A131877 -id:A131877

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A140444

Primes congruent to 1 (mod 14).

+10
7

29, 43, 71, 113, 127, 197, 211, 239, 281, 337, 379, 421, 449, 463, 491, 547, 617, 631, 659, 673, 701, 743, 757, 827, 883, 911, 953, 967, 1009, 1051, 1093, 1163, 1289, 1303, 1373, 1429, 1471, 1499, 1583, 1597, 1667, 1709, 1723, 1877, 1933, 2003, 2017, 2087

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

From Federico Provvedi, May 24 2018: (Start)

Also primes congruent to 1 (mod 7).

For every prime p > 2, primes congruent to 1 (mod p) are also congruent to 1 (mod 2*p).

Conjecture: The monic polynomial P(x) = (x+1)^7/x - 1/x = ((x+1)^7-1)/x is irreducible but factorizable over Galois field (mod a(n)) with exactly 6 distinct irreducible factors of degree 1. Examples:

P(x) == (5 + x) (6 + x) (7 + x) (10 + x) (14 + x) (23 + x) (mod 29)

P(x) == (3 + x) (9 + x) (23 + x) (28 + x) (33 + x) (40 + x) (mod 43)

P(x) == (24 + x) (27 + x) (35 + x) (40 + x) (42 + x) (52 + x) (mod 71)

P(x) == (5 + x) (8 + x) (65 + x) (84 + x) (86 + x) (98 + x) (mod 113)

... (End).

Primes in A131877. - Eric Chen, Jun 14 2018

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

FORMULA

a(n) ~ 6n log n. - Charles R Greathouse IV, Jul 02 2016

MAPLE

select(isprime, select(n->modp(n, 14)=1, [$1..2300])); # Muniru A Asiru, Jun 27 2018

MATHEMATICA

Select[Prime[Range[500]], Mod[#, 14] == 1 &] (* Harvey P. Dale, Mar 21 2011 *)

PROG

(Magma) [p: p in PrimesUpTo(3000)|p mod 14 in {1}]; // Vincenzo Librandi, Dec 18 2010

(PARI) is(n)=isprime(n) && n%14==1 \\ Charles R Greathouse IV, Jul 02 2016

(GAP) Filtered(Filtered([1..2300], n->n mod 14=1), IsPrime); # Muniru A Asiru, Jun 27 2018

CROSSREFS

Cf. A045437, A045458, A045471, A045473.

A090613 gives prime index.

Cf. A090614.

Cf. A131877.

Primes congruent to 1 (mod k): A000040 (k=1), A065091 (k=2), A002476 (k=3 and 6), A002144 (k=4), A030430 (k=5 and 10), this sequence (k=7 and 14), A007519 (k=8), A061237 (k=9 and 18), A141849 (k=11 and 22), A068228 (k=12), A268753 (k=13 and 26), A132230 (k=15 and 30), A094407 (k=16), A129484 (k=17 and 34), A141868 (k=19 and 38), A141881 (k=20), A124826 (k=21 and 42), A212374 (k=23 and 46), A107008 (k=24), A141927 (k=25 and 50), A141948 (k=27 and 54), A093359 (k=28), A141977 (k=29 and 58), A142005 (k=31 and 62), A133870 (k=32).

KEYWORD

nonn,easy,changed

AUTHOR

Juri-Stepan Gerasimov, Jun 26 2008

EXTENSIONS

Simpler definition from N. J. A. Sloane, Jul 11 2008

STATUS

approved

A130154

Triangle read by rows: T(n, k) = 1 + 2*(n-k)*(k-1) (1 <= k <= n).

+10
6

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 9, 7, 1, 1, 9, 13, 13, 9, 1, 1, 11, 17, 19, 17, 11, 1, 1, 13, 21, 25, 25, 21, 13, 1, 1, 15, 25, 31, 33, 31, 25, 15, 1, 1, 17, 29, 37, 41, 41, 37, 29, 17, 1, 1, 19, 33, 43, 49, 51, 49, 43, 33, 19, 1, 1, 21, 37, 49, 57, 61, 61, 57, 49, 37, 21, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,5

COMMENTS

Column k, except for the initial k-1 0's, is an arithmetic progression with first term 1 and common difference 2(k-1). Row sums yield A116731. First column of the inverse matrix is A129779.

