A017053 -id:A017053 - cp's OEIS frontend

A031373 a(n) = prime(7n-1).

13, 41, 71, 103, 139, 179, 223, 257, 293, 347, 383, 431, 463, 509, 569, 607, 647, 691, 743, 797, 839, 883, 941, 991, 1033, 1087, 1123, 1187, 1231, 1289, 1321, 1409, 1451, 1489, 1549, 1597, 1627, 1697, 1747, 1801, 1871, 1913, 1987, 2027

Offset: 1

Jeff Burch

nonn

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. A000040, A017053.

[ NthPrime(7*n-1): n in [1..1000] ]; // Vincenzo Librandi, Apr 10 2011

Prime[7Range[50]-1] (* Harvey P. Dale, Oct 31 2011 *)

A137188 Lucky numbers (A000959) which are congruent to 6 mod 7.

Original entry on oeis.org

13, 69, 111, 195, 223, 237, 307, 321, 349, 391, 433, 475, 489, 517, 559, 601, 615, 643, 685, 699, 727, 741, 769, 867, 895, 937, 979, 993, 1021, 1105, 1147, 1189, 1203, 1231, 1245, 1329, 1357, 1441, 1455, 1497, 1567, 1693, 1749, 1777, 1819, 1833, 1945, 1959

Offset: 1

N. J. A. Sloane, Mar 07 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A000959 and A017053.

A144204 Array A(k,n) = (n+k-2)*(n-1) - 1 (k >= 1, n >= 1) read by antidiagonals.

Original entry on oeis.org

-1, -1, 0, -1, 1, 3, -1, 2, 5, 8, -1, 3, 7, 11, 15, -1, 4, 9, 14, 19, 24, -1, 5, 11, 17, 23, 29, 35, -1, 6, 13, 20, 27, 34, 41, 48, -1, 7, 15, 23, 31, 39, 47, 55, 63, -1, 8, 17, 26, 35, 44, 53, 62, 71, 80, -1, 9, 19, 29, 39, 49, 59, 69, 79, 89, 99, -1, 10, 21, 32, 43, 54, 65, 76, 87

Offset: 1

Jonathan Vos Post, Sep 13 2008

Arises in complete intersection threefolds,
Also can be produced as a triangle read by rows: a(n, k) = nk - (n + k). - Alonso del Arte, Jul 09 2009
Kosta: Let X be a complete intersection of two hypersurfaces F_n and F_k in the projective space P^5 of degree n and k respectively. with n=>k, such that the singularities of X are nodal and F_k is smooth. We prove that if the threefold X has at most (n+k-2)*(n-1) - 1 singular points, then it is factorial.

			From _R. J. Mathar_, Jul 10 2009: (Start)
The rows A(n,1), A(n,2), A(n,3), etc., are :
.-1...0...3...8..15..24..35..48..63..80..99.120.143.168 A067998
.-1...1...5..11..19..29..41..55..71..89.109.131.155.181 A028387
.-1...2...7..14..23..34..47..62..79..98.119.142.167.194 A008865
.-1...3...9..17..27..39..53..69..87.107.129.153.179.207 A014209
.-1...4..11..20..31..44..59..76..95.116.139.164.191.220 A028875
.-1...5..13..23..35..49..65..83.103.125.149.175.203.233 A108195
.-1...6..15..26..39..54..71..90.111.134.159.186.215.246
.-1...7..17..29..43..59..77..97.119.143.169.197.227.259
.-1...8..19..32..47..64..83.104.127.152.179.208.239.272
.-1...9..21..35..51..69..89.111.135.161.189.219.251.285
.-1..10..23..38..55..74..95.118.143.170.199.230.263.298
.-1..11..25..41..59..79.101.125.151.179.209.241.275.311
.-1..12..27..44..63..84.107.132.159.188.219.252.287.324
.-1..13..29..47..67..89.113.139.167.197.229.263.299.337 Cf. A126719.
(End)
As a triangle:
. 0
. 1, 3
. 2, 5, 8
. 3, 7, 11, 15
. 4, 9, 14, 19, 24
. 5, 11, 17, 23, 29, 35
. 6, 13, 20, 27, 34, 41, 48
. 7, 15, 23, 31, 39, 47, 55, 63
. 8, 17, 26, 35, 44, 53, 62, 71, 80

Dimitra Kosta, Factoriality of complete intersection threefolds arXiv:0808.4071 [math.AG]

Row 1 = A067998(n) for n>0. Row 2 = A028387(n) for n>0.Column 1 = -A000012(n). Column 2 = A001477. Column 3 = A005408(k). Column 4 = A016789(k+1). Column 5 = A004767(k+2). Column 6 = A016897(k+3). Column 7 = A016969(k+4). Column 8 = A017053(k+5). Column 9 = A004771(k+6). Column 10 = A017257(k+7).

Cf. A000012, A001477, A004767, A004771, A005408, A016789, A016897, A016969, A017053, A028387, A067998, A126719.

