A017257 -id:A017257 - cp's OEIS frontend

A031375 a(n) = prime(9n-1).

19, 59, 101, 149, 193, 241, 293, 353, 409, 461, 521, 587, 641, 691, 757, 823, 881, 947, 1013, 1063, 1123, 1201, 1277, 1319, 1423, 1471, 1531, 1597, 1657, 1723, 1789, 1873, 1949, 2011, 2083, 2141, 2237, 2293, 2357, 2417, 2503, 2591, 2663

Offset: 1

Jeff Burch

nonn

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

Cf. A000040, A017257.

[ NthPrime(9*n-1): n in [1..1000] ]; // Vincenzo Librandi, Apr 08 2011

Prime[9*Range[50]-1] (* Harvey P. Dale, Aug 19 2020 *)

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.

A168409 a(n) = 8 + 9*floor((n-1)/2).

Original entry on oeis.org

8, 8, 17, 17, 26, 26, 35, 35, 44, 44, 53, 53, 62, 62, 71, 71, 80, 80, 89, 89, 98, 98, 107, 107, 116, 116, 125, 125, 134, 134, 143, 143, 152, 152, 161, 161, 170, 170, 179, 179, 188, 188, 197, 197, 206, 206, 215, 215, 224, 224, 233, 233, 242, 242, 251, 251, 260

Offset: 1

Vincenzo Librandi, Nov 25 2009

Old definition: "a(n)=9*n-a(n-1)-2 (with a(1)=8)".

The sequences generated by the recursive relation b(n) = h*n-b(n-1)+k, with b(1)=c and h, k, c prefixed integers, satisfy the formula b(n) = c + h*floor((n-1)/2) when 2*h+k = 2*c. - Bruno Berselli, Sep 18 2013

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

Cf. A017257.

[8+9*Floor((n-1)/2): n in [1..70]]; // Vincenzo Librandi, Sep 18 2013

RecurrenceTable[{a[1]==8,a[n]==9n-a[n-1]-2},a,{n,60}] (* or *) LinearRecurrence[{1,1,-1},{8,8,17},80] (* or *) With[{c = LinearRecurrence[{2,-1},{8,17},30]},Riffle[c,c]] (* Harvey P. Dale, Aug 18 2013 *)
CoefficientList[Series[(8 + x^2)/((1 + x) (x - 1)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Sep 18 2013 *)

a(n)=(n-1)\2*9 + 8 \\ Charles R Greathouse IV, Jul 21 2016

a(n) = 9*n - a(n-1) - 2, with n>1, a(1)=8.
a(1)=8, a(2)=8, a(3)=17; for n>1, a(n) = a(n-1) +a(n-2) -a(n-3). - Harvey P. Dale, Aug 18 2013
G.f.: x*(8 + x^2)/((1+x)*(x-1)^2). - Vincenzo Librandi, Sep 18 2013
E.g.f.: (1/4)*(-9 + 4*exp(x) + (5 + 18*x)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 21 2016

New definition by Vincenzo Librandi, Sep 18 2013

A295869 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 8.

Original entry on oeis.org

17, 53, 71, 89, 107, 143, 161, 179, 197, 233, 251, 269, 287, 323, 341, 359, 377, 413, 431, 449, 467, 503, 521, 539, 557, 593, 611, 629, 647, 683, 701, 719, 737, 773, 791, 809, 827, 863, 881, 899, 917, 953, 971, 989, 1007, 1043, 1061, 1079, 1097, 1133

Offset: 1

Gary Croft, Mar 24 2018

Numbers == {17, 53, 71, 89} mod 90 with additive sum sequence 17{+36+18+18+18} {repeat ...}. Includes all prime numbers >5 with digital root 8.

			17+36=53; 53+18=71; 71+18=89; 89+18=107; 107+36=143.

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

Intersection of A007775 and A017257.

Filtered([1..1200],n->n mod 2<>0 and n mod 3 <>0 and n mod 5<>0 and n-9*Int((n-1)/9)=8); # Muniru A Asiru, May 30 2018

select(n->modp(n,2)<>0 and modp(n,3)<>0 and modp(n,5)<>0 and n-9*floor((n-1)/9)=8,[$1..1200]); # Muniru A Asiru, May 30 2018

Vec(x*(17 + 36*x + 18*x^2 + 18*x^3 + x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^60)) \\ Colin Barker, Mar 26 2018

Numbers == {17, 53, 71, 89} mod 90.
From Colin Barker, Mar 26 2018: (Start)
G.f.: x*(17 + 36*x + 18*x^2 + 18*x^3 + x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)).
a(n) = (5 + 9*(-1)^n - (9+9*i)*(-i)^n - (9-9*i)*i^n + 90*n) / 4, where i=sqrt(-1).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
(End)

A373864 Reduced Collatz function R applied to the numbers 6n+5: a(n) = R(6n+5), where R(k) = (3k+1)/2^r, with r as large as possible.

Original entry on oeis.org

1, 17, 13, 35, 11, 53, 31, 71, 5, 89, 49, 107, 29, 125, 67, 143, 19, 161, 85, 179, 47, 197, 103, 215, 7, 233, 121, 251, 65, 269, 139, 287, 37, 305, 157, 323, 83, 341, 175, 359, 23, 377, 193, 395, 101, 413, 211, 431, 55, 449, 229, 467, 119

Offset: 0

Jonas Kaiser, Jun 19 2024

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A000265, A017257, A075677, A260658, A373730.

A373864[n_] := #/2^IntegerExponent[#, 2] & [9*n + 8];
Array[A373864, 100, 0] (* Paolo Xausa, Aug 19 2024 *)

PARI
```
a(n) = n=9*n+8; n>>valuation(n,2);
```

a(n) = A000265(A017257(n)).

A300186 Largest digit sum among all n-digit primes.

Original entry on oeis.org

7, 17, 25, 35, 44, 53, 62, 71, 80, 88, 98, 107, 115, 125, 134, 143, 152, 161, 170, 179, 188, 197, 206, 215, 223, 233, 242, 250, 260, 269, 278, 287, 296, 304, 314, 323, 332, 341, 350, 359, 367, 377, 386, 394, 404, 413, 421, 431, 440, 449, 458, 466, 476, 485, 494

Offset: 1

Felix Fröhlich, Feb 28 2018

Largest value of A007605(x) for any integer x in the interval [A090226(n), A090226(n+1)-1].

Trivially, 1 < a(n) < 9*n = A008591(n). The lower bound follows, since a prime > 10 must contain at least two nonzero digits, with the least possible digit sum 2. The upper bound follows because 10^n-1 is always composite and thus the digit sum can be at most A017257(n-1). The maximal possible value is reached by a(n) iff a term t exists in A263431 such that A055642(t) = n-1.

			For n = 2: Among all 2-digit primes, the largest possible digit sum is 8+9 = 17, which is achieved by the prime 89, so a(2) = 17.

Cf. A007605, A008591, A017257, A055642, A090226, A263431.

a(n) = my(r=0); forprime(p=10^(n-1), 10^n, if(sumdigits(p) > r, r=sumdigits(p))); r

More terms from Alois P. Heinz, Feb 28 2018

cp's OEIS Frontend

A031375 a(n) = prime(9n-1).

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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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A168409 a(n) = 8 + 9*floor((n-1)/2).

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A295869 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 8.

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A373864 Reduced Collatz function R applied to the numbers 6n+5: a(n) = R(6n+5), where R(k) = (3k+1)/2^r, with r as large as possible.

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A300186 Largest digit sum among all n-digit primes.

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