A197652 -id:A197652 - cp's OEIS frontend

A260181 Numbers whose last digit is prime.

2, 3, 5, 7, 12, 13, 15, 17, 22, 23, 25, 27, 32, 33, 35, 37, 42, 43, 45, 47, 52, 53, 55, 57, 62, 63, 65, 67, 72, 73, 75, 77, 82, 83, 85, 87, 92, 93, 95, 97, 102, 103, 105, 107, 112, 113, 115, 117, 122, 123, 125, 127, 132, 133, 135, 137, 142, 143, 145, 147

Offset: 1

Wesley Ivan Hurt, Jul 17 2015

Numbers ending in 2, 3, 5 or 7.
The subsequence of primes is A042993. - Michel Marcus, Jul 19 2015
From Wesley Ivan Hurt, Aug 15 2015, Sep 26 2015: (Start)
Ceiling(a(n)/2) = A047201(n).
Complement of (A197652 Union A262389). (End)

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A042993, A047201, A092620, subset of A118950.
Union of A017293, A017305, A017329 and A017353.
First differences are [1,2,2,5,...] = A002522(A140081(n-1)).
Cf. A197652, A262389.

a:=n->(5*n-4-(-1)^n+((3-(-1)^n)/2)*(-1)^((2*n+5-(-1)^n)/4))/2; List([1..60],n->a(n)); # Muniru A Asiru, Feb 16 2018

[(5*n-4-(-1)^n+((3-(-1)^n) div 2)*(-1)^((2*n+5-(-1)^n) div 4))/2: n in [1..70]]; // Vincenzo Librandi, Jul 18 2015

A260181:=n->(5*n-4-(-1)^n+((3-(-1)^n)/2)*(-1)^((2*n+5-(-1)^n)/4))/2: seq(A260181(n), n=1..100);

CoefficientList[Series[(2 + x + 2 x^2 + 2 x^3 + 3 x^4)/((x - 1)^2*(1 + x + x^2 + x^3)), {x, 0, 100}], x]
LinearRecurrence[{1, 0, 0, 1, -1}, {2, 3, 5, 7, 12}, 60] (* Vincenzo Librandi, Jul 18 2015 *)
Table[(5n - 4 - (-1)^n + ((3 - (-1)^n)/2)*(-1)^((2*n + 5 - (-1)^n)/4))/2, {n, 100}] (* Wesley Ivan Hurt, Aug 11 2015 *)

is(n)=my(m=digits(n));isprime(m[#m]) \\ Anders Hellström, Jul 19 2015

A260181(n)=(n--)\4*10+prime(n%4+1) \\ is(n)=isprime(n%10) is much more efficient than the above. - M. F. Hasler, Sep 16 2016

G.f.: x*(2+x+2*x^2+2*x^3+3*x^4) / ((x-1)^2*(1+x+x^2+x^3)).
a(n) = a(n-1)+a(n-4)-a(n-5), n>5.
a(n) = (5*n-4-(-1)^n+((3-(-1)^n)/2)*(-1)^((2*n+5-(-1)^n)/4))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(5*sqrt(5+2*sqrt(5))) - 25*log(5) - 40*log(2) + 5*sqrt(5)*arccoth(843/2))/200. - Amiram Eldar, Jul 30 2024

A219259 Numbers k such that 25*k+1 is a square.

Original entry on oeis.org

0, 23, 27, 96, 104, 219, 231, 392, 408, 615, 635, 888, 912, 1211, 1239, 1584, 1616, 2007, 2043, 2480, 2520, 3003, 3047, 3576, 3624, 4199, 4251, 4872, 4928, 5595, 5655, 6368, 6432, 7191, 7259, 8064, 8136, 8987, 9063, 9960, 10040, 10983, 11067, 12056, 12144

Offset: 1

Bruno Berselli, Nov 19 2012

Equivalently, numbers of the form m*(25*m+2), where m = 0,-1,1,-2,2,-3,3,...
Also, integer values of h*(h+2)/25.
Exponents in the expansion of Product_{n >= 1} (1 - q^(50*n))*(1 - q^(50*n-23))*(1 - q^(50*n-27)) = 1 - q^23 - q^27 + q^96 + q^104 - q^219 - q^231 + + - - .... - Peter Bala, Dec 18 2024

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. similar sequences listed in A219257.

Cf. A047209, A197652.

[n: n in [0..13000] | IsSquare(25*n+1)];

I:=[0,23,27,96,104]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

A219259:=proc(q)
local n;
for n from 1 to q do if type(sqrt(25*n+1), integer) then print(n);
fi; od; end:
A219259(1000); # Paolo P. Lava, Feb 19 2013

Select[Range[0, 13000], IntegerQ[Sqrt[25 # + 1]] &]
CoefficientList[Series[x (23 + 4 x + 23 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)

G.f.: x^2*(23 + 4*x + 23*x^2)/((1 + x)^2*(1 - x)^3).
a(n) = a(-n+1) = (50*n*(n-1) + 21*(-1)^n*(2*n - 1) + 5)/8 + 2.
25*a(n)+1 = A047209(A197652(n+1))^2.
Sum_{n>=2} 1/a(n) = 25/4 - cot(2*Pi/25)*Pi/2. - Amiram Eldar, Mar 17 2022

A262389 Numbers whose last digit is composite.

