A033286 -id:A033286 - cp's OEIS frontend

A066197 Squarefree kernel of (nprime(n))(n+prime(n)).

6, 30, 30, 154, 110, 1482, 714, 114, 138, 11310, 14322, 1554, 3198, 34314, 43710, 7314, 38114, 28914, 109478, 64610, 144102, 175538, 202354, 60342, 59170, 333502, 40170, 22470, 436218, 484770, 622046, 42706, 768570, 817598, 239890, 169422

Offset: 1

Reinhard Zumkeller, Dec 15 2001

nonn

			For n=20 we have: A = n = 20, B = A000040(20) = 71, C = A + B = 20 + 71 = 91 and A*B*C = 129220 with squarefree kernel a(20) = 64610 = 2*5*7*13*71.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Ivars Peterson, The Amazing ABC Conjecture
M. Waldschmidt, Open Diophantine problems, arXiv:math/0312440 [math.NT], 2003-2004.
Eric Weisstein's World of Mathematics, ABC Conjecture
Eric Weisstein's World of Mathematics, Squarefree
Wikipedia, abc conjecture

a066197 n = a007947 $ a033286 n * a014688 n
-- Reinhard Zumkeller, Jul 24 2013

sfk[n_] := Times @@ FactorInteger[n][[All, 1]];
a[n_] := sfk[n Prime[n] (n+Prime[n])];
Array[a, 40] (* Jean-François Alcover, Feb 04 2019 *)

a(n)=my(p=prime(n),f=vecsort(concat(concat(p, factor(n)[,1]), factor(n+p)[,1]),,8)~); prod(i=1,#f,f[i]) \\ Charles R Greathouse IV, Jul 23 2013

a(n) = A007947(A033286(n) * A014688(n)).

A068981 Arithmetic derivative of n*prime(n).

Original entry on oeis.org

1, 5, 8, 32, 16, 71, 24, 236, 147, 213, 42, 604, 54, 401, 391, 1712, 76, 1299, 86, 1724, 751, 1049, 106, 3940, 995, 1541, 2808, 3452, 138, 3533, 158, 10512, 1951, 2675, 1823, 9096, 194, 3461, 2711, 11804, 220, 7463, 234, 9308, 7728, 5021, 258, 25024, 3227, 10355

Offset: 1

Reinhard Zumkeller, Apr 01 2002

			a(10) = A003415(A033286(10)) = A003415(A000040(10)*10) = A003415(29*10) = A003415(29)*10 + 29*A003415(10) = 1*10 + 29*(2*1 + 1*5) = 10 + 29*7 = 213.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A003415, A033286.

a[n_] := n * Prime[n] * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n*Prime[n]]); Array[a, 100] (* Amiram Eldar, May 17 2025 *)

a(n) = n + A000040(n)*A003415(n).

a(n) = A003415(A033286(n)) = A003415(n*A000040(n)).

A079779 a(n) = smallest prime > n*prime(n).

Original entry on oeis.org

3, 7, 17, 29, 59, 79, 127, 157, 211, 293, 347, 449, 541, 607, 709, 853, 1009, 1103, 1277, 1423, 1543, 1741, 1913, 2137, 2437, 2633, 2789, 2999, 3163, 3391, 3943, 4201, 4523, 4729, 5227, 5437, 5813, 6197, 6521, 6947, 7349, 7603, 8219, 8501, 8867, 9157, 9923

Offset: 1

Amarnath Murthy, Feb 03 2003

nonn

a(n) is the smallest prime > A079780(n).

Cf. A033286, A079780, A151800.

Maple
```
seq(nextprime(n*ithprime(n)),n=1..40);
```

for(n=1, 47, print1(nextprime(n*prime(n)+1), ", "))

a(n) = A151800(A033286(n)). - Michel Marcus, Aug 31 2019

More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com) and Klaus Brockhaus, Feb 04 2003

A171520 Integers m that are not the product of k-th prime and k for any k >= 1.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79

Offset: 1

Jaroslav Krizek, Dec 11 2009

Complement of A033286.

Cf. A033286.

More terms from Michel Marcus, Aug 31 2019

A175897 Numbers n with property that nprime(n)+(n+1)prime(n+1) is a perfect square s^2.

Original entry on oeis.org

1681, 3146, 5917, 308950, 10553441, 10553550, 4273262954, 9781980985

Offset: 1

Zak Seidov, Oct 11 2010

Or, numbers n with property that A033286(n)+A033286(n+1) is a perfect square s^2, A033286(n)=n*(n-th prime). [From Zak Seidov, Oct 12 2010]

a(9) > pi(4*10^12). [From Donovan Johnson, Oct 22 2010]

			{n,s}: {1681,6943},{3146,13487},{5917,26299},{308950,1647737}, {10553441,63320647},{10553550,63321299}; no more n's up to 2*10^8.

