José Eduardo Gaboardi de Carvalho

A283735 Expansion of 10 in base Pi.

1, 0, 0, 0, 1, 0, 2, 2, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 0, 0, 1, 1, 1, 1, 2, 1, 0, 2, 0, 1, 2, 0, 1, 0, 2, 2, 1, 2, 0, 2, 1, 1, 0, 0, 1, 1, 1, 2, 1, 2, 0, 0, 3, 0, 1, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 2, 2, 0, 2, 2, 1, 2, 1, 0, 0, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 3, 0, 0, 2, 0, 1, 0, 1

Offset: 3

José Eduardo Gaboardi de Carvalho, Mar 15 2017

			100.0102212222..._{Pi}

José Eduardo Gaboardi de Carvalho, Java program

CF. A000796, A303877.

RealDigits[10, Pi, 100][[1]] (* Indranil Ghosh, Mar 15 2017 *)

A276665 In the '3x+1' problem, these values for the starting value set new records for both the number of steps and the highest point of trajectory before reaching 1.

Original entry on oeis.org

1, 2, 3, 7, 27, 703, 26623

Offset: 1

José Eduardo Gaboardi de Carvalho, Sep 12 2016

Both the 3x+1 steps and the halving steps are counted.

If it exists, a(7) > 14727207461063895711 (A006877(148)). - Hugo Pfoertner, Jan 12 2025

José Eduardo Gaboardi de Carvalho, Java program

Intersection of A006877 and A006884.

A276199 Smallest prime that begins with at least n digits of Pi.

Original entry on oeis.org

3, 31, 31469, 314107, 314159, 314159, 314159207, 3141592603, 31415926541, 314159265307, 314159265359, 3141592653581, 314159265358909, 3141592653589711, 31415926535897921, 314159265358979347, 3141592653589793239, 3141592653589793239, 314159265358979323861

Offset: 1

José Eduardo Gaboardi de Carvalho, Aug 24 2016

			a(7) = 314159207, begins with first 7 digits of Pi = 3.141592653...

Dana Jacobsen, Table of n, a(n) for n = 1..997 (first 28 terms from José Eduardo Gaboardi de Carvalho)

Cf. A005042.

use ntheory ":all"; sub a276199 { my $l=shift; my $p="3".substr(Pi($l+20),2,$l-1); for my $dig (0 .. 20) { my $add = "0" x $dig; do { return "$p$add" if is_prime("$p$add"); } while length(++$add) == $dig; } } # Dana Jacobsen, Aug 30 2016

A275614 Decimal expansion of number with continued fraction expansion 0, 1, 2, 4, 8, 16, 32, ... (powers of 2).

Original entry on oeis.org

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

Offset: 0

José Eduardo Gaboardi de Carvalho, Aug 03 2016

			0.69159419242197808428289286692643...

Cf. A000079, A096641, A214070.

terms=sqrtint(bitprecision(1.))+2; \\ terms needed at current precision
t=contfracpnqn(vector(terms,i,.5<Charles R Greathouse IV, Aug 04 2016

From Amiram Eldar, Feb 08 2022: (Start)
Equals 1/A214070.
Equals 1/(1 + A096641). (End)

Definition corrected by Jianing Song, Mar 29 2025

A269588 Numbers n such that n^2 ends with the digits of n reversed (A004086(n)).

Original entry on oeis.org

1, 5, 6, 963, 9867, 65766, 69714, 6317056, 90899553, 169605719, 4270981082, 96528287587, 465454256742, 692153612536, 182921919071841, 655785969669834, 650700037578750084, 125631041500927357539, 673774165549097456624, 16719041449406813636569

Offset: 1

José Eduardo Gaboardi de Carvalho, Mar 01 2016

a(29)>10^32 (if it exists)

			6317056^2 = 39905196507136 which ends with 6507136, so 6317056 is a term.

Robert Gerbicz, Table of n, a(n) for n = 1..28
IBM, "Ponder This" puzzle for March 2016

Subsequence of A115761.

Cf. A003226, A004086.

Select[Range[10^7], Function[k, Take[IntegerDigits[#^2], -Length@ k] == Reverse@ k]@ IntegerDigits@ # &] (* Michael De Vlieger, Mar 04 2016 *)

isA269588(n)=dn = digits(n); rn = subst(Polrev(dn), x, 10); nbd = #dn; (n^2 - rn) % 10^nbd == 0; \\ Michel Marcus, Mar 01 2016

\\ printA269588len(d) prints all terms of the sequence with d digits
rev(n) = eval(concat(Vecrev(Str(n))));
{ printA269588len(d) = my(l, u, n); l=ceil(d/2); u=floor(d/2); for(y=0, 10^l-1, n=rev(y^2 % 10^u)*10^l+y; if(#Str(n)==d && Mod(n, 10^d)^2==rev(n), print(n)); ); }
\\ Max Alekseyev, Mar 07 2016

a(18)-a(20) from Max Alekseyev, Mar 07 2016
a(21)-a(27) from Robert Gerbicz, Apr 03 2016
a(28) from Dieter Beckerle, Jun 09 2016

A243916 Largest safe prime less than 2^n.

Original entry on oeis.org

7, 11, 23, 59, 107, 227, 503, 1019, 2039, 4079, 8147, 16223, 32603, 65267, 130787, 262127, 524243, 1048343, 2097143, 4194287, 8388287, 16776899, 33553799, 67108187, 134217323, 268435019, 536870723, 1073740439, 2147483579, 4294967087

Offset: 3

José Eduardo Gaboardi de Carvalho, Jun 18 2014

nonn

Largest safe prime (A005385) p=2*q+1, q also prime (A005384), that can be represented using n binary digits.

Gokberk Yaltirakli, Table of n, a(n) for n = 3..1024 (terms 3..100 from Harvey P. Dale)

Cf. A005385, A005384, A000079.

lsp[n_]:=Module[{sp=NextPrime[2^n,-1]},While[!PrimeQ[(sp-1)/2],sp= NextPrime[ sp,-1]];sp]; Array[lsp,35,3] (* Harvey P. Dale, Feb 10 2019 *)

from sympy import isprime
def a(n):
    if n<3: return 0
    i=2**n - 1
    while True:
        if isprime(i) and isprime((i - 1)/2): return i
        else: i-=2 # Indranil Ghosh, Jun 12 2017, after Antti Karttunen's Scheme Code

cp's OEIS Frontend

User: José Eduardo Gaboardi de Carvalho

A283735 Expansion of 10 in base Pi.

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Mathematica

A276665 In the '3x+1' problem, these values for the starting value set new records for both the number of steps and the highest point of trajectory before reaching 1.

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A276199 Smallest prime that begins with at least n digits of Pi.

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Perl

A275614 Decimal expansion of number with continued fraction expansion 0, 1, 2, 4, 8, 16, 32, ... (powers of 2).

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PARI

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Extensions

A269588 Numbers n such that n^2 ends with the digits of n reversed (A004086(n)).

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Mathematica

PARI

PARI

Extensions

A243916 Largest safe prime less than 2^n.

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Mathematica

Python