A130569 -id:A130569 - cp's OEIS frontend

A292576 Permutation of the natural numbers partitioned into quadruples [4k-1, 4k-3, 4k-2, 4k], k > 0.

3, 1, 2, 4, 7, 5, 6, 8, 11, 9, 10, 12, 15, 13, 14, 16, 19, 17, 18, 20, 23, 21, 22, 24, 27, 25, 26, 28, 31, 29, 30, 32, 35, 33, 34, 36, 39, 37, 38, 40, 43, 41, 42, 44, 47, 45, 46, 48, 51, 49, 50, 52, 55, 53, 54, 56, 59, 57, 58, 60, 63, 61, 62

Offset: 1

Guenther Schrack, Sep 19 2017

nonn

Partition the natural number sequence into quadruples starting with (1,2,3,4); swap the second and third elements, then swap the first and the second element; repeat for all quadruples.

Guenther Schrack, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
Index entries for sequences that are permutations of the natural numbers

Inverse: A056699(n+1) - 1 for n > 0.
Sequence of fixed points: A008586(n) for n > 0.
Subsequences:
   elements with odd index: A042964(A103889(n)) for n > 0.
   elements with even index: A042948(n) for n > 0.
   odd elements: A166519(n) for n>0.
   indices of odd elements: A042963(n) for n > 0.
   even elements: A005843(n) for n>0.
   indices of even elements: A014601(n) for n > 0.
Sum of pairs of elements:
   a(n+2) + a(n) = A163980(n+1) = A168277(n+2) for n > 0.
Difference between pairs of elements:
   a(n+2) - a(n) = (-1)^A011765(n+3)*A091084(n+1) for n > 0.
Compound relations:
   a(n) = A284307(n+1) - 1 for n > 0.
   a(n+2) - 2*a(n+1) + a(n) = (-1)^A011765(n)*A132400(n+1) for n > 0.
Compositions:
   a(n) = A116966(A080412(n)) for n > 0.
   a(A284307(n)) = A256008(n) for n > 0.
   a(A042963(n)) = A166519(n-1) for n > 0.
   A256008(a(n)) = A056699(n) for n > 0.

a = [3 1 2 4]; % Generate b-file
max = 10000;
for n := 5:max
   a(n) = a(n-4) + 4;
end;

for(n=1, 10000, print1(n + ((-1)^(n*(n-1)/2)*(2 - (-1)^n) - (-1)^n)/2, ", "))

a(1)=3, a(2)=1, a(3)=2, a(4)=4, a(n) = a(n-4) + 4 for n > 4.
O.g.f.: (2*x^3 + x^2 - 2*x + 3)/(x^5 - x^4 - x + 1).
a(n) = n + ((-1)^(n*(n-1)/2)*(2-(-1)^n) - (-1)^n)/2.
a(n) = n + (cos(n*Pi/2) - cos(n*Pi) + 3*sin(n*Pi/2))/2.
a(n) = n + n mod 2 + (ceiling(n/2)) mod 2 - 2*(floor(n/2) mod 2).
Linear recurrence: a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
First Differences, periodic: (-2, 1, 2, 3), repeat; also (-1)^A130569(n)*A068073(n+2) for n > 0.

A290812 Odd composite numbers m such that k^(m - 1) == 1 (mod m) and gcd(k^((m - 1)/2) - 1, m) = 1 for some integer k in the interval [2, sqrt(m) + 1].

Original entry on oeis.org

91, 247, 325, 343, 485, 703, 871, 901, 931, 949, 1099, 1111, 1157, 1247, 1261, 1271, 1387, 1445, 1525, 1649, 1765, 1807, 1891, 1975, 2047, 2059, 2071, 2117, 2501, 2701, 2863, 2871, 3277, 3281, 3365, 3589, 3845, 4069, 4141, 4187, 4291, 4371, 4411, 4525

Offset: 1

Arkadiusz Wesolowski, Aug 11 2017

nonn

If the condition "odd composite numbers" in the definition is replaced by "odd numbers", then every odd prime number is in the sequence.

This is not a subsequence of A290543 (for example, 65683 is missing in A290543).

			91 is in the sequence because:
1) it is an odd composite number.
2) k^90 == 1 (mod 91) and gcd(k^45 - 1, 91) = 1 with k = 10 < sqrt(91) + 1.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from A. Wesolowski)
Wikipedia, Pocklington primality test.
Index entries for sequences related to pseudoprimes.

Cf. A130569, A290543.

lst:=[]; for n in [3..4525 by 2] do if not IsPrime(n) then for a in [2..Floor(Sqrt(n)+1)] do if Modexp(a, n-1, n) eq 1 and GCD(a^Truncate((n-1)/2)-1, n) eq 1 then Append(~lst, n); break; end if; end for; end if; end for; lst;

Select[Range[3, 4525, 2], Function[n, And[CompositeQ@ n, AnyTrue[Range[2, Sqrt[n] + 1], And[PowerMod[#, n - 1, n] == 1, CoprimeQ[#^((n - 1)/2) - 1, n]] &]]]] (* Michael De Vlieger, Aug 16 2017 *)

is(n) = if(n > 1 && n%2==1 && !ispseudoprime(n), for(x=2, sqrt(n)+1, if(Mod(x, n)^(n-1)==1 && gcd(x^((n-1)/2)-1, n)==1, return(1)))); 0 \\ Felix Fröhlich, Aug 18 2017

cp's OEIS Frontend

A292576 Permutation of the natural numbers partitioned into quadruples [4k-1, 4k-3, 4k-2, 4k], k > 0.

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