Jonathan D. B. Hodgson - cp's OEIS frontend

A181325 Primes that can be written as the sum of a Fibonacci number (A000045) and a triangular number (A000217).

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 61, 67, 71, 79, 83, 89, 107, 113, 137, 139, 149, 157, 167, 173, 179, 191, 193, 199, 211, 223, 233, 239, 269, 277, 281, 311, 313, 331, 353, 359, 379, 383, 389, 397, 409, 419, 433, 443, 461, 467, 499, 509

Offset: 0

Jonathan D. B. Hodgson, Oct 13 2010

nonn

			29 is in the sequence since 29 is prime and 29 = 21 + 8 = T(6) + F(7); 43 is not in the sequence, despite being prime, since there exist no m and n such that 43 = T(m) + F(n).

Cf. A000040, A000045, A000217, A107259.

A181173 Primes whose base 5 representation does not contain a 0.

Original entry on oeis.org

2, 3, 7, 11, 13, 17, 19, 23, 31, 37, 41, 43, 47, 59, 61, 67, 71, 73, 83, 89, 97, 107, 109, 113, 157, 163, 167, 173, 181, 191, 193, 197, 199, 211, 223, 233, 239, 241, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 359, 367, 373, 409, 419, 421, 431, 433

Offset: 1

Jonathan D. B. Hodgson, Oct 08 2010

			23 = 43 (base 5) which contains no 0.

Cf. A082555.

The following stores the first 200 digits of the sequence in K: for i from 1 to 200 do if i=i then x[i]:=convert(ithprime(i),base,5) else x[i]:=0 end if: end do: S:={}: for i from 1 to 200 do if evalb(`in`(0, x[i]))=false then S:=S union {i} fi od; for i from 1 to nops(S)do z[i]:=ithprime(S[i]) od: K:=[seq((z[i]),i=1..nops(S))];

A181285 Primes of the form 5^k - 4.

Original entry on oeis.org

3121, 78121, 30517578121, 710542735760100185871124267578121, 413590306276513837435704346034981426782906055450439453121

Offset: 1

Jonathan D. B. Hodgson, Oct 12 2010

nonn

			3121 = 5^5 - 4 is prime and therefore is in the sequence.

Cf. A059613, A135535, A228028.

n:=1000: S:={}: for i from 1 to n do if type(5^i-4,prime)=true then S:=S union {5^i-4} end if od; S;

Select[5^Range[90]-4,PrimeQ] (* Harvey P. Dale, Aug 23 2013 *)

A181172 Primes whose base 4 representation does not contain a 0.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 89, 101, 103, 107, 109, 127, 149, 151, 157, 167, 173, 181, 191, 223, 229, 233, 239, 251, 347, 349, 359, 367, 373, 379, 383, 409, 421, 431, 439, 443, 479, 487, 491, 503, 509, 599, 601, 607, 613, 617, 619

Offset: 1

Jonathan D. B. Hodgson, Oct 08 2010

This sequence contains all Mersenne primes (i.e. this is a supersequence of A000668). - Iain Fox, Dec 25 2017

			53 = 311 (base 4), which contains no 0.

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

Cf. A082555, A000668 (subsequence).

Cf. A073779 (number of 0's in base-3 representation of n-th prime), A181173 (primes whose base 5 representation does not contain a 0). - Klaus Brockhaus, Oct 10 2010

[ p: p in PrimesUpTo(620) | not exists(t){d: d in Intseq(p, 4) | d eq 0 } ]; // Klaus Brockhaus, Oct 10 2010

The following code will store the first 200 terms into a sequence K. for i from 1 to 200 do if i=i then x[i]:=convert(ithprime(i),base,4) else x[i]:=0 end if: end do: S:={}: for i from 1 to 200 do if evalb(`in`(0, x[i]))=false then S:=S union {i} fi od; for i from 1 to nops(S)do z[i]:=ithprime(S[i]) od: K:=[seq((z[i]),i=1..nops(S))];
# Alternative:
select(t -> isprime(t) and not has(convert(t,base,4),0), [2,seq(i,i=3..10^4,2)]); # Robert Israel, Dec 24 2017

Select[Prime@ Range@ 120, DigitCount[#, 4, 0] == 0 &] (* Michael De Vlieger, Dec 24 2017 *)

lista(nn) = forprime(p=2, nn, if(!setsearch(Set(digits(p, 4)), 0), print1(p, ", "))) \\ Iain Fox, Dec 25 2017

A181122 Decimal expansion of Sum_{k>=0} (-1)^k/(5k+1).

