Simon Plouffe - cp's OEIS frontend

A381788 Greedy expansion of Pi-3 in a base with place values 1/(10^k-1), k >= 1, using digits {0,1,2,...,8,9,A=10}.

1, 3, 0, 1, 7, 8, 5, 0, 1, 4, 6, 6, 5, 9, 4, 7, 1, 5, 1, 9, 5, 6, 1, 3, 4, 8, 9, 3, 4, 2, 2, 7, 5, 2, 2, 9, 0, 3, 8, 6, 2, 8, 1, 1, 5, 8, 3, 5, 3, 1, 1, 9, 8, 2, 3, 5, 2, 0, 8, 9, 4, 1, 8, 2, 4, 8, 6, 3, 1, 2, 5, 9, 1, 2, 9, 1, 5, 5, 5, 0, 6, 9, 6, 8, 0, 7, 7, 9, 7, 4, 0, 9, 8, 2, 8, 5, 7, 4, 1, 9, 5, 5, 7, 5, 2, 8, 3, 1, 1, 0, 8, 8, 5

Offset: 1

Simon Plouffe, Mar 07 2025

From Pontus von Brömssen, Mar 13 2025: (Start)
Since the ratio of successive place values is less than 1/10, a digit A=10 is sometimes needed. For example, if 10*A073668-37/33 < x < 1/9, the expansion of x must have an A at the second position after the radix point (for any choice of digits, not only greedy).
The expansion is not unique without specifying greedy choice of digits. For example, the number 11/1000 can be represented both as 0.010898908982... and (non-greedily) as 0.00A989899171... in this system.
For a random number, the probability that the digit A occurs decreases exponentially with the position in the expansion (with greedy choice of digits), so it seems very unlikely that 10 is a term of this sequence.
(End)

Cf. A000796, A073668.

BASEN:= proc(x, b, sgn, k)
local i, j, v, premier, fin, lll, liste, w, baz;
    baz := evalf(b);
    v := abs(frac(evalf(x)));
    fin := trunc(evalf(Digits/log10(b)));
    lll := [seq(i^k*(baz^i + sgn), i = 1 .. fin)];
    liste := [];
    for i to fin do w := trunc(v*lll[i]); v := v - w/lll[i]; liste := [op(liste), w] end do;
    RETURN(liste)
end;
BASEN(Pi-3,10,-1,0);

Sum_{k>=1} a(k)/(10^k - 1) = Pi - 3.

Edited by N. J. A. Sloane, Mar 18 2025

A363271 Vertical sum of n in base 10.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 6, 7, 8, 9, 10, 11, 12

Offset: 1

Simon Plouffe, May 24 2023

The Sum_{n>=1} a(n)/10^n = 10/81, it is the vertical sum of each integer. The pattern is easy to see but apparently impossible for a program to find any closed form or recurrence. The sequence is generated by adding each integer with an offset of 1 at each step.
If you sum integers with each term divided by 10^n, at n = 9 there are 2 terms in the column 9 + 1 = 10 which is a(10).
Here is the actual sum:
  .100000000000000000000
  .020000000000000000000
  .003000000000000000000
  .000400000000000000000
  .000050000000000000000
  .000006000000000000000
  .000000700000000000000
  .000000080000000000000
  .000000009000000000000
  .000000001000000000000
  .000000000110000000000
  .000000000012000000000
  .000000000001300000000
  .000000000000140000000
  .000000000000015000000
  .000000000000001600000
  .000000000000000170000
  .000000000000000018000
  .000000000000000001900
  .000000000000000000200
  .000000000000000000021
  .000000000000000000002
  ...
By adding each column we get a(n), which explains why a(9) = 10.

			The original sequence is 1 2 3 4 5 6 7 8 9 10 11 12 ... but when we sum digit per digit (in base 10) the sequence is not a rational fraction.

Cf. A021085 (10/81), A089400 (binary analog).

p:=proc(v) local n, aa, nn, s, k, t;
    aa := v;
    nn := nops(aa);
    s := [seq(1 + aa[k]/10^k,
        k = 1 .. nops(aa))];
    [seq(sum(trunc(10*frac(10^t*s[k])),
        k = 1 .. nops(aa)),
        t = 0 .. nops(aa))]
end;
# enter a sequence like a(n) = [1, 2, 3, 4, ...] it will return a sequence r such that sum(r(n)/10^n) is equal to sum(a(n)/10^n).

A345986 Numbers k > 0 such that the k-th power of the Dedekind eta-function is lacunary.

Original entry on oeis.org

2, 4, 6, 8, 10, 14, 26

Offset: 1

Simon Plouffe and N. J. A. Sloane, Jul 10 2021, following a suggestion from Francis Sanchez

S. Krishnamoorthy and T. Dalal, On the vanishing of coefficients of eta^26, arXiv:2411.14990 [math.NT], 2024.
J.-P. Serre, Sur la lacunarité des puissances de eta, Glasgow Math. Journal, 27 (1985), 203-221.

Cf. A007706.

A344013 a(1)=1; thereafter a(n) = A169639(a(n-1)).