A := proc(k,n) (n+k-2)*(n-1)-1 ; end: for d from 1 to 13 do for n from 1 to d do printf("%d,",A(d-n+1,n)) ; od: od: # R. J. Mathar, Jul 10 2009

a[n_, k_] := a[n, k] = n*k - (n + k); ColumnForm[Table[a[n, k], {n, 10}, {k, n}], Center] (* Alonso del Arte, Jul 09 2009 *)

A[k,n] = (n+k-2)*(n-1) - 1.

Antidiagonal sum: Sum_{n=1..d} A(d-n+1,n) = d*(d^2-2d-1)/2 = -A110427(d). - R. J. Mathar, Jul 10 2009

Duplicate of 6th antidiagonal removed by R. J. Mathar, Jul 10 2009
Keyword:tabl added by R. J. Mathar, Jul 23 2009
Edited by N. J. A. Sloane, Sep 14 2009. There was a comment that the defining formula was wrong, but it is perfectly correct.

A253671 a(n) = floor(A000111(n)/A000111(n-1)).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 8, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 15, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 29, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 36, 37, 38, 38, 39, 40, 40

Offset: 1

Paul Curtz, Jan 08 2015, with the help of Jean-François Alcover

1, 2, 3, 4, ... first appear at n = 1, 3, 5, 7, 8, 10, 11, 13, ... . a(500) = 318.
Numbers appearing only once: interleave 4+7*n, 6+7*n, 9+7*n = 4, 6, 9, 11, 13, 16, ... .
This is a nondecreasing sequence.
The ratio a(n)/n asymptotically tends to 7/11 = 0.6363... - Jean-François Alcover, Jul 21 2015

			Floor of 1/1, 1/1, 2/1, 5/2, 16/5, 61/16, ... .
1=1*1+0, 1=1*1+0, 2=2*1+0, 5=2*2+1, 16=3*5+1, 61=3*16+13, 272=4*61+28, ... .

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

Cf. A000111, A010727, A017005, A017029, A017053.

max = 500; ee = Table[2^n*EulerE[n, 1] + EulerE[n] - 1, {n, 0, max}]; A000111 = Table[Differences[ee, n] // First // Abs, {n, 0, max}]; Table[Quotient[A000111[[n + 1]], A000111[[n]]], {n, 1, max}] (* Jean-François Alcover, Jan 08 2015 *)

Vec(x*(x^14-x^13+x^12-x^11+x^10+x^9+x^7+x^6+x^4+x^2+1)/((x-1)^2*(x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)) + O(x^100)) \\ Colin Barker, Jan 22 2015

# requires python 3.2 or higher
from itertools import accumulate
A253671_list, blist, l1, l2 = [1], [1], 1, 1
for n in range(10**2):
    blist = list(reversed(list(accumulate(reversed(blist))))) + [0] if n % 2 else [0]+list(accumulate(blist))
    l2, l1 = l1, sum(blist)
    A253671_list.append(l1//l2) # Chai Wah Wu, Jan 29 2015

a(n+2) = a(n+1) + (0, 1, 0, followed by a sequence of period 11: repeat 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1).
a(n+12) = a(n+1) + (6, 7, 6, followed by 7's = A010727).
a(n) = a(n-1) + a(n-11) - a(n-12) for n>15. - Colin Barker, Jan 22 2015
G.f.: x*(x^14-x^13+x^12-x^11+x^10+x^9+x^7+x^6+x^4+x^2+1) / ((x-1)^2*(x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)). - Colin Barker, Jan 22 2015

A358053 a(n) = 14*n - 1.

Original entry on oeis.org

13, 27, 41, 55, 69, 83, 97, 111, 125, 139, 153, 167, 181, 195, 209, 223, 237, 251, 265, 279, 293, 307, 321, 335, 349, 363, 377, 391, 405, 419, 433, 447, 461, 475, 489, 503, 517, 531, 545, 559, 573, 587, 601, 615, 629, 643, 657, 671, 685, 699, 713, 727, 741, 755, 769, 783, 797

Offset: 1

Leo Tavares, Oct 27 2022

This sequence can be illustrated by a star outline with a central row of counters.

Subsequence of primes is A045473. - Bernard Schott, Jan 25 2023

Paolo Xausa, Table of n, a(n) for n = 1..10000
Leo Tavares, Illustration: Mid-line Stars.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008596, A003154, A017053, A045473, A045944, A202804.

14*Range[100] - 1 (* Paolo Xausa, Oct 04 2024 *)

a(n) = 14*n - 1.
a(n) = A003154(n+1) - A202804(n-1).
a(n) = A003154(n+1) - 2*A045944(n-1).
From Elmo R. Oliveira, Apr 03 2025: (Start)
G.f.: x*(13 + x)/(x - 1)^2.
E.g.f.: exp(x)*(14*x - 1) + 1.
a(n) = A017053(2*n-1).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

cp's OEIS Frontend

A031373 a(n) = prime(7n-1).

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A137188 Lucky numbers (A000959) which are congruent to 6 mod 7.

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A144204 Array A(k,n) = (n+k-2)*(n-1) - 1 (k >= 1, n >= 1) read by antidiagonals.

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A253671 a(n) = floor(A000111(n)/A000111(n-1)).

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A358053 a(n) = 14*n - 1.

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