A047235 -id:A047235 - cp's OEIS frontend

A291050 Decimal expansion of Pi^2 / 27.

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

Offset: 0

Dimitris Valianatos, Oct 20 2017

			0.365540903744050319216092259254672264270877755823732986163457384304446104534...

Index entries for transcendental numbers

Cf. A002388, A047235.

RealDigits[Pi^2/27,10,160][[1]] (* Harvey P. Dale, Apr 29 2019 *)

PARI
```
Pi^2 / 27
```

Equals Sum_{n>=1} 1/A047235(n)^2.

Equals A002388 / 27. - Felix Fröhlich, Oct 22 2017

A360735 Even integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has only two elements.

Original entry on oeis.org

16, 22, 26, 32, 44, 46, 52, 56, 58, 62, 70, 74, 76, 82, 86, 88, 92, 100, 106, 112, 116, 118, 122, 128, 130, 136, 140, 142, 146, 148, 152, 158, 160, 166, 170, 172, 176, 182, 184, 194, 196, 200, 202, 206, 212, 214, 218, 224, 226, 232, 236, 242, 244, 250, 254, 256, 262, 266, 268

Offset: 1

Bernard Schott, Feb 19 2023

nonn

Similar sequence with odd integers d is A040976 \ {0}.
Terms are even numbers that are not divisible by 3 and that are not also in A206037.
These longest corresponding APs are of the form (q, q+d) with q odd primes (see examples).
This subsequence of A359408 corresponds to the second case '2 is one less than prime 3' (see A173919); the first case is linked to A040976.
A342309(d) gives the first element of the smallest such AP with 2 elements whose common difference is a(n) = d.

			d = 16 is a term because the first longest APs of primes with common difference 16 are (3, 19), (7,23), (13, 29), ... and all have 2 elements because next elements should be respectively 35, 39 and 45 that are all composite; the first such AP that starts with A342309(16) = 3 is (3, 19).
d = 22 is a term because the first longest APs of primes with common difference 22 are (7, 29), (19, 41), (31, 53), ... and all have 2 elements because next elements should be respectively 51, 63 and 75 that are all composite; the first such AP that starts with A342309(22) = 7 is (7, 29).

Equals A359408 \ A040976.

Cf. A047235, A123556, A206037, A342309.

filter := d -> (irem(d, 2) = 0) and (irem(d, 3) <> 0) and not isprime(3+d) or isprime(3+d) and not isprime(3+2*d) : select(filter, [`$`(1 .. 270)]);
isA360735 := d -> isA047235(d) and not isA206037(d): # Peter Luschny, Mar 03 2023

Select[Range[2, 270, 2], Mod[#, 3] > 0 && Nand @@ PrimeQ[{# + 3, 2*# + 3}] &] (* Amiram Eldar, Mar 03 2023 *)

isok(d) = !(d%2) && (d%3) && !(isprime(d+3) && isprime(2*d+3)); \\ Michel Marcus, Mar 03 2023

If m is a term then A123556(m) = 2, but the converse is false: a counterexample is A123556(11) = 2 and 11 is not a term.

A091738 Primes arising in the second row of array in A091734.

Original entry on oeis.org

3, 7, 19, 29, 43, 53, 71, 79, 101, 107, 131, 139, 163, 173, 193, 199, 229, 239, 263, 271, 293, 311, 337, 349, 373, 383, 409, 421, 443, 457, 479, 491, 521, 541, 569, 577, 601, 613, 641, 647, 673, 683, 719, 733, 757, 769, 809, 821, 839, 857, 881, 887, 929, 941

Offset: 1

Giovanni Teofilatto, Mar 06 2004

nonn

a(n) = prime( A047235(n)) . [From R. J. Mathar, Apr 22 2010]

Terms beyond 193 from R. J. Mathar, Apr 22 2010

A248323 Numbers n which appear at least twice in A037278(n), concatenation of their divisors written in base 10.