Original entry on oeis.org

1, 35, 125, 157, 206, 148, 197, 293, 333, 286, 302, 177, 246, 245, 236, 230, 178, 244, 275, 295, 342, 274, 326, 247, 253, 281, 285, 293, 333, 286, 302, 177, 246, 245, 236, 230, 178, 244, 275, 295, 342, 274, 326, 247, 253, 281, 285, 293, 333, 286, 302, 177, 246, 245, 236, 230, 178, 244, 275

Offset: 1

Simon Plouffe and N. J. A. Sloane, Jun 10 2021

A French analog of A345126 (British English) and A345157 (US English).

Enters a cycle of length 20 starting from a(8). - Chai Wah Wu, Jun 12 2021

			1 = un -> 35 = trente-cinq -> 125 -> cent vingt cinq = 157.

Cf. A073327, A169639, A345240, A345126, A345157.

a[1]=1;a[n_]:=a[n]=Total@LetterNumber@StringDelete[IntegerName[a[n-1],"French"],{" ","-"}];Array[a,100] (* Giorgos Kalogeropoulos, Jun 11 2021 *)

from num2words import num2words
from unidecode import unidecode
A344013_list = [1]
for _ in range(10):
    A344013_list.append(sum(ord(s)-96 for s in unidecode(num2words(A344013_list[-1],lang='fr')) if s.isalpha())) # Chai Wah Wu, Jun 11 2021

More terms from Chai Wah Wu, Jun 11 2021

A342756 Rounded value of z(n)*prime(n), where z(n) = imaginary part of n-th nontrivial zero of the Zeta function and prime(n) = n-th prime.

Original entry on oeis.org

28, 63, 125, 213, 362, 489, 696, 823, 1104, 1443, 1642, 2089, 2433, 2616, 3060, 3555, 4103, 4396, 5072, 5477, 5792, 6550, 7033, 7781, 8614, 9342, 9749, 10258, 10773, 11449, 13173, 13814, 14682, 15433, 16669, 17262, 18248, 19363, 20269

Offset: 1

Simon Plouffe, Apr 11 2021

nonn

Empirical: a(n) ~ 2*Pi*n^2.

			z(1) = 14.134... and prime(1) = 2, a(1) = round(14.134...*2) = 28.

Simon Plouffe, Table of n, a(n) for n = 1..100000
David Platt, Zeros of the Zeta function.
David Platt, Computing pi(x) analytically, Mathematics of computation, 84 (2015), 1521-1535.
S. Plouffe, Pi, the primes and the Lambert W function [Slides of a talk]

A333127 a(n) = round(c^n) with prime generating constant c = 55237.07504296764715433124... .

Original entry on oeis.org

3051134459, 168535743094673, 9309421488742788613, 514225213380301008078907, 28404296700473737832215645327, 1568970268386786190461323870128523, 86665328455065998699156259015013574567, 4787139251495919953666696192531499395660543, 264427570076016102911609907247384217082828701473

Offset: 2

Simon Plouffe, Mar 08 2020

The exact value of c = 55237.0750429676471543312478152861741726374... has 5000 decimal digits (cf. A335321).

			{c^2} = 3051134459, {c^3} = 168535743094673, where {c^n} = nearest integer to c^n.

Hugo Pfoertner, Table of n, a(n) for n = 2..211
Simon Plouffe, The calculation of p(n) and pi(n), arXiv:2002.12137 [math.NT], 2020. See Appendix.
Simon Plouffe, Constant c and list of the 632 primes, Jun 01 2020.

Cf. A332308, A335321.

a(n) = round(c^n), is prime for n = 2..633, with c = 55237.07504296764715433124...

A332308 a(n) = round(c^n), where c is the prime generating constant c = 31622.77671855956934118197870614288... .

Original entry on oeis.org

1000000007, 31622776952311, 1000000014783746303, 31622777186062677745609, 1000000022175619536498921059, 31622777419814234539614807614633, 1000000029567492824611472390607319403, 31622777653565793061482767695810547093627, 1000000036959366167363813218134876470482703123

Offset: 2

Simon Plouffe, Mar 07 2020

The exact value of c = 31622.776718559569341 ... has 4096 decimal digits (cf. A335320).

			round(c^2) = 1000000007, round(c^3) = 31622776952311.

Hugo Pfoertner, Table of n, a(n) for n = 2..222
Simon Plouffe, The calculation of p(n) and pi(n), arXiv:2002.12137 [math.NT], 2020. See Appendix.
Simon Plouffe, A formula for primes

Cf. A333127, A335320.

a(n) = round(c^n), gives primes for n = 2..388.

Edited by Georg Fischer, Jun 27 2020

A306317 Prime numbers generated by the formula a(n) = round(2^(d^n)), where d is the real constant 1.30076870414817691055252567828266106688423996320151467218595488...

