A113244 -id:A113244 - cp's OEIS frontend

A113243 Differences of nonzero tetranacci numbers.

0, 1, 2, 3, 4, 6, 7, 11, 13, 14, 21, 25, 27, 28, 41, 48, 52, 54, 55, 79, 93, 100, 104, 106, 107, 152, 179, 193, 200, 204, 206, 207, 293, 345, 372, 386, 393, 397, 399, 400, 565, 665, 717, 744, 758, 765, 769, 771, 772, 1089, 1282, 1382, 1434, 1461, 1475, 1482

Offset: 1

Jonathan Vos Post, Oct 19 2005; corrected Oct 20 2005

R. J. Mathar, Table of n, a(n) for n = 1..197

Cf. A000078, A113244.

isA113243 := proc(n)
    local i,j ;
    for j from 3 do
        for i from 3 to j do
            if A000078(j) - A000078(i) = n then
                return true;
            elif A000078(j) - A000078(i) < n then
                break ;
            end if;
        end do:
        if A000078(j) - A000078(j-1) > n then
            return false;
        end if;
    end do:
end proc:
for n from 0 to 10000 do
    if isA113243(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Oct 04 2014

Union[Flatten[Differences/@Subsets[Drop[LinearRecurrence[{1, 1, 1,1}, {0,0, 0, 1}, 16],3],{2}]]] (* James C. McMahon, Jun 23 2024 *)

{a(n)} = { A000078(i) - A000078(j) such that i>=j>=3 }.

A113293 First differences of Lucas 3-step numbers.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 14, 18, 20, 28, 32, 36, 38, 50, 60, 64, 68, 70, 92, 110, 120, 124, 128, 130, 170, 202, 220, 230, 234, 238, 240, 312, 372, 404, 422, 432, 436, 440, 442, 574, 684, 744, 776, 794, 804, 808, 812, 814, 1056, 1258, 1368, 1428, 1460, 1478, 1488

Offset: 1

Jonathan Vos Post, Oct 23 2005

There are no primes in this sequence, except 2, as all values are odd, so all differences are even. Semiprimes include: a(3) = 4, a(4) = 6, a(6) = 10, a(7) = 14, a(13) = 38, a(26) = 202, a(35) = 422, a(44) = 794, a(54) = 1478, a(59) = 1942, a(66) = 2746, a(94) = 9326.

			a(0) = 0 because A001644(2)-A001644(0) = 3 - 3 = 0.
a(1) = 2 because A001644(2)-A001644(1) = 3 - 1 = 2.
a(2) = 4 because A001644(3)-A001644(2) = 7 - 3 = 4.
a(3) = 6 because A001644(3)-A001644(1) = 7 - 1 = 6.
a(75) = 5000 because A001644(14)-A001644(7) = 5071 - 71 = 5000.

Eric Weisstein's World of Mathematics, Lucas n-Step Number.
Noe, T. D. and Post, J. V., Primes in Fibonacci n-step and Lucas n-Step Sequences." J. Integer Seq. 8, Article 05.4.4, 2005.

Cf. A000040, A001358, A001644, A113188-A113194, A113238, A113239, A113244.

{a(n)} = { | A001644(i) - A001644(j) | such that i>=j}

A113294 First differences of Lucas 4-step numbers.

Original entry on oeis.org

1, 3, 4, 8, 11, 12, 19, 22, 23, 25, 36, 44, 47, 48, 73, 84, 92, 95, 96, 140, 165, 176, 184, 187, 188, 268, 316, 341, 352, 360, 363, 364, 517, 609, 657, 682, 693, 701, 704, 705, 998, 1174, 1266, 1314, 1339, 1350, 1358, 1361, 1362, 1923, 2264, 2440, 2532, 2580

Offset: 1

Jonathan Vos Post, Oct 23 2005

Lucas 4-step numbers are also known as "Tetranacci Lucas numbers" or "Tetranacci numbers with different initial conditions" in A073817. Primes in this sequence are A113295. In this sequence are: 13340261 = 11 * 19 * 29 * 31 * 71 is a product of 5 distinct 2-digit primes; 95550683 = 269 * 593 * 599 is a product of 3 distinct 3-digit primes.

			a(0) = 1 because A073817(0)-A001644(2) = 4 - 3 = 1.
a(1) = 3 because A073817(3)-A001644(0) = 7 - 4 = 3.
a(2) = 4 because A073817(3)-A001644(2) = 7 - 3 = 4.
a(3) = 8 because A073817(4)-A001644(3) = 15 - 7 = 8.
a(122) = 70000 because A073817(17)-A001644(3) = 70007 - 7 = 70000.

Eric Weisstein's World of Mathematics, Lucas n-Step Number.
Noe, T. D. and Post, J. V., Primes in Fibonacci n-step and Lucas n-Step Sequences." J. Integer Seq. 8, Article 05.4.4, 2005.

Cf. A000040, A073817, A113188-A113194, A113238, A113239, A113244, A113293, A113295.

{a(n)} = { | A073817(i) - A073817(j) | such that i>=j }

A113295 Prime differences of Lucas 4-step numbers.

Original entry on oeis.org

3, 11, 19, 23, 47, 73, 701, 1361, 4363, 9067, 9749, 17477, 18743, 18839, 36293, 70003, 116101, 134917, 366437, 465061, 498749, 501013, 1844033, 3590099, 13305307, 13341259, 13341619, 36229121, 49069367, 49570721, 95550661, 351427309

Offset: 1

Jonathan Vos Post, Oct 23 2005

These are primes from the sequence A113294, which is differences of Lucas 4-step numbers, also known as "Tetranacci Lucas numbers" or "Tetranacci numbers with different initial conditions" in A073817. Also in the difference set sequence are: 13340261 = 11 * 19 * 29 * 31 * 71 is a product of 5 distinct 2-digit primes; 95550683 = 269 * 593 * 599 is a product of 3 distinct 3-digit primes.

			a(1) = 3 because A073817(0)-A001644(1) = 4 - 1 = 3, a prime.
a(2) = 11 because A073817(4)-A001644(0) = 15 - 4 = 11, a prime.
a(3) = 19 because A073817(5)-A001644(3) = 26 - 7 = 19, a prime.
a(4) = 23 because A073817(5)-A001644(2) = 26 - 3 = 23, a prime.
a(16) = 70003 because A073817(17)-A001644(0) = 70007 - 4 = 70003, a prime.

Eric Weisstein's World of Mathematics, Lucas n-Step Number.
Noe, T. D. and Post, J. V., Primes in Fibonacci n-step and Lucas n-Step Sequences." J. Integer Seq. 8, Article 05.4.4, 2005.

Cf. A000040, A073817, A113188-A113194, A113238, A113239, A113244, A113293, A113294.

{a(n)} = Intersection of { | A073817(i) - A073817(j) | such that i>=j} and A000040. {a(n)} = Prime elements of { | A073817(i) - A073817(j) | such that i>=j}. {a(n)} = Prime elements of A113294.

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A113243 Differences of nonzero tetranacci numbers.

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A113293 First differences of Lucas 3-step numbers.

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A113294 First differences of Lucas 4-step numbers.

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A113295 Prime differences of Lucas 4-step numbers.

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