Displaying 1-10 of 50 results found.
Sums of twin prime pairs.
+10
41
8, 12, 24, 36, 60, 84, 120, 144, 204, 216, 276, 300, 360, 384, 396, 456, 480, 540, 564, 624, 696, 840, 864, 924, 1044, 1140, 1200, 1236, 1284, 1320, 1620, 1644, 1656, 1716, 1764, 2040, 2064, 2100, 2124, 2184, 2304, 2460, 2556, 2580, 2604, 2640, 2856, 2904
COMMENTS
(p^q)+(q^p) calculated modulo pq, where (p,q) is the n-th twin prime pair. Example: (599^601)+(601^599) == 1200 mod (599*601). - Sam Alexander, Nov 14 2003 El'hakk makes the following claim (without any proof): (q^p)+(p^q) = 2*cosh(q arctanh( sqrt( 1-((2/p)^2) ) )) + 2cosh(p arctanh( sqrt( 1-((2/q)^2) ) )) mod p*q. - Sam Alexander, Nov 14 2003 Also: Numbers N such that N/2-1 and N/2+1 both are prime. - M. F. Hasler, Jan 03 2013 Excluding the first term, all remaining terms have digital root 3, 6 or 9. - J. W. Helkenberg, Jul 24 2013 Except for the first term, this sequence is a subsequence of A005101 (Abundant numbers) and of A008594 (Multiples of 12). - Ivan N. Ianakiev, Jul 04 2021
EXAMPLE
a(3) = 24 because the twin primes 11 and 13 add to 24.
MAPLE
ZL:=[]:for p from 1 to 1451 do if (isprime(p) and isprime(p+2)) then ZL:=[op(ZL), p+(p+2)]; fi; od; print(ZL); # Zerinvary Lajos, Mar 07 2007
MATHEMATICA
Select[Table[Prime[n] + 1, {n, 230}], PrimeQ[ # + 1] &] *2 (* Ray Chandler, Oct 12 2005 *) Total/@Select[Partition[Prime[Range[300]], 2, 1], #[[2]]-#[[1]]==2&] (* Harvey P. Dale, Oct 23 2022 *)
PROG
(PARI) is_ A054735(n)={!bittest(n, 0)&&isprime(n\2-1)&&isprime(n\2+1)} \\ M. F. Hasler, Jan 03 2013 (PARI) pp=1; forprime(p=1, 1482, if( p==pp+2, print1(p+pp, ", ")); pp=p) \\ Following a suggestion by R. J. Cano, Jan 05 2013 (Haskell)
Multiples of 12 containing a 12 in their decimal representation.
+10
35
12, 120, 312, 612, 912, 1128, 1200, 1212, 1224, 1236, 1248, 1260, 1272, 1284, 1296, 1512, 1812, 2112, 2124, 2412, 2712, 3012, 3120, 3312, 3612, 3912, 4128, 4212, 4512, 4812, 5112, 5124, 5412, 5712, 6012, 6120, 6312, 6612, 6912, 7128, 7212, 7512, 7812
MATHEMATICA
Select[12*Range[700], SequenceCount[IntegerDigits[#], {1, 2}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 25 2017 *)
PROG
(Haskell)
import Data.List (isInfixOf)
a121032 n = a121032_list !! (n-1)
a121032_list = filter ((isInfixOf "12") . show) a008594_list
(PARI) is(n)=if(n%12, return(0)); while(n>11, if(n%100==12, return(1)); n\=10); 0 \\ Charles R Greathouse IV, Feb 12 2017
CROSSREFS
Cf. A121041, A011531, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040.