Position[Total/@Partition[Table[n Prime[n],{n,310000}],2,1],? (IntegerQ[ Sqrt[#]]&)]//Flatten (* The program generates the first four terms of the sequence. To generate more, Increase the Range constant, but the program may take a long time to run. *) (* _Harvey P. Dale, Feb 05 2022 *)

isok(n) = issquare(n*prime(n)+(n+1)*prime(n+1)); \\ Michel Marcus, Oct 19 2013

a(7)-a(8) from Donovan Johnson, Oct 22 2010

A228529 a(n) = prime(n*prime(n)).

Original entry on oeis.org

3, 13, 47, 107, 257, 397, 653, 881, 1279, 1889, 2293, 3119, 3847, 4423, 5323, 6563, 7937, 8819, 10391, 11833, 12889, 14831, 16477, 18713, 21599, 23603, 25189, 27409, 29063, 31511, 37159, 39869, 43321, 45589, 50923, 53281, 57271, 61561, 65173, 69821, 74383

Offset: 1

Omar E. Pol, Oct 20 2013

nonn

			For n = 2, prime(2*prime(2)) = prime(2*3) = prime(6) = 13, so a(2) = 13.

Cf. A000040, A006450, A033286, A036689, A080697, A217622, A230098, A230325, A230285, A230460.

Table[Prime[n*Prime[n]], {n, 100}] (* T. D. Noe, Oct 22 2013 *)

a(n) = prime(n*prime(n)); \\ Michel Marcus, Oct 22 2013

a(n) = A000040(A033286(n)).

A237684 a(n) = floor(n*prime(n) / Sum_{i<=n} prime(i)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Jaroslav Krizek, Feb 21 2014

nonn

a(n) = 1 for n = 8 and 1 <= n <= 6.
a(n) = 2 for n = 7 and 9 <= n < 10^11 (verified terms).
Conjectures:
(1): a(n) = 1 or 2 for all n.
(2): sequence of numbers n sorted by decreasing values of function f(n) = n*prime(n) / Sum_{i<=n} prime(i): 48, 35, 31, 25, 17, 49, 33, 69, 32, 26, 43, 38, 12, 63, 102, 67, 68, 37, ... The last term of this sequence is 1.
(3): maximal value of function f(n) is for n = 48: f(48) = 10704/4661 = 2.29650289637416...
(4): minimal value of function f(n) is for n = 1: f(1) = 1.

			a(8) = floor(8*prime(8) / Sum_{i<=8} prime(i)) = floor(8*19 / 77) = 1.

Jaroslav Krizek, Table of n, a(n) for n = 1..87
Manuel Kauers and Christoph Koutschan, Some D-finite and some Possibly D-finite Sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023, p. 4.

Cf. A007504, A033286.

Block[{$MaxExtraPrecision = 1000, a, t = 0, nn = 120}, Do[(t += #; Set[a[i], Floor[i*#/t]]) &[Prime[i]], {i, nn}]; Array[a, nn] ] (* Michael De Vlieger, Mar 10 2023 *)

a(n) = floor(A033286(n) / A007504(n)).

A304194 Numbers k such that k = Product (p_j^e_j) = Product (pi(p_j)*p_j), where pi() = A000720.

Original entry on oeis.org

1, 2, 12, 56, 180, 304, 336, 936, 1696, 1824, 2484, 5040, 5328, 6664, 8384, 8512, 9900, 10176, 13176, 14040, 25632, 26208, 27360, 33372, 33712, 37260, 39808, 39984, 47488, 50304, 51072, 52200, 65232, 69552, 79920, 126900, 128448, 142272, 149184, 152640, 162648, 167776, 184064, 193752, 197640

Offset: 1

Ilya Gutkovskiy, May 07 2018

nonn

Numbers k such that A007947(k)*A156061(k) = k or A156061(k) = A003557(k).

			9900 is a term because 9900 = 2^2 * 3^2 * 5^2 * 11 = prime(1)^2 * prime(2)^2 * prime(3)^2 * prime(5) = 1*prime(1) * 2*prime(2) * 3*prime(3) * 5*prime(5).

Eric Weisstein's World of Mathematics, Distinct Prime Factors
Index entries for sequences computed from indices in prime factorization

Cf. A000720, A003557, A005117, A007947, A008478, A033286, A046022, A048768, A078779, A109298, A156061.

a[n_] := Times @@ (PrimePi[#[[1]]] #[[1]] & /@ FactorInteger[n]); a[1] = 1; Select[Range[200000], a[#] == # &]

isok(n) = {my(f=factor(n)); prod(k=1, #f~, primepi(f[k,1])*f[k,1]) == n;} \\ Michel Marcus, May 08 2018

A318367 a(n) = Sum_{d|n} (-1)^(n/d+1)dprime(d).

