A374663 -id:A374663 - 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

A375791 a(n) = A375516(n+1) / A375516(n).

Original entry on oeis.org

2, 2, 3, 4, 25, 201, 40201, 1212060151, 1305857607493406801, 1534737681943564047120326770001682121, 11777098761887521784975815904636471022877972047160405176265171997646882601

Offset: 0

Rémy Sigrist and N. J. A. Sloane, Aug 29 2024

nonn

			a(7) = A375516(8) / A375516(7) = 11752718467440661200 / 9696481200 = 1212060151.

Alois P. Heinz, Table of n, a(n) for n = 0..14

Cf. A374663, A375516.

s:= proc(n) s(n):= `if`(n=0, 0, s(n-1)+1/(n*b(n))) end:
b:= proc(n) b(n):= 1+floor(1/((1-s(n-1))*n)) end:
a:= n-> denom(s(n+1))/denom(s(n)):
seq(a(n), n=0..10);  # Alois P. Heinz, Oct 19 2024

s[n_] := s[n] = If[n == 0, 0, s[n-1] + 1/(n*b[n])];
b[n_] := b[n] = 1 + Floor[1/((1 - s[n-1])*n)];
a[n_] := Denominator[s[n+1]]/Denominator[s[n]];
Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Jul 02 2025, after Alois P. Heinz *)

{ r = 1; for (n = 1, 11, a = floor(1/(r*n))+1; d = denominator(r); r -= 1/(n*a); print1 (denominator(r)/d", ");); }

from itertools import count, islice
from math import gcd
def A375791_gen(): # generator of terms
    p, q = 0, 1
    for k in count(1):
        m = q//(k*(q-p))+1
        p, q = p*k*m+q, k*m*q
        p //= (r:=gcd(p,q))
        q //= r
        yield k*m//r
A375791_list = list(islice(A375791_gen(),11)) # Chai Wah Wu, Aug 30 2024

A376050 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} 1/((2k-1)a(k)) < 1.

Original entry on oeis.org

2, 1, 2, 3, 6, 172, 137534, 106557767317, 10018727448950607892211, 218107864753736742334588510315735629277159621, 43040465365773907074907163986022284668974202910116417170603263409796800986397420975160781

Offset: 1

N. J. A. Sloane, Sep 13 2024

It appears that S(n) = (e(n)-1)/e(n) for all n != 4, where e(n) = A376051(n). Exceptionally, S(4) = (e(4)-2)/e(4).

a(15) has 1420 decimal digits, too large for a b-file. - Robert Israel, Oct 13 2024

Rémy Sigrist and N. J. A. Sloane, Dampening Down a Divergent Series, Manuscript in preparation, September 2024.

Robert Israel, Table of n, a(n) for n = 1..14

Cf. A374663, A375516, A375531, A375532, A375781, A375522, A376048, A376049, A376051.

S:= 1:R:= NULL:
for i from 1 to 11 do
  r:= ceil(1/((2*i-1)*S));
  if r *(2*i-1) = 1/S then r:= r+1 fi;
  R:= R,r;
  S:= S - 1/((2*i-1)*r)
od:
R; # Robert Israel, Oct 13 2024

A376053 Numerator of the sum S(n) defined in A376052.

Original entry on oeis.org

1, 8, 71, 248, 3043, 43024, 89051, 764441, 451021514, 25508567769, 411827311870583771, 525058386770138717020639964821, 528134692562568161116953143877712480332943632586669596859, 2267693117789905604207315326366543773113615946806750184592188584359364943382168221068055512231683584106110223751

Offset: 1

N. J. A. Sloane, Sep 14 2024

			The initial values of S(n) are 1/3, 8/15, 71/105, 248/315, 3043/3465, 43024/45045, 89051/90090, ...

Cf. A374663, A375516, A375531, A375532, A375781, A375522, A376048-A376051, A376052, A376054, A376055

A376054 Denominator of sum S(n) defined in A376052.

Original entry on oeis.org

3, 15, 105, 315, 3465, 45045, 90090, 765765, 451035585, 25508568085, 411827311870584610, 525058386770138717020639964850, 528134692562568161116953143877712480332943632586669596900

Offset: 1

N. J. A. Sloane, Sep 14 2024

			The initial values of S(n) are 1/3, 8/15, 71/105, 248/315, 3043/3465, 43024/45045, 89051/90090, ...

Cf. A374663, A375516, A375531, A375532, A375781, A375522, A376048-A376051, A376052, A376053, A376055

A376059 a(n) is the denominator of the sum S(n) defined in A376058.

Original entry on oeis.org

1, 2, 6, 78, 18330, 1679962830, 22578200883132834030, 6627077016548303724729207245056971365730, 922281145448518091883798423085535218757314338662318933097843039655721026758456630

Offset: 0

N. J. A. Sloane, Sep 14 2024

			The first few values of S(n) are 0, 1/2, 5/6, 77/78, 18329/18330, 1679962829/1679962830, 22578200883132834029/22578200883132834030, ...