Original entry on oeis.org

11, 12, 42, 84, 124, 135, 248, 325, 366, 510, 550, 555, 624, 650, 714, 1010, 1111, 1525, 1734, 2510, 3913, 4020, 5100, 5500, 5610, 5625, 8040, 11111, 13515, 16575, 21175, 24104, 25500, 28160, 34170, 35250, 35610, 36800, 37444, 44919, 50100, 51020, 51102, 51250, 52000

Offset: 1

Eric Angelini and M. F. Hasler, Oct 04 2014

Overlapping is allowed, so a(1) = 11 is in the sequence, with concatenated divisors A037278(11) = "111".
All repunits (10^k-1)/9 = A000042(k) = A002275(k) with even k = number of digits (as to be divisible by 11) but not multiples of 3, i.e., k in A047235, have divisors {1, 11, ..., 1010...101, 1111...111} and therefore are in this sequence.
Numbers n = floor(10^(8+3k)/7), k>=0, also belong to this sequence; for k>=2m, the number n appears (at least) m+2 times in A037278(n). [Found by extending results from Hans Havermann.]
The smallest terms that appear more than twice in the concatenation are 1111, 400200, 800400, 28571428, all 3 times, and 42857142, 4 times. - Hans Havermann, Oct 05 2014

			The divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42, and "42" appears twice in their concatenation A037278(42) = "12367142142".

Chai Wah Wu, Table of n, a(n) for n = 1..340
E. Angelini, Divisors showing a string, SeqFan list, Oct 03 2014

Cf. A037278.

is(n)={d=concat(apply(digits,divisors(n)));n=digits(n);for(j=0,#d-#n-1,for(i=1,#n,d[i+j]==n[i]||next(2));return(1))}

from sympy import divisors
import re
A248323_list = [n for n in range(1,10**7) if len(list(re.finditer('(?='+str(n)+')',''.join([str(d) for d in divisors(n)])))) > 1]
# Chai Wah Wu, Nov 01 2014

A086382 k divides F(k*n^2+1)-F(k+1) for 1<=k<=a(n) where F(k) is the k-th Fibonacci number.

Original entry on oeis.org

2, 1, 2, 10, 1, 10, 2, 1, 2, 12, 1, 10, 2, 1, 2, 10, 1, 12, 2, 1, 2, 10, 1, 10, 2, 1, 2, 16, 1, 12, 2, 1, 2, 10, 1, 10, 2, 1, 2, 16, 1, 10, 2, 1, 2, 10, 1, 12, 2, 1, 2, 10, 1, 10, 2, 1, 2, 12, 1, 12, 2, 1, 2, 10, 1, 10, 2, 1, 2, 36, 1, 10, 2, 1, 2, 10, 1, 12, 2, 1, 2, 10, 1, 10, 2, 1, 2, 12, 1, 12

Offset: 2

Benoit Cloitre, Sep 06 2003

nonn

Record values: a(2) = 2, a(5) = 10, a(11) = 12, a(29) = 16, a(71) = 36, a(3079) = 58. The next record a(n), if any has n > 10^5. - Robert Israel, Oct 14 2024

Robert Israel, Table of n, a(n) for n = 2..10000

fibmod:= proc(k,m) uses LinearAlgebra:-Modular;
  local M;
  M:= Mod(m,<<0,1>|<1,1>>,integer[8]);
  MatrixPower(m,M,k)[1,2]
end proc:
f:= proc(n) local k;
   for k from 2 do if fibmod(k*n^2+1,k) <> fibmod(k+1,k) then return k-1 fi od
end proc:
map(f, [$2..100]); # Robert Israel, Oct 14 2024

a(n)=if(n<0,0,m=1; while((fibonacci(m*n^2+1)-fibonacci(m+1))%m==0,m++); m-1)

a(3n)=1; a( A047235(n))=2

A119981 a(n) = 1 iff number congruent to {2, 4} mod 6 is equal to prime minus 3, otherwise a(n)=0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0

Offset: 1

Giovanni Teofilatto, Aug 03 2006

a(n) = 1 iff A047235(n) + 3 = A086801(n).

Cf. A047235, A086801.

A181156 Odd Fibonacci numbers F which have a proper Fibonacci divisor G such that F/G is a Lucas number or a product of Lucas numbers.

