Amir H. Farrahi - cp's OEIS frontend

A185949 Smallest prime ending in 10^n+1 in its base-10 representation.

11, 101, 21001, 1810001, 2100001, 61000001, 2010000001, 11100000001, 61000000001, 1810000000001, 14100000000001, 151000000000001, 5010000000000001, 9100000000000001, 271000000000000001, 1110000000000000001, 24100000000000000001, 261000000000000000001, 3910000000000000000001, 11100000000000000000001

Offset: 1

Amir H. Farrahi, Feb 07 2011

Robert Israel, Table of n, a(n) for n = 1..995

f:= proc(n) local p;
  for p from 10^n+1 by 10^(n+1) do
    if isprime(p) then return p fi
  od
end proc:
map(f, [$1..30]); # Robert Israel, May 03 2018

Table[k=0; While[!PrimeQ[p=FromDigits[Join[IntegerDigits[k], IntegerDigits[10^n+1]]]], k++]; p, {n,20}]

from sympy import isprime as is_prime
# This implementation assumes function is_prime(n)
# returns True if n is prime, or False otherwise:
for n in range (1, 100):
    pattern = 10**n + 1
    for j in range (0, 10000000):
        if (j == 0):
            num = "%d" % (pattern)
        else:
            num = "%d%d" % (j, pattern)
        if (is_prime(int(num))):
            print(num)
            break

A185940 a(n) = 1 - 2^(n+1) + 3^(n+2).

Original entry on oeis.org

24, 74, 228, 698, 2124, 6434, 19428, 58538, 176124, 529394, 1590228, 4774778, 14332524, 43013954, 129074628, 387289418, 1161999324, 3486260114, 10459304628, 31378962458, 94138984524, 282421147874, 847271832228, 2541832273898, 7625530376124, 22876658237234

Offset: 1

Amir H. Farrahi, Feb 06 2011

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A000079, A000244, A066280.

[1 - 2^(n+1) + 3^(n+2): n in [1..40]]; // Vincenzo Librandi, Apr 05 2011

A185940:=n->1-2^(n+1)+3^(n+2): seq(A185940(n), n=1..40); # Wesley Ivan Hurt, Jul 23 2017

CoefficientList[Series[-2*x*(12 - 35*x + 24*x^2)/(-1 + 6*x - 11*x^2 + 6*x^3), {x,0,50}], x] (* or *) LinearRecurrence[{6, -11, 6}, {24, 74, 228}, 50] (* G. C. Greubel, Feb 25 2017 *)

x='x+O('x^50); Vec(-2*x*(12 - 35*x + 24*x^2) / (-1 + 6*x - 11*x^2 + 6*x^3)) \\ G. C. Greubel, Feb 25 2017

a(n) = 1 - A000079(n+1) + A000244(n+2)
From Alexander R. Povolotsky, Jan 07 2011: (Start)
G.f.: 2*x*(12 - 35*x + 24*x^2) / (1 - 6*x + 11*x^2 - 6*x^3)
a(n+2) = -6*a(n) + 5*a(n+1)+2. (End)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3). - G. C. Greubel, Feb 25 2017
E.g.f.: exp(x) - 2*exp(2*x) + 9*exp(3*x) - 8. - G. C. Greubel, Jul 23 2017

Corrected and edited by Bruno Berselli, Apr 04 2011

A185939 a(n) = 9n^2 - 6n + 2.

Original entry on oeis.org

5, 26, 65, 122, 197, 290, 401, 530, 677, 842, 1025, 1226, 1445, 1682, 1937, 2210, 2501, 2810, 3137, 3482, 3845, 4226, 4625, 5042, 5477, 5930, 6401, 6890, 7397, 7922, 8465, 9026, 9605, 10202, 10817, 11450

Offset: 1

Amir H. Farrahi, Feb 06 2011

Group the set of natural numbers in set of 3 (1, 2, 3; 4, 5, 6; 7, 8, 9; ...) In each group, multiply the first two numbers and then add the third number to the result to get the corresponding entry in our sequence.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