Studied by Paul Curtz circa 1993.

From Rogério Serôdio, Dec 19 2017: (Start)

T(n, k) gives the number of distinct sums of 2(k-1) elements in {1,1,2,2,...,n-1,n-1}. For example, T(6, 2) = the number of distinct sums of 2 elements in {1,1,2,2,3,3,4,4,5,5}, and because each sum from the smallest 1 + 1 = 2 to the largest 5 + 5 = 10 appears, T(6, 2) = 10 - 1 = 9. [In general:

2*Sum_{j=1..(k-1)} (n-j) - (2*(Sum_{j=1..k-1} j) - 1) = 2*(n*(k-1) - 4*(k-1)*k/2 + 1 = 2*(k-1)*(n-k) + 1 = T(n, k). - Wolfdieter Lang, Dec 20 2017]

T(n, k) is the number of lattice points with abscissa x = 2*(k-1) and even ordinate in the closed region bounded by the parabola y = x*(2*(n-1) - x) and the x axis. [That is, (1/2)*y(2*(k-1)) + 1 = T(n, k). - Wolfdieter Lang, Dec 20 2017]

Pascal's triangle (A007318, but with apex in the middle) is formed using the rule South = West + East; the rascal triangle A077028 uses the rule South = (West*East + 1)/North; the present triangle uses a similar rule: South = (West*East + 2)/North. See the formula section for this recurrence. (End)

LINKS

G. C. Greubel, Rows n = 1..100 of triangle, flattened

FORMULA

T(n, k) = 1 + 2*(n-k)*(k-1) (1 <= k <= n).

G.f.: G(t,z) = t*z*(3*t*z^2 - z - t*z + 1)/((1-t*z)*(1-z))^2.

Equals = 2 * A077028 - A000012 as infinite lower triangular matrices. - Gary W. Adamson, Oct 23 2007

T(n, 1) = 1 and T(n, n) = 1 for n >= 1; T(n, k) = (T(n-1, k-1)*T(n-1, k) + 2)/T(n-2, k-1), for n > 2 and 1 < k < n. See a comment above. - Rogério Serôdio, Dec 19 2017

G.f. column k (with leading zeros): (x^k/(1-x)^2)*(1 + (2*k-3)*x), k >= 1. See the g.f. of the triangle G(t,z) above: (d/dt)^k G(t,x)/k!|_{t=0}. - Wolfdieter Lang, Dec 20 2017

EXAMPLE

The triangle T(n, k) starts:

n\k 1 2 3 4 5 6 7 8 9 10 ...

1: 1

2: 1 1

3: 1 3 1

4: 1 5 5 1

5: 1 7 9 7 1

6: 1 9 13 13 9 1

7: 1 11 17 19 17 11 1

8: 1 13 21 25 25 21 13 1

9: 1 15 25 31 33 31 25 15 1

10: 1 17 29 37 41 41 37 29 17 1

... reformatted. - Wolfdieter Lang, Dec 19 2017

MAPLE

T:=proc(n, k) if k<=n then 2*(n-k)*(k-1)+1 else 0 fi end: for n from 1 to 14 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form

MATHEMATICA

Flatten[Table[1+2(n-k)(k-1), {n, 0, 20}, {k, n}]] (* Harvey P. Dale, Jul 13 2013 *)

PROG

(PARI) T(n, k) = 1 + 2*(n-k)*(k-1) \\ Iain Fox, Dec 19 2017

(PARI) first(n) = my(res = vector(binomial(n+1, 2)), i = 1); for(r=1, n, for(k=1, r, res[i] = 1 + 2*(r-k)*(k-1); i++)); res \\ Iain Fox, Dec 19 2017

(Magma) [1 + 2*(n-k)*(k-1): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 25 2019

(Sage) [[1 + 2*(n-k)*(k-1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Nov 25 2019

(GAP) Flat(List([1..12], n-> List([1..n], k-> 1 + 2*(n-k)*(k-1) ))); # G. C. Greubel, Nov 25 2019

CROSSREFS

Cf. A116731, A129779. A007318, A077028.