Cf. A000045, A374663, A375516, A375531, A375532, A375781, A375522, A376048-A376058.

RecurrenceTable[{a[n+1] == Fibonacci[n+1]*a[n]^2 + a[n], a[0] == 1}, a, {n, 0, 8}] (* Amiram Eldar, Sep 15 2024 *)

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

A376184 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} b(k)/a(k) < 1, where {b(k)} is the sequence b(1)=5/4, b(2i)=3/2, b(2i+1)=6/5 (i>0).

Original entry on oeis.org

2, 5, 17, 341, 92753, 10753782821, 92515075960384748177, 10698799099944699918936107506299150093941, 91571441744782016867976366392607084634231243149599342901251284090792487979854033

Offset: 1

N. J. A. Sloane, Sep 15 2024

This sequence and A376062 were discovered by Rémy Sigrist on Sep 09 2024. The two sequences {b(1)=7/6, b(k)=5/4 for k>1} and {b(1)=5/4, b(2*k)=3/2, b(2*k+1)=6/5 for k>0} are the first sequences {b(i)} discovered with the property that the sums S(n) do not converge to numbers of the form (e_n - 1)/e_n as n-> oo.

			The initial values of S(n) are 5/8, 37/40, 677/680, 231877/231880, 21507565637/21507565640, 231287689900961870437/231287689900961870440, ...

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. A004168, A082732, A374663, A375516, A375531, A375532, A375781, A375522, A376048-A376062, A376186.

A376185 a(n) = denominator of the sum S(n) defined in A376062.

Original entry on oeis.org

12, 48, 624, 97968, 2399530224, 1439436326371902768, 517994234419759747473589427583418224, 67079506723028253472357256785558488997471406450171845011442457607246768

Offset: 1

N. J. A. Sloane, Sep 15 2024

			The initial values of S(n) are 7/12, 43/48, 619/624, 97963/97968, 2399530219/2399530224, 1439436326371902763/1439436326371902768 ...

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, A375516, A375531, A375532, A375781, A375522, A376048-A376062.

1/a(n) = 1/a(n-1) - 1/(4*A376062(n)) for n >= 2.

A376186 a(n) = denominator of the sum S(n) defined in A376184.

Original entry on oeis.org

8, 40, 680, 231880, 21507565640, 231287689900961870440, 21397598199889399837872215012598300187880, 228928604361955042169940915981517711585578107873998357253128210226981219949635080

Offset: 1

N. J. A. Sloane, Sep 15 2024

			The initial values of S(n) are 5/8, 37/40, 677/680, 231877/231880, 21507565637/21507565640, 231287689900961870437/231287689900961870440, ...

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. A004168, A082732, A374663, A375516, A375531, A375532, A375781, A375522, A376048-A376062, A376184, A376185.

A376768 a(1) = 2; thereafter, a(n) = c*A376767(n-1) + 1, where c=4 if n is even, c=2 if n is odd.

Original entry on oeis.org

2, 5, 11, 221, 24311, 1182000821, 698562969831336611, 975980445639153806859347417915257421, 476268915135000629546288739757931900755190480621127468115605921390156911

Offset: 1

N. J. A. Sloane, Nov 03 2024

nonn

This is the sequence of dampening coefficients for the divergent series 1 + 2 + 1 + 2 + 1 + 2 + ...

Paolo Xausa, Table of n, a(n) for n = 1..12
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

Cf. A374663, A376767.

A376767[n_] := A376767[n] = If[n == 1, 1, If[OddQ[n], 2, 4]*#^2 + # & [A376767[n-1]]];
A376768[n_] := If[n == 1, 2, If[OddQ[n], 2, 4]*A376767[n-1] + 1];
Array[A376768, 10] (* Paolo Xausa, Dec 16 2024 *)

Definition corrected by N. J. A. Sloane, Dec 15 2024 - thanks to Paolo Xausa for noticing the error.

cp's OEIS Frontend

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

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A375791 a(n) = A375516(n+1) / A375516(n).

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A376050 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} 1/((2*k-1)*a(k)) < 1.

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A376053 Numerator of the sum S(n) defined in A376052.

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A376054 Denominator of sum S(n) defined in A376052.

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A376059 a(n) is the denominator of the sum S(n) defined in A376058.

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A376184 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} b(k)/a(k) < 1, where {b(k)} is the sequence b(1)=5/4, b(2*i)=3/2, b(2*i+1)=6/5 (i>0).

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A376185 a(n) = denominator of the sum S(n) defined in A376062.

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A376186 a(n) = denominator of the sum S(n) defined in A376184.

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A376768 a(1) = 2; thereafter, a(n) = c*A376767(n-1) + 1, where c=4 if n is even, c=2 if n is odd.

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A376050 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} 1/((2k-1)a(k)) < 1.

A376184 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} b(k)/a(k) < 1, where {b(k)} is the sequence b(1)=5/4, b(2i)=3/2, b(2i+1)=6/5 (i>0).