Original entry on oeis.org

3, 21, 55, 377, 987, 6765, 17711, 121393, 317811, 2178309, 5702887, 39088169, 102334155, 701408733, 1836311903, 12586269025, 32951280099, 225851433717, 591286729879, 4052739537881, 10610209857723, 72723460248141, 190392490709135, 1304969544928657, 3416454622906707

Offset: 1

Vladimir Shevelev, Oct 07 2010

nonn

A conjectural statement that, for odd prime p, the ratio F_{p^2}/F_{p} is never a Lucas number or a product of some Lucas numbers, yields that
a) an odd Fibonacci number F is in the sequence iff for its maximal proper Fibonacci divisor G, we have: ind G does not equal sqrt(ind F) and F/G does not have a proper Fibonacci divisor > 3;
b) an odd Fibonacci number F is in the sequence iff its index has one of the forms: 6k+2 or 6k+4 (see A047235).

			F = 3 has the proper Fibonacci divisor G=1, and F/G = 3 is a Lucas number.
F = 317811 has the proper Fibonacci divisors 3, 13, and 377, and F/377 = 843 is a Lucas number.

Cf. A000045, A000032, A014437, A014447.

A255846 a(n) = 2*n^2 + 14.

Original entry on oeis.org

14, 16, 22, 32, 46, 64, 86, 112, 142, 176, 214, 256, 302, 352, 406, 464, 526, 592, 662, 736, 814, 896, 982, 1072, 1166, 1264, 1366, 1472, 1582, 1696, 1814, 1936, 2062, 2192, 2326, 2464, 2606, 2752, 2902, 3056, 3214, 3376, 3542, 3712, 3886, 4064, 4246, 4432

Offset: 0

Avi Friedlich, Mar 08 2015

This is the case k=7 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2.

Equivalently, numbers m such that 2*m - 28 is a square.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A117619.
Subsequence of A047235 and A047451.
Cf. similar sequences listed in A255843.

Magma
```
[2*n^2+14: n in [0..50]];
```
Mathematica
```
Table[2 n^2 + 14, {n, 0, 50}]
```
PARI
```
vector(50, n, n--; 2*n^2+14)
```
Sage
```
[2*n^2+14 for n in (0..50)]
```

G.f.: 2*(7 - 13*x + 8*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A117619(n).
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(7)*Pi*coth(sqrt(7)*Pi))/28.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(7)*Pi*cosech(sqrt(7)*Pi))/28. (End)
E.g.f.: 2*exp(x)*(7 + x + x^2). - Elmo R. Oliveira, Jan 25 2025

Edited by Bruno Berselli, Mar 13 2015

A299647 Positive solutions to x^2 == -2 (mod 11).

Original entry on oeis.org

3, 8, 14, 19, 25, 30, 36, 41, 47, 52, 58, 63, 69, 74, 80, 85, 91, 96, 102, 107, 113, 118, 124, 129, 135, 140, 146, 151, 157, 162, 168, 173, 179, 184, 190, 195, 201, 206, 212, 217, 223, 228, 234, 239, 245, 250, 256, 261, 267, 272, 278, 283, 289, 294, 300, 305, 311, 316

Offset: 1

Bruno Berselli, Mar 06 2018

Positive numbers congruent to {3, 8} mod 11.

Equivalently, interleaving of A017425 and A017485.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Subsequence of A106252, A279000.
Cf. A017425, A017485.
Cf. A017497: positive solutions to x == -2 (mod 11).
Cf. A017437: positive solutions to x^3 == -2 (mod 11).
Nonnegative solutions to x^2 == -2 (mod j): A005843 (j=2), A001651 (j=3), A047235 (j=6), A156638 (j=9), this sequence (j=11).

List([1..60], n -> 5*n-2+(2*n-(-1)^n-3)/4);

[(11(2n-1)-(-1)^n)>>2 for n in 1:60] # Peter Luschny, Mar 07 2018

[5*n-2+(2*n-(-1)^n-3)/4: n in [1..60]];

Table[5 n - 2 + (2 n - (-1)^n - 3)/4, {n, 1, 60}]
CoefficientList[ Series[(3 + 5x + 3x^2)/((x - 1)^2 (x + 1)), {x, 0, 57}], x] (* or *)
LinearRecurrence[{1, 1, -1}, {3, 8, 14}, 58] (* Robert G. Wilson v, Mar 08 2018 *)

makelist(5*n-2+(2*n-(-1)^n-3)/4, n, 1, 60);

vector(60, n, nn; 5*n-2+(2*n-(-1)^n-3)/4)

[5*n-2+(2*n-(-1)**n-3)/4 for n in range(1, 60)]

[5*n-2+(2*n-(-1)^n-3)/4 for n in (1..60)]