Original entry on oeis.org

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

Offset: 0

Jonathan D. B. Hodgson, Oct 05 2010

			0.88831357265178863804075522702037934627811083077545817120597...

Gheorghe Coserea, Table of n, a(n) for n = 0..1000
H. Wilf, Accelerated series for universal constants, by the WZ method, Discrete Mathematics and Theoretical Computer Science 3(4) (1999), 189-192.

Cf. A002162, A003881, A024396, A113476, A181048, A181049, A181122, A193534, A262246.

(int(1/(1+x^5),x=0..1));
evalf(LerchPhi(-1,1,1/5)/5) ; # R. J. Mathar, Oct 16 2011

(Sqrt[8 + 8/Sqrt[5]]*Pi + 2*Sqrt[5]*ArcCoth[3/Sqrt[5]] + Log[16])/20 // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Feb 13 2013 *)

default(realprecision, 106);
eval(vecextract(Vec(Str(sumalt(n=0, (-1)^(n)/(5*n+1)))), "3..-2")) \\ Gheorghe Coserea, Oct 06 2015

Sum_{k>=0} (-1)^k/(5k+1) = Integral_{x=0..1}dx/(1+x^5) = (1/10)*sqrt(10-2*sqrt(5))*arctan((3/4)*sqrt(10-2*sqrt(5)) + (1/4)*sqrt(10-2*sqrt(5))*sqrt(5)) + (1/20)*sqrt(10-2*sqrt(5))*arctan(-(1/4)*sqrt(10-2*sqrt(5)) + (1/4)*sqrt(10-2*sqrt(5))*sqrt(5)) + (1/20)*sqrt(10-2*sqrt(5))*sqrt(5)*arctan(-(1/4)*sqrt(10-2*sqrt(5)) + (1/4)*sqrt(10-2*sqrt(5))*sqrt(5)) + (1/20)*log(2)*sqrt(5) + (1/5)*log(2) - (1/20)*log(7-3*sqrt(5))*sqrt(5).
Equals Pi*sqrt(phi)/5^(5/4) + log(phi)/sqrt(5) + log(2)/5, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 01 2015
From Peter Bala, Feb 19 2024: (Start)
Equals (1/2)*Sum_{n >= 0} n!*(5/2)^n/(Product_{k = 0..n} 5*k + 1) = (1/2)*Sum_{n >= 0} n!*(5/2)^n/A008548(n+1) (apply Euler's series transformation to Sum_{k >= 0} (-1)^k/(5*k + 1)).
Continued fraction: 1/(1 + 1^2/(5 + 6^2/(5 + 11^2/(5 + ... + (5*n + 1)^2/(5 + ... ))))).
The slowly converging series representation Sum_{n >= 0} (-1)^n/(5*n + 1) for the constant can be accelerated to give the following faster converging series:
1/2 + (5/2)*Sum_{n >= 0} (-1)^n/((5*n + 1)(5*n + 6)) and
17/24 + (25/2)*Sum_{n >= 0} (-1)^n/((5*n + 1)(5*n + 6)*(5*n + 11)).
These two series are the cases r = 1 and r = 2 of the general result: for r >= 0, the constant equals
C(r) + ((5/2)^r)*r!*Sum_{n >= 0} (-1)^n/((5*n + 1)*(5*n + 6)*...*(5*n + 5*r + 1)), where C(r) is the rational number (1/2)*Sum_{k = 0..r-1} (5/2)^k*k!/(1*6*11*...*(5*k + 1)). The general result can be proved by the WZ method as described in Wilf. (End)
From Peter Bala, Mar 03 2024: (Start)
Equals hypergeom([1/5, 1], [6/5], -1).
Gauss's continued fraction: 1/(1 + 1^2/(6 + 5^2/(11 + 6^2/(16 + 10^2/(21 + 11^2/(26 + 15^2/(31 + 16^2/(36 + 20^2/(41 + 21^2/(46 + ... )))))))))). (End)

A180658 Decimal expansion of the value of the continued fraction 2+2^2/(2^3+2^4/(2^5+2^6...