Original entry on oeis.org

2, 4, 17, 20, 57, 67, 121, 116, 224, 239, 343, 371, 535, 487, 777, 660, 1005, 958, 1275, 1095, 1669, 1401, 1911, 1715, 2482, 2097, 3005, 2295, 3163, 2987, 3939, 3156, 4879, 3727, 5391, 4502, 5811, 4925, 7063, 5271, 7341, 6619, 8215, 6433, 9849, 7249, 9919, 8691, 11244, 9264

Offset: 1

Ilya Gutkovskiy, Aug 24 2018

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

Cf. A033286, A061150, A061151, A061152.

f:= proc(n) local d; add((-1)^(n/d+1)*d*ithprime(d), d = numtheory:-divisors(n)); end proc:map(f, [$1..100]); # Robert Israel, Aug 01 2023

Table[Sum[(-1)^(n/d + 1) d Prime[d], {d, Divisors[n]}], {n, 50}]
nmax = 50; Rest[CoefficientList[Series[Sum[k Prime[k] x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 50; Rest[CoefficientList[Series[Log[Product[(1 + x^k)^Prime[k], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]

a(n) = sumdiv(n, d, (-1)^(n/d+1)*d*prime(d)); \\ Michel Marcus, Aug 25 2018

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

L.g.f.: log(Product_{k>=1} (1 + x^k)^prime(k)) = Sum_{n>=1} a(n)*x^n/n.

A330087 Permanent of a square matrix M(n) whose general element M_{i,j} is defined by i*prime(j).

Original entry on oeis.org

1, 2, 24, 1080, 120960, 33264000, 15567552000, 12967770816000, 15768809312256000, 29377291748732928000, 85194146071325491200000, 319563241913541917491200000, 1702632952915351336393113600000, 11797543730750469409867884134400000, 99429698562764956186366527484723200000

Offset: 0

Stefano Spezia, Dec 01 2019

nonn

det(M(0)) = 1, det(M(1)) = 2 and det(M(n)) = 0 for n > 1.
The trace of the matrix M(n) is A014285(n).
The antitrace of the matrix M(n) is A014148(n).
The antidiagonal of the matrix M(n) is the n-th row of the triangle A309131.

			For n = 1 the matrix M(1) is
  2
with permanent a(1) = 2.
For n = 2 the matrix M(2) is
  2, 3
  4, 6
with permanent a(2) = 24.
For n = 3 the matrix M(3) is
  2,  3,  5
  4,  6, 10
  6,  9, 15
with permanent a(3) = 1080.

Vaclav Kotesovec, Table of n, a(n) for n = 0..36

Cf. A000040, A014148, A014285, A033286, A309131.

with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> i*ithprime(j)))):
seq(a(n), n=0..14);  # Alois P. Heinz, Dec 04 2019

M[i_, j_, n_] := i*Prime[j]; a[n_] := If[n==0,1,Permanent[Table[M[i, j, n], {i, n}, {j, n}]]]; Array[a, 14, 0]

a(n) = matpermanent(matrix(n, n, i, j, i*prime(j))); \\ Michel Marcus, Dec 04 2019

a(0) = 1 prepended by Michel Marcus, Dec 04 2019

cp's OEIS Frontend

A066197 Squarefree kernel of (n*prime(n))*(n+prime(n)).

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Haskell

Mathematica

PARI

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A068981 Arithmetic derivative of n*prime(n).

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Mathematica

Formula

A079779 a(n) = smallest prime > n*prime(n).

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Comments

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Programs

Maple

PARI

Formula

Extensions

A171520 Integers m that are not the product of k-th prime and k for any k >= 1.

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Author

Keywords

Comments

Crossrefs

Extensions

A175897 Numbers n with property that n*prime(n)+(n+1)*prime(n+1) is a perfect square s^2.

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Author

Keywords

Comments

Examples

Programs

Mathematica

PARI

Extensions

A228529 a(n) = prime(n*prime(n)).

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Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A237684 a(n) = floor(n*prime(n) / Sum_{i<=n} prime(i)).

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Examples

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Crossrefs

Programs

Mathematica

Formula

A304194 Numbers k such that k = Product (p_j^e_j) = Product (pi(p_j)*p_j), where pi() = A000720.

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Author

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Comments

Examples

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Crossrefs

Programs

A066197 Squarefree kernel of (nprime(n))(n+prime(n)).

A175897 Numbers n with property that nprime(n)+(n+1)prime(n+1) is a perfect square s^2.

A318367 a(n) = Sum_{d|n} (-1)^(n/d+1)dprime(d).