Original entry on oeis.org

4, 6, 8, 9, 14, 16, 18, 19, 24, 26, 28, 29, 34, 36, 38, 39, 44, 46, 48, 49, 54, 56, 58, 59, 64, 66, 68, 69, 74, 76, 78, 79, 84, 86, 88, 89, 94, 96, 98, 99, 104, 106, 108, 109, 114, 116, 118, 119, 124, 126, 128, 129, 134, 136, 138, 139, 144, 146, 148, 149

Offset: 1

Wesley Ivan Hurt, Sep 21 2015

Numbers ending in 4, 6, 8 or 9.
Union of A017317, A017341, A017365 and A017377.
Subsequence of A118951 (numbers containing at least one composite digit).
Complement of (A197652 Union A260181).

Gerald Hillier and Didier Lachieze, Last Digit Composite.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A017317, A017341, A017365, A017377.

Cf. A118951, A197652, A260181 (last digit is prime).

[(5*n+1-(-1)^n+(3+(-1)^n)*(-1)^((2*n-3-(-1)^n) div 4) div 2) div 2: n in [1..70]]; // Vincenzo Librandi, Sep 21 2015

A262389:=n->(5*n+1-(-1)^n+(3+(-1)^n)*(-1)^((2*n-3-(-1)^n)/4)/2)/2: seq(A262389(n), n=1..100);

Table[(5n+1-(-1)^n+(3+(-1)^n)*(-1)^((2n-3-(-1)^n)/4)/2)/2, {n, 100}]
LinearRecurrence[{1, 0, 0, 1, -1}, {4, 6, 8, 9, 14}, 80] (* Vincenzo Librandi, Sep 21 2015 *)
CoefficientList[Series[(4 + 2*x + 2*x^2 + x^3 + x^4)/((x - 1)^2*(1 + x + x^2 + x^3)), {x, 0, 80}], x] (* Wesley Ivan Hurt, Sep 21 2015 *)
Select[Range[200],CompositeQ[Mod[#,10]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 21 2019 *)

G.f.: x*(4+2*x+2*x^2+x^3+x^4)/((x-1)^2*(1+x+x^2+x^3)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (5*n+1-(-1)^n+(3+(-1)^n)*(-1)^((2*n-3-(-1)^n)/4)/2)/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(10-2*sqrt(5))*Pi - sqrt(5)*arccoth(3/sqrt(5)) - 4*log(2))/20. - Amiram Eldar, Jul 30 2024

Name edited by Jon E. Schoenfield, Feb 15 2018

A293292 Numbers with last digit less than 5 (in base 10).

Original entry on oeis.org

0, 1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 30, 31, 32, 33, 34, 40, 41, 42, 43, 44, 50, 51, 52, 53, 54, 60, 61, 62, 63, 64, 70, 71, 72, 73, 74, 80, 81, 82, 83, 84, 90, 91, 92, 93, 94, 100, 101, 102, 103, 104, 110, 111, 112, 113, 114, 120, 121, 122, 123, 124, 130

Offset: 1

Bruno Berselli, Oct 05 2017

Equivalently, numbers k such that floor(k/5) = 2*floor(k/10).
After 0, partial sums of A010122 starting from the 2nd term.
The sequence differs from A007091 after a(25).
Also numbers k such that floor(k/5) is even. - Peter Luschny, Oct 05 2017

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A010122, A239229, A257145, A293481 (complement).
Sequences of the type floor(n/d) = (10/d)*floor(n/10), where d is a factor of 10: A008592 (d=1), A197652 (d=2), this sequence (d=5), A001477 (d=10).
Sequences of the type n + r*floor(n/r): A005843 (r=1), A042948 (r=2), A047240 (r=3), A047476 (r=4), this sequence (r=5).