CoefficientList[Series[-x*(x + 5)*(2*x + 1)/(x - 1)^3, {x,0,50}], x] (* or *) LinearRecurrence[{3, -3, 1}, {5, 26, 65}, 50] (* G. C. Greubel, Feb 25 2017 *)
Table[9n^2-6n+2,{n,40}] (* or *) #[[1]]#[[2]]+#[[3]]&/@Partition[Range[111],3]  (* Harvey P. Dale, Apr 08 2022 *)

x='x+O('x^50); Vec(-x*(x+5)*(2*x+1)/(x-1)^3) \\ G. C. Greubel, Feb 25 2017

G.f. -x*(x+5)*(2*x+1) / (x-1)^3 . - Alexander R. Povolotsky, Feb 06 2011
a(n) = a(n-1) + 18*n - 15, a(1) = 5. - Vincenzo Librandi, Feb 07 2011
a(n) = (2*n-1)^2 + (2*n)^2 + (n-1)^2. - Bruno Berselli, Feb 06 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - G. C. Greubel, Feb 25 2017
E.g.f.: (9*x^2 + 3*x + 2)*exp(x) - 2. - G. C. Greubel, Jul 23 2017

A185826 Sum of the next n natural numbers raised to the n-th power.

Original entry on oeis.org

1, 25, 3375, 1336336, 1160290625, 1870414552161, 5026507568359375, 20882706457600000000, 126834469112674289266929, 1078732544346879404306640625, 12415028528548173886807771291871, 188031682201497672618081000000000000, 3661926425131437024691115607984619140625

Offset: 1

Amir H. Farrahi, Feb 05 2011

nonn

Write the natural numbers in groups, with group sizes incremented by one, each time: 1; 2,3; 4,5,6; 7,8,9,10; ... and add the numbers in each group, then take the i-th power for the i-th group to get the i-th entry in the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..158

Cf. A006003.

Module[{nn=15},#[[1]]^#[[2]]&/@Thread[{Total/@TakeList[Range[(nn(nn+1))/2],Range[ nn]],Range[nn]}]] (* or *) Table[((n+n^3)/2)^n,{n,20}] (* Harvey P. Dale, Apr 24 2022 *)

num = 100
n = 0
a = []
for i in range(1, num):
    sum = 0
    for j in range(1, i+1):
        sum = sum + (n+j)
    n = n + i
    a.append(sum**i)

a(n) = A006003(n)^n.

a(n) = ((n + n^3)/2)^n. - Harvey P. Dale, Apr 24 2022

Edited by N. J. A. Sloane, Feb 05 2011

A181428 a(n) = prime(n+2) + prime(n+1) - prime(n).

Original entry on oeis.org

6, 9, 13, 17, 19, 23, 25, 33, 37, 39, 47, 47, 49, 57, 65, 67, 69, 77, 77, 81, 89, 93, 103, 109, 107, 109, 113, 115, 131, 145, 141, 145, 151, 161, 159, 169, 173, 177, 185, 187, 193, 203, 199, 203, 213, 235, 239, 233, 235, 243, 247, 253, 267, 269, 275, 277, 279, 287, 287, 295, 317, 325, 317, 319, 335, 351, 353

Offset: 1

Amir H. Farrahi, Jan 29 2011

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

A181428 := proc(n)
    ithprime(n+1)-ithprime(n)+ithprime(n+2) ;
end proc:
seq(A181428(n),n=1..80) ; # R. J. Mathar, Sep 10 2016

#[[3]]+#[[2]]-#[[1]]&/@Partition[Prime[Range[100]],3,1]  (* Harvey P. Dale, Feb 01 2011 *)
ListConvolve[{1, 1, -1}, Prime[Range[100]]]

first(n)=my(v=primes(n+2)); for(k=1,n, v[k]=v[k+2]+v[k+1]-v[k]); v[1..n] \\ Charles R Greathouse IV, Feb 23 2017

cp's OEIS Frontend

User: Amir H. Farrahi

A185949 Smallest prime ending in 10^n+1 in its base-10 representation.

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A185940 a(n) = 1 - 2^(n+1) + 3^(n+2).

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A185939 a(n) = 9*n^2 - 6*n + 2.

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A185826 Sum of the next n natural numbers raised to the n-th power.

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A181428 a(n) = prime(n+2) + prime(n+1) - prime(n).

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A185939 a(n) = 9n^2 - 6n + 2.