A375529 -id:A375529 - cp's OEIS frontend

A375530 a(n) is the denominator of Sum_{k = 1..n} prime(k) / A375529(k).

1, 3, 30, 4530, 143650830, 226991170700228730, 669824890486184912549321336826596430, 7627311526552103393330686732733999706332372434754669475019405844335259730

Offset: 0

Rémy Sigrist and N. J. A. Sloane, Sep 04 2024

			The first few fractions Sum_{k = 1..n} prime(k) / A375529(k) are 0/1, 2/3, 29/30, 4529/4530, 143650829/143650830, 226991170700228729/226991170700228730, ...

Alois P. Heinz, Table of n, a(n) for n = 0..11
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]

Cf. A374663, A374983/A375516, A375529.

a:= proc(n) option remember; `if`(n=0, 1,
      ithprime(n)*a(n-1)^2+a(n-1))
    end:
seq(a(n), n=0..7);  # Alois P. Heinz, Oct 21 2024

a(n) = prime(n)*a(n-1)^2 + a(n-1), with a(0) = 1.

a(0)=1 prepended by Alois P. Heinz, Oct 21 2024

A376934 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, Sum_{k = 1..n} 1/(k*a(k)) < Pi.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 6, 80, 21896, 604163887, 677040508659246685, 447405708743254015046365510044832005, 309471557529368331206803181535934923519436869019793750609292014082198479

Offset: 1

Scott R. Shannon, Oct 11 2024

nonn

			a(17) = 604163887 as Sum_{k = 1..17} 1/(k*a(k)) = 1/(1*1) + 1/(2*1) + ... + 1/(16*21896) + 1/(17*604163887) = 3056398076673607759/972881723918332800, which is ~8.2*10^-20 less than Pi.

Scott R. Shannon, Table of n, a(n) for n = 1..27

Cf. A000796, A270744, A375781, A375529, A074631, A270752.

A376941 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, Sum_{k = 1..n} 1/(k*a(k)) < Sum_{k = 1..oo} 1/k^2 = Pi^2/6.

Original entry on oeis.org

1, 1, 3, 8, 78, 18826, 848809436, 3078251288697343844, 37949774653961921717972183675013581047, 4221065664206126654046840821317530741151656832301121739091602085731664210467

Offset: 1

Scott R. Shannon, Oct 12 2024

nonn

			a(7) = 848809436 as Sum_{k = 1..7} 1/(k*a(k)) = 1/(1*1) + 1/(2*1) + ... + 1/(6*18826) + 1/(7*848809436) = 430556991329920237/261747263922187680, which is ~4.1*10^-20 less than Pi^2/6.

Scott R. Shannon, Table of n, a(n) for n = 1..18

Cf. A013661, A376934, A270744, A375781, A375529, A074631, A270752.

cp's OEIS Frontend

A375530 a(n) is the denominator of Sum_{k = 1..n} prime(k) / A375529(k).

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A376934 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, Sum_{k = 1..n} 1/(k*a(k)) < Pi.

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A376941 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, Sum_{k = 1..n} 1/(k*a(k)) < Sum_{k = 1..oo} 1/k^2 = Pi^2/6.

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