A141290
Triangle read by rows, descending antidiagonals of a (1, 3, 5, ...) * (1, 4, 16, ...) multiplication table.
Original entry on oeis.org
1, 3, 4, 5, 12, 16, 7, 20, 48, 64, 9, 28, 80, 192, 256, 11, 36, 112, 320, 768, 1024, 13, 44, 144, 448, 1280, 3072, 4096, 15, 52, 176, 576, 1792, 5120, 12288, 16384, 17, 60, 208, 704, 2304, 7168, 20480, 49152, 65536, 19, 68, 240, 832, 2816, 9216, 28672, 81920, 196608, 262144
Offset: 1
Given the multiplication table (1, 3, 5, ...) * (1, 4, 16, ...); i.e., odd numbers as column headings, powers of 4 along the left border:
1, 3, 5, 7, ...
4, 12, 20, 28, ...
16, 48, 80, 112, ...
64, 192, 320, 448, ...
...
Rows of the triangle = descending antidiagonals of the array, getting:
1;
3, 4;
5, 12, 16;
7, 20, 48, 64;
9, 28, 80, 192, 256;
11, 36, 112, 320, 768, 1024;
13, 44, 144, 448, 1280, 3072, 4096;
15, 52, 176, 576, 1792, 5120, 122288, 16384;
...
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A[n_,k_]:=(2k-1)*4^(n-1); Table[A[k,n-k+1],{n,10},{k,n}]//Flatten (* Stefano Spezia, May 21 2024 *)
A172285
a(n) = (5*2^n - 5*(-1)^n - 3*n*(-1)^n) / 9.
Original entry on oeis.org
0, 2, 1, 6, 7, 20, 33, 74, 139, 288, 565, 1142, 2271, 4556, 9097, 18210, 36403, 72824, 145629, 291278, 582535, 1165092, 2330161, 4660346, 9320667, 18641360, 37282693, 74565414, 149130799, 298261628, 596523225, 1193046482, 2386092931, 4772185896
Offset: 0
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[(5*2^n - 5*(-1)^n - 3*n*(-1)^n) / 9: n in [0..40]]; // Vincenzo Librandi, Aug 05 2011
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A172295 := proc(n) (5*2^n - 5*(-1)^n - 3*n*(-1)^n) / 9 ; end proc: seq(A172295(n), n=0..100) ; # R. J. Mathar, Feb 02 2010
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Table[(5*2^n - 5*(-1)^n - 3*n*(-1)^n)/9, {n, 0, 40}] (* Wesley Ivan Hurt, Aug 27 2015 *)
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first(m)=vector(m,i,i--;(5*2^i -5*(-1)^i - 3*i*(-1)^i ) / 9) \\ Anders Hellström, Aug 27 2015
Definition replaced by explicit formula; g.f. added -
R. J. Mathar, Feb 02 2010
A243528
Integers n such that p = 4n + 1, q = 4p + 3, r = 4q + 5, s = 4r + 7 and t = 4s + 9 are all prime.
Original entry on oeis.org
1564, 4057, 4654, 5884, 26599, 30139, 37204, 66532, 74227, 80812, 98137, 113929, 124249, 138604, 245524, 249847, 250879, 299767, 309469, 315277, 340504, 346279, 359467, 362674, 367069, 401407, 410332, 435049, 437377, 438799, 537844, 550582, 579814, 587047
Offset: 1
First 3 values of n, p, q, r, s and t:
{1564, 6257, 25031, 100129, 400523, 1602101},
{4057, 16229, 64919, 259681, 1038731, 4154933},
{4654, 18617, 74471, 297889, 1191563, 4766261}.
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A141291:=func; [n: n in [1..10^6] | forall{i: i in [1..5] | IsPrime(4^i*n + A141291(i))}]; // Bruno Berselli, Jun 06 2014
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pqrstQ[n_]:=Module[{p=4n+1,q,r,s},q=4p+3;r=4q+5;s=4r+7;AllTrue[{p,q,r,s,4s+9},PrimeQ]]; Select[Range[590000],pqrstQ] (* Harvey P. Dale, Jan 18 2024 *)
Showing 1-3 of 3 results.
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