Original entry on oeis.org

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

Offset: 1

Jonathan D. B. Hodgson, Sep 15 2010

			2.4710128596979097485957862398868620614877702... [From _R. J. Mathar_, Sep 19 2010]

More digits from R. J. Mathar, Sep 19 2010

A180659 Decimal expansion of the continued fraction F(0)+F(1)/(F(2)+F(3)/(F(4)+F(5)/ ... where F(n) is the n-th Fibonacci number.

Original entry on oeis.org

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

Offset: 0

Jonathan D. B. Hodgson, Sep 15 2010

			0.6416600010214519998536699491955...

Offset set to 0 by R. J. Mathar, Sep 23 2010

A180660 Decimal expansion of the constant whose continued fraction representation is [Phi^0; Phi^1, Phi^2, Phi^3, Phi^4, ...] where Phi is the golden ratio (A001622) and the exponents cycle through all nonnegative integers.

Original entry on oeis.org

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

Offset: 1

Jonathan D. B. Hodgson, Sep 15 2010

			1.50777857...

Cf. A001622

Cf. A100609

More digits from R. J. Mathar, Sep 19 2010

A180661 Decimal expansion of the constant whose continued fraction representation is [Pi^0; Pi^1, Pi^2, Pi^3, Pi^4, ...] where Pi is the ratio of a circle's circumference to its diameter (A000796) and the exponents cycle through all nonnegative integers.

Original entry on oeis.org

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

Offset: 1

Jonathan D. B. Hodgson, Sep 15 2010

			1.30839602...

Cf. A000796, A100609

More digits from R. J. Mathar, Sep 19 2010

A181050 Decimal expansion of the constant 1+3/(5+7/(9+11/(13+...))), using all odd integers in this generalized continued fraction.

Original entry on oeis.org

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

Offset: 1

Jonathan D. B. Hodgson, Oct 01 2010

The (simple) continued fraction of this constant is [1;1,1,9,1,1,17,1,1,25,...], every 3rd term being of the form 8n+1.

			1.524965344417894912821223094...

Cf. A113011.

r:= (n, i)-> n+ `if`(i<1, 1, (n+2)/r(n+4, i-1)):
s:= convert(evalf(r(1, 80)/10, 130), string):
seq(parse(s[n+1]), n=1..120); # Alois P. Heinz, Oct 16 2011

digits = 105; f[n_] := f[n] = Fold[#2 + (#2+2)/#1 &, 4*n+1, Range[4*n-3, 1, -4] ] // RealDigits[#, 10, digits]& // First; f[digits]; f[n = 2*digits]; While[f[n] != f[n/2], n = 2*n]; f[n] (* Jean-François Alcover, Feb 21 2014 *)

cp's OEIS Frontend

User: Jonathan D. B. Hodgson

A181325 Primes that can be written as the sum of a Fibonacci number (A000045) and a triangular number (A000217).

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A181173 Primes whose base 5 representation does not contain a 0.

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Maple

A181285 Primes of the form 5^k - 4.

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Maple

Mathematica

A181172 Primes whose base 4 representation does not contain a 0.

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Magma

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PARI

A181122 Decimal expansion of Sum_{k>=0} (-1)^k/(5k+1).

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Maple

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A180658 Decimal expansion of the value of the continued fraction 2+2^2/(2^3+2^4/(2^5+2^6...

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A180659 Decimal expansion of the continued fraction F(0)+F(1)/(F(2)+F(3)/(F(4)+F(5)/ ... where F(n) is the n-th Fibonacci number.

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A180660 Decimal expansion of the constant whose continued fraction representation is [Phi^0; Phi^1, Phi^2, Phi^3, Phi^4, ...] where Phi is the golden ratio (A001622) and the exponents cycle through all nonnegative integers.

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A180661 Decimal expansion of the constant whose continued fraction representation is [Pi^0; Pi^1, Pi^2, Pi^3, Pi^4, ...] where Pi is the ratio of a circle's circumference to its diameter (A000796) and the exponents cycle through all nonnegative integers.

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A181050 Decimal expansion of the constant 1+3/(5+7/(9+11/(13+...))), using all odd integers in this generalized continued fraction.

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