0, 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156, 169, 182, 195, 208, 221, 234, 247, 260, 273, 286, 299, 312, 325, 338, 351, 364, 377, 390, 403, 416, 429, 442, 455, 468, 481, 494, 507, 520, 533, 546, 559, 572, 585, 598, 611, 624, 637, 650, 663, 676
-1, 23, 47, 71, 95, 119, 143, 167, 191, 215, 239, 263, 287, 311, 335, 359, 383, 407, 431, 455, 479, 503, 527, 551, 575, 599, 623, 647, 671, 695, 719, 743, 767, 791, 815, 839, 863, 887, 911, 935, 959, 983, 1007, 1031, 1055, 1079, 1103, 1127, 1151, 1175, 1199
COMMENTS
a(n) is also the denominator of the finite algebraic formula for the number of partitions of n, if n >= 1. The formula is p(n) = Tr(n)/(24*n - 1), n >= 1. See theorem 1.1 of the Bruinier-Ono paper in the link. For the numerators see A183011. It appears that a(n) is also the denominator of the coefficient of the third term in the n-th Bruinier-Ono "partition polynomial" H_n(x). See the Bruinier-Ono paper, chapter 5 "Examples". For the numerators see A183007. - Omar E. Pol, Jul 13 2011 Also exponents in the formula q^(-1) + q^23 + 2*q^47 + 3*q^71 + 5*q^95 + 7*q^119 + 11*q^143 + 15*q^167 + ... in which the coefficients are the partition numbers (see A000041, Example section). - Omar E. Pol, Feb 27 2013
EXAMPLE
G.f. = -1 + 23*x + 47*x^2 + 71*x^3 + 95*x^4 + 119*x^5 + 143*x^6 + 167*x^7 + ...
9, 21, 33, 45, 57, 69, 81, 93, 105, 117, 129, 141, 153, 165, 177, 189, 201, 213, 225, 237, 249, 261, 273, 285, 297, 309, 321, 333, 345, 357, 369, 381, 393, 405, 417, 429, 441, 453, 465, 477, 489, 501, 513, 525, 537, 549, 561, 573, 585, 597, 609, 621, 633
COMMENTS
Numbers k such that k mod 2 = (k+1) mod 3 = 1 and (k+2) mod 4 != 1. - Klaus Brockhaus, Jun 15 2004 For n > 3, the number of squares on the infinite 3-column chessboard at <= n knight moves from any fixed point. - Ralf Stephan, Sep 15 2004
FORMULA
Sum_{n>=0} (-1)^n/a(n) = (Pi + log(3-2*sqrt(2)))/(12*sqrt(2)). - Amiram Eldar, Dec 12 2021
MATHEMATICA
LinearRecurrence[{2, -1}, {9, 21}, 60] (* Harvey P. Dale, Apr 14 2019 *)
PROG
(Sage) [i+9 for i in range(525) if gcd(i, 12) == 12] # Zerinvary Lajos, May 21 2009 (Haskell)
a(n) = Fibonacci(n) mod 12.
+10
20
0, 1, 1, 2, 3, 5, 8, 1, 9, 10, 7, 5, 0, 5, 5, 10, 3, 1, 4, 5, 9, 2, 11, 1, 0, 1, 1, 2, 3, 5, 8, 1, 9, 10, 7, 5, 0, 5, 5, 10, 3, 1, 4, 5, 9, 2, 11, 1, 0, 1, 1, 2, 3, 5, 8, 1, 9, 10, 7, 5, 0, 5, 5, 10, 3, 1, 4, 5, 9, 2, 11, 1, 0, 1, 1, 2, 3, 5, 8, 1, 9, 10, 7, 5, 0, 5, 5, 10, 3, 1, 4, 5, 9, 2, 11, 1, 0, 1, 1
COMMENTS
Sequence has been applied by several composers to 12-tone equal temperament pitch structure. The complete Fibonacci mod 12 system (a set of 10 periodic sequences) exhausts all possible ordered dyads; that is, every possible combination of two pitches is found in these sets.
a(nonexistent) = 6;
LINKS
Index entries for linear recurrences with constant coefficients, signature (1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1).
FORMULA
Has period of 24, restricted period 12 and multiplier 5.
a(n) = (a(n-1) + a(n-2)) mod 12, a(0) = 0, a(1) = 1.
MAPLE
with(combinat, fibonacci); A089911 := proc(n) fibonacci(n) mod 12; end;
PROG
(Haskell)
a089911 n = a089911_list !! n
a089911_list = 0 : 1 : zipWith (\u v -> (u + v) `mod` 12)
(tail a089911_list) a089911_list
0, 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168, 182, 196, 210, 224, 238, 252, 266, 280, 294, 308, 322, 336, 350, 364, 378, 392, 406, 420, 434, 448, 462, 476, 490, 504, 518, 532, 546, 560, 574, 588, 602, 616, 630, 644, 658, 672, 686, 700, 714, 728
FORMULA
a(n) = 14n = 2*a(n-1) - a(n-2). G.f.: 14x/(x-1)^2. - R. J. Mathar, Jun 23 2009
MATHEMATICA
CoefficientList[Series[14 x / (x - 1)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Jun 10 2013 *)
2, 14, 26, 38, 50, 62, 74, 86, 98, 110, 122, 134, 146, 158, 170, 182, 194, 206, 218, 230, 242, 254, 266, 278, 290, 302, 314, 326, 338, 350, 362, 374, 386, 398, 410, 422, 434, 446, 458, 470, 482, 494, 506, 518, 530, 542, 554, 566, 578, 590, 602, 614, 626, 638
COMMENTS
Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 40 ).