O.g.f.: x*(3 + 5*x + 3*x^2)/((1 + x)*(1 - x)^2).
E.g.f.: (-1 + 12*exp(x) - 11*exp(2*x) + 22*x*exp(2*x))*exp(-x)/4.
a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).
a(n) = 5*n - 2 + (2*n - (-1)^n - 3)/4.
a(n) = 4*n - 1 + floor((n - 1)/2) + floor((3*n - 1)/3).
a(n+k) - a(n) = 11*k/2 + (1 - (-1)^k)*(-1)^n/4.
a(n+k) + a(n) = 11*(2*n + k - 1)/2 - (1 + (-1)^k)*(-1)^n/4.
E.g.f.: 3 + ((22*x - 11)*exp(x) - exp(-x))/4. - David Lovler, Aug 08 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(5*Pi/22)*Pi/11. - Amiram Eldar, Feb 27 2023
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = cosec(3*Pi/22)/2.
Product_{n>=1} (1 + (-1)^n/a(n)) = sec(5*Pi/22)*sin(2*Pi/11). (End)

A308378 Numbers k such that phi(2k+1) = phi(2k+2).

Original entry on oeis.org

0, 1, 7, 127, 247, 487, 1312, 1627, 1852, 2593, 5857, 6682, 9157, 11467, 12772, 23107, 24607, 24667, 28822, 32767, 82087, 92317, 99157, 107887, 143497, 153697, 159637, 194122, 198742, 207637, 245767, 284407, 294703, 343492, 420127

Offset: 1

Torlach Rush, May 24 2019

nonn

For n > 0, 2*a(n) + 1 is a term of A020884. This is because 2*a(n) + 1 is odd and every odd number is the difference of the squares of two consecutive numbers and hence are coprime.
For n > 0, (2*a(n) + 1) * (2*a(n) + 2) is a term of A024364. This is because (2*a(n) + 1) * (2*a(n) + 2) = 2*((a(n) + 1)^2 + (a(n) + 1) * a(n)) and gcd((a(n) + 1), a(n)) = 1.
For n > 0, a(n) is congruent to 1 or 4 mod 6.
2*a(n) + 1 is congruent to 1 or 3 mod 6 and is a term of A047241.
2*a(n) + 2 is congruent to 2 or 4 mod 6 and is a term of A047235.

			0 is a term because phi(1) = phi(2) = 1.
1 is a term because phi(3) = phi(4) = 2.
7 is a term because phi(15) = phi(16) = 8.

Amiram Eldar, Table of n, a(n) for n = 1..5416 (calculated using the b-file at A001274)

Cf. A000010, A004767, A020884, A024364, A047235, A047241, A299535.
Subset of A047234.
Subset of A001274.

Select[Range[0, 9999], EulerPhi[2# + 1] == EulerPhi[2# + 2] &] (* Alonso del Arte, Jul 05 2019 *)
Select[(#-1)/2&/@SequencePosition[EulerPhi[Range[900000]],{x_,x_}][[All,1]],IntegerQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 24 2019 *)

lista(nn) = for(n=0, nn, if(eulerphi(2*n+1) == eulerphi(2*n+2), print1(n, ", ")));
lista(430000)

a(n) = (A299535(n) - 2) / 2.

cp's OEIS Frontend

A291050 Decimal expansion of Pi^2 / 27.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A360735 Even integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has only two elements.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A091738 Primes arising in the second row of array in A091734.

Views

Author

Keywords

Formula

Extensions

A248323 Numbers n which appear at least twice in A037278(n), concatenation of their divisors written in base 10.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

A086382 k divides F(k*n^2+1)-F(k+1) for 1<=k<=a(n) where F(k) is the k-th Fibonacci number.

Views

Author

Keywords

Comments

Links

Programs

Maple

PARI

Formula

A119981 a(n) = 1 iff number congruent to {2, 4} mod 6 is equal to prime minus 3, otherwise a(n)=0.

Views

Author

Keywords

Comments

Crossrefs

A181156 Odd Fibonacci numbers F which have a proper Fibonacci divisor G such that F/G is a Lucas number or a product of Lucas numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

A255846 a(n) = 2*n^2 + 14.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

Extensions

A299647 Positive solutions to x^2 == -2 (mod 11).