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 1-6 of 6 results.

A220427 G.f.: exp( Sum_{n>=1} A005064(n)*x^n/n ), where A005064(n) = sum of cubes of primes dividing n.

Original entry on oeis.org

1, 0, 4, 9, 10, 61, 65, 239, 440, 791, 2172, 3211, 8018, 14292, 27174, 56064, 96092, 195616, 345831, 643733, 1189397, 2102921, 3864549, 6804894, 12150956, 21419322, 37460309, 65511385, 113436266, 195931822, 336547491, 575446427, 979007055, 1660337942, 2800856388
Offset: 0

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Author

Paul D. Hanna, Dec 14 2012

Keywords

Examples

			G.f.: A(x) = 1 + 4*x^2 + 9*x^3 + 10*x^4 + 61*x^5 + 65*x^6 + 239*x^7 +...
where
log(A(x)) = 8*x^2/2 + 27*x^3/3 + 8*x^4/4 + 125*x^5/5 + 35*x^6/6 + 343*x^7/7 + 8*x^8/8 + 27*x^9/9 + 133*x^10/10 + 1331*x^11/11 + 35*x^12/12 + 2197*x^13/13 + 351*x^14/14 + 152*x^15/15 +...+ A005064(n)*x^n/n +...

Crossrefs

Cf. A005064, A219224.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(k=1,n+1,sumdiv(k,d,isprime(d)*d^3)*x^k/k)+x*O(x^n)),n)}
for(n=0,40,print1(a(n),", "))

A291647 Expansion of Product_{k>=1} (1 + x^prime(k))^prime(k).

Original entry on oeis.org

1, 0, 2, 3, 1, 11, 3, 20, 21, 20, 64, 35, 112, 117, 160, 269, 284, 477, 598, 819, 1116, 1495, 1899, 2718, 3389, 4596, 6121, 7627, 10460, 13128, 17350, 22506, 28696, 37063, 47779, 60249, 78642, 98783, 126058, 160758, 200795, 257750, 321768, 407930, 511526, 640636, 802816, 1005618, 1252820, 1567454, 1946162
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 28 2017

Keywords

Comments

Number of partitions of n into distinct prime parts, where prime(k) different parts of size prime(k) are available (2a, 2b, 3a, 3b, 3c, ...).

Examples

			a(6) = 3 because we have [3a, 3b], [3a, 3c] and [3b, 3c].

Links

Crossrefs

Cf. A000040, A000586, A007441, A026007, A061151, A061152, A219224, A262736.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 + x^Prime[k])^Prime[k], {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^A000040(k))^A000040(k).

A301971 a(n) = [x^n] Product_{k>=1} 1/(1 - x^prime(k))^n.

Original entry on oeis.org

1, 0, 2, 3, 10, 30, 77, 252, 682, 2136, 6182, 18766, 56173, 169351, 512990, 1551828, 4720170, 14348289, 43751984, 133502873, 408029510, 1248460587, 3823949824, 11724787763, 35980251181, 110510334780, 339674840715, 1044812449722, 3215861978150, 9904301974294, 30521063942312, 94103983534015
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 29 2018

Keywords

Comments

Number of partitions of n into prime parts of n kinds.

Links

Crossrefs

Cf. A000607, A008485, A030009, A117278, A219224, A259254, A291647, A298436, A299168.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[1/(1 - x^Prime[k])^n, {k, 1, n}], {x, 0, n}], {n, 0, 31}]

A318969 Expansion of exp(Sum_{k>=1} ( Sum_{p|k, p prime} p^k ) * x^k/k).

Original entry on oeis.org

1, 0, 2, 9, 6, 643, 182, 118953, 6019, 242630, 2243190, 25938251679, 78106516, 23349992199606, 288964822371, 46755212195033, 226472341461312, 48661337027901364945, 18066374340919781, 104224677113940850317679, 440728415311733637734, 208546898802899685866735, 972477473959172989443327
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 06 2018

Keywords

Crossrefs

Cf. A000607, A023881, A200768, A219224, A220427, A318913, A318968.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[Sum[Sum[Boole[PrimeQ[d]] d^k, {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
nmax = 22; CoefficientList[Series[Product[1/(1 - Prime[k]^Prime[k] x^Prime[k])^(1/Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[Boole[PrimeQ[d]] d^k, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 22}]

Formula

G.f.: Product_{k>=1} 1/(1 - prime(k)^prime(k)*x^prime(k))^(1/prime(k)).

A300521 Expansion of Product_{k>=1} (1 - x^prime(k))^prime(k).

Original entry on oeis.org

1, 0, -2, -3, 1, 1, 3, 0, 9, 8, 4, -31, -12, -13, 20, -13, 48, -17, 74, -87, 8, -143, 175, -174, 349, -164, 369, -651, 520, -1004, 1142, -1218, 1652, -1739, 3291, -3933, 3546, -5743, 6170, -8022, 11435, -13230, 17196, -18706, 22958, -31884, 38420, -49802, 58916
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 08 2018

Keywords

Crossrefs

Cf. A000040, A000607, A005063, A007441, A030009, A046675, A073592, A219224, A291647, A291696.

Programs

  • Mathematica
    nmax = 48; CoefficientList[Series[Product[(1 - x^Prime[k])^Prime[k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[-Sum[DivisorSum[k, Boole[PrimeQ[#]] #^2 &] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 - x^A000040(k))^A000040(k).
G.f.: exp(-Sum_{k>=1} A005063(k)*x^k/k).

A358452 The inverse Euler transform of p(n) = n if n is prime, otherwise 1.

Original entry on oeis.org

1, 1, 1, 1, -3, 3, -3, 5, -8, 5, -11, 36, -45, 41, -72, 142, -223, 311, -493, 851, -1243, 1823, -3204, 5336, -7906, 12083, -20134, 33133, -51685, 81568, -133556, 215363, -340155, 547916, -895895, 1442323, -2300704, 3718260, -6056908, 9787064, -15755664, 25541623
Offset: 0

Views

Author

Peter Luschny, Nov 21 2022

Keywords

Comments

Conjecture: signum(a(n)) + (-1)^n = 0 for n >= 3.

Links

Crossrefs

Cf. A089026, A219224.

Programs

  • Maple
    # The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> ifelse(isprime(n), n, 1)):
seq(a(n), n = 0..41);
# Using EULERi the sequence is returned without a(0) and has offset 1.
f := n -> ifelse(isprime(n), n, 1): EULERi([seq(f(n), n = 1..41)]);
Showing 1-6 of 6 results.

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