Magma
```
[n: n in [0..130] | n mod 10 lt 5];
```

[n: n in [0..130] | IsEven(Floor(n/5))];

Magma
```
[n+5*Floor(n/5): n in [0..70]];
```

select(k -> type(floor(k/5), even), [$0..130]); # Peter Luschny, Oct 05 2017

Table[n + 5 Floor[n/5], {n, 0, 70}]
Reap[For[k = 0, k <= 130, k++, If[Floor[k/5] == 2*Floor[k/10], Sow[k]]]][[2, 1]] (* or *) LinearRecurrence[{1, 0, 0, 0, 1, -1}, {0, 1, 2, 3, 4, 10}, 66] (* Jean-François Alcover, Oct 05 2017 *)

concat(0, Vec(x^2*(1 + x + x^2 + x^3 + 6*x^4) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)) + O(x^70))) \\ Colin Barker, Oct 05 2017

select(k->floor(k/5) == 2*floor(k/10), vector(1000, k, k)) \\ Colin Barker, Oct 05 2017

[k for k in range(131) if (k//5) % 2 == 0] # Peter Luschny, Oct 05 2017

def A293292(n): return (n-1<<1)-(n-1)%5 # Chai Wah Wu, Oct 29 2024

[k for k in (0..130) if 2.divides(floor(k/5))] # Peter Luschny, Oct 05 2017

G.f.: x^2*(1 + x + x^2 + x^3 + 6*x^4)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-5) - a(n-6).
a(n) = (n-1) + 5*floor((n-1)/5) = 10*floor((n-1)/5) + ((n-1) mod 5).
a(n) = A257145(n+2) - A239229(n-1). - R. J. Mathar, Oct 05 2017
a(n) = 2n-2-((n-1) mod 5). - Chai Wah Wu, Oct 29 2024

Definition by David A. Corneth, Oct 05 2017

A266297 Numbers whose last digit is a square.

Original entry on oeis.org

0, 1, 4, 9, 10, 11, 14, 19, 20, 21, 24, 29, 30, 31, 34, 39, 40, 41, 44, 49, 50, 51, 54, 59, 60, 61, 64, 69, 70, 71, 74, 79, 80, 81, 84, 89, 90, 91, 94, 99, 100, 101, 104, 109, 110, 111, 114, 119, 120, 121, 124, 129, 130, 131, 134, 139, 140, 141, 144, 149

Offset: 1

Wesley Ivan Hurt, Dec 26 2015

Numbers ending in 0, 1, 4 and 9.
Union of A008592, A017281, A017317 and A017377. - Hurt
None of these numbers are prime in Z[phi] (where phi = 1/2 + sqrt(5)/2 is the golden ratio), since the numbers in this sequence that are prime in Z can be expressed in the form (a - b sqrt(5))(a + b sqrt(5)). - Alonso del Arte, Dec 30 2015
Union of A197652 and A016897. - Wesley Ivan Hurt, Dec 31 2015
Union of A146763 and A090771. - Wesley Ivan Hurt, Jan 01 2016

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A008592, A016897, A017281, A017317, A017377, A090771, A091084, A141158, A146763, A197652.

[(10*n-11+(-1)^n+(4+2*(-1)^n)*(-1)^((2*n-1+(-1)^n) div 4))/4: n in [1..60]]; // Vincenzo Librandi, Dec 27 2015

A266297:=n->(10*n-11+(-1)^n+(4+2*(-1)^n)*(-1)^((2*n-1+(-1)^n)/4))/4: seq(A266297(n), n=1..100);

Table[(10 n - 11 + (-1)^n + (4 + 2 (-1)^n)*(-1)^((2 n - 1 + (-1)^n)/4))/4, {n, 50}] (* G. C. Greubel, Dec 27 2015 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 1, 4, 9, 10}, 60] (* Vincenzo Librandi, Dec 27 2015 *)
CoefficientList[Series[x*(1 + 3*x + 5*x^2 + x^3)/((x - 1)^2*(1 + x + x^2 + x^3)), {x, 0, 100}], x] (* Wesley Ivan Hurt, Dec 30 2015 *)
Flatten[Table[10n + {0, 1, 4, 9}, {n, 0, 19}]] (* Alonso del Arte, Dec 30 2015 *)
Select[Range[0,150],MemberQ[{0,1,4,9},Mod[#,10]]&] (* Harvey P. Dale, Jul 30 2019 *)

is(n) = issquare(n%10); \\ Altug Alkan, Dec 29 2015

G.f.: x^2*(1 + 3*x + 5*x^2 + x^3)/((x - 1)^2*(1 + x + x^2 + x^3)).
a(n) = a(n - 1) + (n - 4) - a(n - 5) for n > 5.
a(n) = (10n - 11 + (-1)^n + (4 + 2(-1)^n) * (-1)^((2n - 1 + (-1)^n)/4))/4.
a(n+1) - a(n) = A091084(n+1) for n>0.
Sum_{n>=2} (-1)^n/a(n) = (14*sqrt(5)*arccoth(sqrt(5)) - 2*Pi*sqrt(1-2/sqrt(5)) + 16*log(2) + 5*log(5))/40. - Amiram Eldar, Jul 30 2024

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A260181 Numbers whose last digit is prime.

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A219259 Numbers k such that 25*k+1 is a square.

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A262389 Numbers whose last digit is composite.

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A293292 Numbers with last digit less than 5 (in base 10).

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A266297 Numbers whose last digit is a square.

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