Displaying 1-10 of 421 results found.
page
1
2
3
4
5
6
7
8
9
10
... 43
Least number which satisfies the requirements of A380943 in exactly n ways.
+0
4
2, 37, 373, 19937, 257931013, 4199993923
COMMENTS
The requirements of A380943 are that primes, p_n, written in decimal representation by the concatenation of primes p and q such that the concatenation of q and p also forms a prime.
MATHEMATICA
f[n_] := Block[{cnt = 0, id = IntegerDigits@ n, k = 1, len, p, q, qp}, len = Length@ id; While[k < len, p = Take[id, k]; q = Take[id, -len + k]; qp = FromDigits[ Join[q, p]]; If[ PrimeQ[FromDigits[p]] && PrimeQ[FromDigits[q]] && PrimeQ[qp] && IntegerLength[qp] == len, cnt++]; k++]; cnt]; a[n_]:=Module[{k=1}, While[f[Prime[k]]!=n, k++]; Prime[k]]; Array[a, 4, 0]
Consecutive states of the linear congruential pseudo-random number generator 7^13*s mod 10^11 when started at 1.
+0
0
1, 96889010407, 47754305649, 5019889143, 18113311201, 58918668807, 8009274449, 41680190743, 64272062401, 49842407207, 88954803249, 44998412343, 99404253601, 47356225607, 72862892049, 19678553943, 78037884801, 24356124007, 88405540849, 55224615543, 68300956001
COMMENTS
Periodic with period 5*10^8.
A very early generation method.
FORMULA
a(n) = 7^13 * a(n-1) mod 10^11.
MAPLE
a:= proc(n) option remember; `if`(n<2, n,
irem(7^13*a(n-1), 10^11))
end:
2, 5, 13, 37, 101, 271, 727, 1931, 5003, 12547, 30449, 71761, 165037, 372149, 826303, 1813219, 3944921, 8533073, 18393821, 39588071, 85192381, 183479291, 395667617, 854417989, 1847225579, 3996807053, 8650687127, 18721431499, 40496966207, 87538925959, 189076973699
FORMULA
a(n) = Sum_{i=0..n} A111107(n-i) * binomial(n,i).
Consecutive states of the linear congruential pseudo-random number generator 5^13*s mod 2^35 when started at s=1.
+0
0
1, 1220703125, 30903841977, 6589172397, 2335288753, 33236884229, 14113929449, 5184031389, 9287939169, 11535683445, 23833284121, 2547937165, 6466389009, 29109403877, 12213593161, 31057406333, 2513210561, 12977872725, 17811893113, 2710136941, 28076457585
COMMENTS
Periodic with period 2^33.
REFERENCES
W. C. Bulnren, Discrete System Simulation, Prentice-Hall, 1982 (see p. 155).
FORMULA
a(n) = 5^13*a(n-1) mod 2^35.
PROG
(Python)
def a(n):
m = 1 << 35
G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x*A(x)^3) ).
+0
0
1, 1, 2, 8, 44, 298, 2359, 21112, 209175, 2262121, 26431042, 331096188, 4419824468, 62565545535, 935341395343, 14716294925179, 242945752432294, 4197094127399756, 75698807290515322, 1422350601250404765, 27788515730656558613, 563512508612712699574, 11841983002490204813514
PROG
(PARI) a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-2*j+k, j)/(3*n-2*j+k)*a(n-j, j)));
G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x*A(x)^4) ).
+0
0
1, 1, 2, 9, 56, 432, 3935, 40820, 471633, 5980210, 82329140, 1220547845, 19359684220, 326799737576, 5844913732057, 110341722975077, 2191461358459051, 45656013573862832, 995196646595460516, 22644288881875546322, 536706817952488705651, 13225669497771610891404
PROG
(PARI) a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(4*n-3*j+k, j)/(4*n-3*j+k)*a(n-j, j)));
G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x*A(x)^5) ).
+0
0
1, 1, 2, 10, 69, 592, 6052, 70870, 928497, 13404514, 210892157, 3584892350, 65390514877, 1272723903336, 26307949481077, 575201364472316, 13255835789428863, 320999903683710948, 8145524458876305526, 216062918679078474529, 5977572987203090333399
PROG
(PARI) a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(5*n-4*j+k, j)/(5*n-4*j+k)*a(n-j, j)));
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A384649.
+0
0
1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 9, 0, 1, 4, 9, 22, 56, 0, 1, 5, 14, 40, 134, 432, 0, 1, 6, 20, 64, 240, 1012, 3935, 0, 1, 7, 27, 95, 381, 1779, 9039, 40820, 0, 1, 8, 35, 134, 565, 2780, 15596, 92246, 471633, 0, 1, 9, 44, 182, 801, 4071, 23950, 156597, 1051558, 5980210, 0
FORMULA
A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(4*n-3*j+k,j)/(4*n-3*j+k) * A(n-j,j).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 2, 5, 9, 14, 20, 27, ...
0, 9, 22, 40, 64, 95, 134, ...
0, 56, 134, 240, 381, 565, 801, ...
0, 432, 1012, 1779, 2780, 4071, 5718, ...
0, 3935, 9039, 15596, 23950, 34515, 47786, ...
PROG
(PARI) a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(4*n-3*j+k, j)/(4*n-3*j+k)*a(n-j, j)));
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A384145.
+0
0
1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 8, 0, 1, 4, 9, 20, 44, 0, 1, 5, 14, 37, 108, 298, 0, 1, 6, 20, 60, 198, 716, 2359, 0, 1, 7, 27, 90, 321, 1290, 5554, 21112, 0, 1, 8, 35, 128, 485, 2064, 9821, 48838, 209175, 0, 1, 9, 44, 175, 699, 3091, 15452, 84888, 476714, 2262121, 0
FORMULA
A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(3*n-2*j+k,j)/(3*n-2*j+k) * A(n-j,j).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 2, 5, 9, 14, 20, 27, ...
0, 8, 20, 37, 60, 90, 128, ...
0, 44, 108, 198, 321, 485, 699, ...
0, 298, 716, 1290, 2064, 3091, 4434, ...
0, 2359, 5554, 9821, 15452, 22805, 32315, ...
PROG
(PARI) a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-2*j+k, j)/(3*n-2*j+k)*a(n-j, j)));
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A162661.
+0
0
1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 7, 0, 1, 4, 9, 18, 33, 0, 1, 5, 14, 34, 84, 189, 0, 1, 6, 20, 56, 159, 472, 1249, 0, 1, 7, 27, 85, 265, 882, 3057, 9237, 0, 1, 8, 35, 122, 410, 1460, 5615, 22190, 74972, 0, 1, 9, 44, 168, 603, 2256, 9166, 40053, 177149, 659042, 0
FORMULA
A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(2*n-j+k,j)/(2*n-j+k) * A(n-j,j).
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 2, 5, 9, 14, 20, 27, ...
0, 7, 18, 34, 56, 85, 122, ...
0, 33, 84, 159, 265, 410, 603, ...
0, 189, 472, 882, 1460, 2256, 3330, ...
0, 1249, 3057, 5615, 9166, 14015, 20540, ...
PROG
(PARI) a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-j+k, j)/(2*n-j+k)*a(n-j, j)));
Search completed in 0.123 seconds
| 