Original entry on oeis.org

2, 3, 5, 7, 13, 29, 79, 293, 1619, 14947, 269237, 11570443, 1540936027, 893681319109, 3513374197622981, 166491395148719076277, 201072926144898161374940903, 16390008340104365722976984827792343, 320076519482444467256811692239892862140322229, 7781106039755041703318535124896118983796534882794414187099

Offset: 1

Simon Plouffe, Feb 06 2019

nonn

The exponent d = 1.3007687... is the smallest found.

Simon Plouffe, Table of n, a(n) for n = 1..24
Simon Plouffe, A set of formulas for primes, arXiv:1901.01849 [math.NT], 2019.
Simon Plouffe, Une formule pour les nombres premiers, viXra:1902.0036.

Cf. A323176, A323065, A323611.

# Computes the values according to the formula, v = 2..., e = 1.30076870414817691055252567828266106688423996320151467218595488..., m the # number of terms. Returns the real and the rounded values (primes). In this case 23 terms will be generated
val := proc(s, e, m)
local ll, v, n, kk;
    v := s;
    ll := [];
    for n to m do
        v := v^e; ll := [op(ll), v]
    end do;
    return [ll, map(round, ll)]
end;

a(n) = round(2^(d^n)), where d is a real constant starting 1.30076870414817691055252567828266106688423996320151467218595488...

A323065 Prime numbers generated by the formula a(n) = round(c(n)), where c(n) = c(n-1)^d for n >= 2 starting with c(1) = C. C and d are the real constants given below.

Original entry on oeis.org

3, 5, 7, 11, 19, 41, 103, 331, 1423, 8819, 86477, 1504949, 53691233, 4703173021, 1267699542037, 1394588856899951, 8916055416478425247

Offset: 1

Simon Plouffe, Jan 20 2019

C = 3.346835535932430816866371614510056305833213572055338155233562507
and exponent
d = 1.251295195638613270470338478487766898374146819139632632235793814.

			c(1) = 3.3468, a(1) = 3; c(2) = 4.53390554, a(2) = 5; c(3) = 6.6288905, a(3) = 7; ...; c(n) = c(n-1)^d and a(n) = {c(n)} is the value rounded to the nearest integer.

Simon Plouffe, A set of formulas for primes, arXiv:1901.01849 [math.NT], 2019.

Cf. A323176.

# Computes the values according to the formula, s = 3.34683553..., d = 1.2512951, m the number of terms. Returns the real and the rounded values (primes).
val := proc(s, d, m)
local ll, v, n;
    v := s;
    ll := [v];
    for n to m-1 do
        v := v^d; ll := [op(ll), v]
    end do;
    return [ll, map(round, ll)]
end:

A323611 Prime numbers generated by the formula a(n) = round(c(n)), where c(n) = c(n-1)^(3/2) for n >= 2 starting with c(1) = C and C the real constant given below.

Original entry on oeis.org

2, 3, 5, 11, 37, 223, 3331, 192271, 84308429, 774116799347, 681098209317971743, 562101323304225290104514179, 13326678220145859782825116625722145759009, 1538448162271607869601834587431948506238982765193425993274489

Offset: 1

Simon Plouffe, Jan 20 2019

nonn

C = 2.038239154782068767463490862609548251448624778443173613879675732.

			c(1) = 2.038239154782068, c(2) = 2.9099311279, c(3) = 4.96391190457, c(4) = 11.05951540, ... so a(1) = {c(1)} = 2, a(2) = {c(2)} = 3, a(3) = {c(3)} = 5, ...
c(n) = c(n-1)^(3/2) and a(n) = {c(n)} is the value rounded to the nearest integer.

Simon Plouffe, A set of formulas for primes, arXiv:1901.01849 [math.NT], 2019.

Cf. A323176, A323065.

# Computes the values according to the formula, c = 2.03823915478..., e = 3/2, m the number of terms. Returns the real and the rounded values (primes).
val := proc(c, e, m)
local ll, v, n;
    v := c;
    ll := [v];
    for n to m-1 do
        v := v^e; ll := [op(ll), v]
    end do;
    return [ll, map(round, ll)]
end:

cp's OEIS Frontend

User: Simon Plouffe

A381788 Greedy expansion of Pi-3 in a base with place values 1/(10^k-1), k >= 1, using digits {0,1,2,...,8,9,A=10}.

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A363271 Vertical sum of n in base 10.

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A345986 Numbers k > 0 such that the k-th power of the Dedekind eta-function is lacunary.

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A344013 a(1)=1; thereafter a(n) = A169639(a(n-1)).

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A342756 Rounded value of z(n)*prime(n), where z(n) = imaginary part of n-th nontrivial zero of the Zeta function and prime(n) = n-th prime.

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A333127 a(n) = round(c^n) with prime generating constant c = 55237.07504296764715433124... .

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A332308 a(n) = round(c^n), where c is the prime generating constant c = 31622.77671855956934118197870614288... .

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A306317 Prime numbers generated by the formula a(n) = round(2^(d^n)), where d is the real constant 1.30076870414817691055252567828266106688423996320151467218595488...

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A323065 Prime numbers generated by the formula a(n) = round(c(n)), where c(n) = c(n-1)^d for n >= 2 starting with c(1) = C. C and d are the real constants given below.

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