3307
Offset: 1

Views

Author

Amiram Eldar, Dec 21 2024

Keywords

Examples

			Fractions begin with 1, 0, 1, 1/2, 3/2, 1/2, 3/2, 7/6, 5/3, 2/3, 5/3, 7/6, ...

Links

Crossrefs

Cf. A000041, A000688, A063966, A084911, A370897, A379359, A379362 (denominators).

Programs

  • Mathematica
    Numerator[Accumulate[Table[(-1)^(n+1)/FiniteAbelianGroupCount[n], {n, 1, 100}]]]
  • PARI
    f(n) = vecprod(apply(numbpart, factor(n)[, 2]));
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / f(k); print1(numerator(s), ", "))};

Formula

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A000688(k)).
a(n)/A379362(n) ~ D * c * n, where D = A084911, c = 2/(1 + Sum_{k>=1} 1/(P(k)*2^k)) - 1 = 0.18634377034863729099..., and P(k) = A000041(k).

A242180 Least prime divisor of q(n) which does not divide any q(k) with k < n, or 1 if such a primitive prime divisor does not exist, where q(.) is the strict partition function given by A000009.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 5, 1, 1, 1, 1, 1, 1, 11, 1, 1, 19, 23, 1, 1, 1, 89, 13, 61, 71, 1, 1, 37, 1, 1, 17, 1, 7, 1, 1, 167, 1, 1, 491, 53, 1, 31, 1, 227, 1, 1, 1, 97, 1, 59, 241, 29, 1, 953, 1063, 1777, 1, 367, 1, 1
Offset: 1

Views

Author

Zhi-Wei Sun, May 06 2014

Keywords

Comments

Conjecture: a(n) > 1 for all n > 203.

Examples

			a(8) = 1 since q(8) = 2*3 with 2 = q(3) and 3 = q(5).
a(23) = 13 since q(23) = 2^3*13 with 13 not dividing q(1)*q(2)*...*q(22), but 2 divides q(3) = 2.

Links

Crossrefs

Cf. A000009, A000040, A084911, A194261, A272170, A272171, A272173.

Programs

  • Mathematica
    f[n_]:=FactorInteger[PartitionsQ[n]]
pp[n_]:=Table[Part[Part[f[n],k],1],{k,1,Length[f[n]]}]
Do[If[PartitionsQ[n]<2,Goto[cc]];Do[Do[If[Mod[PartitionsQ[i],Part[pp[n],k]]==0,Goto[aa]],{i,1,n-1}];Print[n," ",Part[pp[n],k]];Goto[bb];Label[aa];Continue,{k,1,Length[pp[n]]}];Label[cc];Print[n," ",1];Label[bb];Continue,{n,1,60}]
Showing 1-5 of 5 results.

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