Column sequences (no leading zeros): A000012, A016813, A016921, A017077, A017281, A017533, A131877, A158057, A161705, A215145.

KEYWORD

nonn,tabl,easy

AUTHOR

Emeric Deutsch, May 22 2007

EXTENSIONS

Edited by Wolfdieter Lang, Dec 19 2017

STATUS

approved

A147587

a(n) = 14*n + 7.

+10
5

7, 21, 35, 49, 63, 77, 91, 105, 119, 133, 147, 161, 175, 189, 203, 217, 231, 245, 259, 273, 287, 301, 315, 329, 343, 357, 371, 385, 399, 413, 427, 441, 455, 469, 483, 497, 511, 525, 539, 553, 567, 581, 595, 609, 623, 637, 651, 665, 679, 693, 707, 721, 735

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

a(n+3) = 14*n + 49 is the sum of seven consecutive odd numbers starting with 2*n+1. - Wesley Ivan Hurt, Apr 11 2015

Numbers k such that 3^k + 1 is divisible by 547. - Bruno Berselli, Aug 22 2018

Sum of the numbers from 2*(n-1) to 2*(n+2). - Bruno Berselli, Oct 25 2018

LINKS

Table of n, a(n) for n=0..52.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

FORMULA

a(n) = a(n-1) + 14.

a(n) = A132355(2*n+2) - A132355(2*n+1) = 7*A005408(n).

a(n) = 28*n - a(n-1) for n>0, a(0)=7. - Vincenzo Librandi, Nov 24 2010

From Wesley Ivan Hurt, Apr 11 2015: (Start)

G.f.: 7*(1 + x)/(1 - x)^2.

a(n) = 2*a(n-1) - a(n-2). (End)

Sum_{n>=0} (-1)^n/a(n) = Pi/28 (A132744). - Amiram Eldar, Dec 13 2021

MAPLE

A147587:=n->14*n+7: seq(A147587(n), n=0..100); # Wesley Ivan Hurt, Apr 11 2015

MATHEMATICA

Range[7, 1000, 14] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)

PROG

(Magma) [14*n+7 : n in [0..100]]; // Wesley Ivan Hurt, Apr 11 2015

(PARI) a(n)=14*n+7 \\ Charles R Greathouse IV, Oct 16 2015

CROSSREFS

Cf. A008596, A131877, A132744.

KEYWORD

nonn,easy

AUTHOR

Paul Curtz, Nov 08 2008

EXTENSIONS

More terms from Vincenzo Librandi, Oct 23 2009

STATUS

approved

A131876

2*A131873 - 1.

+10
2

1, 15, 7, 29, 15, 13, 43, 23, 21, 19, 57, 31, 29, 27, 25, 71, 39, 37, 35, 33, 31, 85, 47, 45, 43, 41, 39, 37, 99, 55, 53, 51, 49, 47, 45, 43, 113, 63, 61, 59, 57, 55, 53, 51, 49, 127, 71, 69, 67, 65, 63, 61, 59, 57, 55, 141, 79, 77, 75, 73, 71, 69, 67, 65, 63, 61, 155, 87, 85, 83, 81, 79, 77, 75, 73, 71, 69, 67, 169, 95, 93, 91, 89, 87, 85, 83, 81, 79, 77, 75, 73, 183, 103, 101, 99, 97, 95, 93, 91, 89, 87, 85, 83, 81, 79

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Left column = 14n + 1, A131877: (1, 15, 29, 43, 57, 21, ...).

Right border = 6n + 1, A016921: (1, 7, 13, 19, ...).

Row sums = A131878 (1, 22, 57, 106, 169, ...).

LINKS

Table of n, a(n) for n=0..104.

FORMULA

2*A131873 - 1 as infinite lower triangular matrices.

EXAMPLE

First few rows of the triangle:

15, 7;

29, 15, 13;

43, 23, 21, 19;

57, 31, 29, 27, 25;

71, 39, 37, 35, 33, 31;

85, 47, 45, 43, 41, 39, 37;

...

CROSSREFS

Cf. A131877, A131878, A131873.

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Jul 22 2007

EXTENSIONS

More terms from Russ Cox, Apr 18 2024

STATUS

approved

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