FORMULA
G.f.: 2*(1 + 5*x)/(1-x)^2.
E.g.f.: 2*(1 + 6*x)*exp(x). (End)
Sum_{n>=0} (-1)^n/a(n) = Pi/12 + sqrt(3)*log(2 + sqrt(3))/12. - Amiram Eldar, Dec 12 2021
PROG
(GAP) List([0..60], n-> 2*(6*n+1) ); # G. C. Greubel, Sep 18 2019
4, 16, 28, 40, 52, 64, 76, 88, 100, 112, 124, 136, 148, 160, 172, 184, 196, 208, 220, 232, 244, 256, 268, 280, 292, 304, 316, 328, 340, 352, 364, 376, 388, 400, 412, 424, 436, 448, 460, 472, 484, 496, 508, 520, 532, 544, 556, 568, 580, 592, 604, 616, 628
COMMENTS
Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 46 ).
Number of 6 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (11;0) and (01;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1<i2, j1<j2 and these elements are in the same relative order as those in the triple (x,y,z). In general, the number of m X n 0-1 matrices in question is given by 2^m+2m(n-1). Cf. m=2: A008574; m=3: A016933; m=4: A022144; m=5: A017293. - Sergey Kitaev, Nov 13 2004 Except for 4, exponents e such that x^e-x^2+1 is reducible.
If Y and Z are 2-blocks of a (3n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Oct 28 2007 Terms are perfect squares iff n is a generalized octagonal number ( A001082), then n = k*(3*k-2) and a(n) = (2*(3k-1))^2. - Bernard Schott, Feb 26 2023
FORMULA
Sum_{n>=0} (-1)^n/a(n) = sqrt(3)*Pi/36 + log(2)/12. - Amiram Eldar, Dec 12 2021 O.g.f.: 4*(1 + 2*x)/(1 - x)^2.
E.g.f.: 4*exp(x)*(1 + 3*x). (End)
Triangle read by rows: T(n,k) = n^2 - k^2, 0 <= k < n.
+10
15
1, 4, 3, 9, 8, 5, 16, 15, 12, 7, 25, 24, 21, 16, 9, 36, 35, 32, 27, 20, 11, 49, 48, 45, 40, 33, 24, 13, 64, 63, 60, 55, 48, 39, 28, 15, 81, 80, 77, 72, 65, 56, 45, 32, 17, 100, 99, 96, 91, 84, 75, 64, 51, 36, 19, 121, 120, 117, 112, 105, 96, 85, 72, 57, 40, 21
COMMENTS
All numbers m occur A034178(m) times. The row polynomials T(n,x) appear in the calculation of the column g.f.s of triangle A120070 (used to find the frequencies of the spectral lines of the hydrogen atom).
FORMULA
Row polynomials: T(n,x) = n^2*Sum_{m=0..n} x^m - Sum_{m=0..n} m^2*x^m = Sum_{k=0..n-1} T(n,k)*x^k, n >= 1.
Sum_{k=0..n} T(n, k) = A002412(n) (row sums). Sum_{k=0..n-1} (-1)^k * T(n, k) = A000384(floor((n+1)/2)). Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A128624(n). Sum_{k=0..floor((n-1)/2)} (-1)^k*T(n-k, k) = (1/2)*n*(n+1 - (-1)^n*cos(n*Pi/2)). (End)
EXAMPLE
n=3: T(3,x) = 9+8*x+5*x^2.
Triangle begins:
1;
4, 3;
9, 8, 5;
16, 15, 12, 7;
25, 24, 21, 16, 9;
36, 35, 32, 27, 20, 11;
49, 48, 45, 40, 33, 24, 13;
64, 63, 60, 55, 48, 39, 28, 15;
81, 80, 77, 72, 65, 56, 45, 32, 17;
PROG
(Magma) [n^2-k^2: k in [0..n-1], n in [1..15]]; // G. C. Greubel, Mar 12 2024 (SageMath) flatten([[n^2-k^2 for k in range(n)] for n in range(1, 16)]) # G. C. Greubel, Mar 12 2024
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