A054554 -id:A054554 - cp's OEIS frontend

A163586 a(1) = 3. For n > 1, Ulam's spiral is started with a(n-1), and the primes p on the NE spoke are considered. a(n) is the minimal p that is the lesser of a twin prime pair.

3, 5, 17, 29, 41, 71, 101, 191, 281, 311, 617, 857, 1319, 1451, 1481, 1721, 2027, 2267, 2729, 2969, 2999, 3461, 3767, 4229, 4241, 4271, 20021, 22091, 22961, 23201, 23291, 27581, 31121, 31151, 31181, 32441, 32531, 32561, 32801, 32831, 33071, 33827, 35897

Offset: 1

Milton L. Brown (miltbrown(AT)earthlink.net), Jul 31 2009

nonn

			Spiral for a(2)=5:
 21 20 19 18 17
 22 .9 .8 .7 16
 23 10 .5 .6 15
 24 11 12 13 14
 25 26 27 28 29
a(2)+4*k^2-2*k with k=1 is not taken since 7 is no lesser twin prime, but 17 (k=2) is.

Cf. A001359, A054554, A163633.

twinp(p) = if(isprime(p) && isprime(p + 2),1,0);
ulamNW(p, k) = p + 4*k^2-2*k;
p = 1; for(n=1, 64, k=1; while (!twinp(ulamNW(p,k)), k++); p = ulamNW(p, k); print1(", ", p)) \\ Georg Fischer, Dec 03 2024

The terms on the NE spoke are given by a(n - 1) + 4*k^2 - 2*k for k >= 1.

Edited by Georg Fischer, Dec 03 2024

A226940 a(0)=0; if a(n-1) is odd, a(n) = n + a(n-1), otherwise a(n) = n - a(n-1).

Original entry on oeis.org

0, 1, 3, 6, -2, 7, 13, 20, -12, 21, 31, 42, -30, 43, 57, 72, -56, 73, 91, 110, -90, 111, 133, 156, -132, 157, 183, 210, -182, 211, 241, 272, -240, 273, 307, 342, -306, 343, 381, 420, -380, 421, 463, 506, -462, 507, 553, 600, -552, 601, 651, 702, -650, 703, 757

Offset: 0

Enrico Santilli, Jun 23 2013

Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Cf. A081348 (second bisection); A002939, A054554, A054569, A068377.

[IsZero(n) select 0 else IsOdd(Self(n)) select n+Self(n) else n-Self(n): n in [0..60]]; // Bruno Berselli, Jul 01 2013

LinearRecurrence[{0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1}, {0, 1, 3, 6, -2, 7, 13, 20, -12, 21, 31, 42}, 60] (* Bruno Berselli, Jul 01 2013 *)

makelist(coeff(taylor(x*(1+3*x+6*x^2-2*x^3+4*x^4+4*x^5+2*x^6-6*x^7+3*x^8+x^9)/((1-x)^3*(1+x)^3*(1+x^2)^3), x, 0, n), x, n), n, 0, 60); /* Bruno Berselli, Jul 01 2013 */

G.f.: x*(1 +3*x +6*x^2 -2*x^3 +4*x^4 +4*x^5 +2*x^6 -6*x^7 +3*x^8 +x^9)/((1-x)^3*(1+x)^3*(1+x^2)^3). [Bruno Berselli, Jul 01 2013]
a(n) = 3*a(n-4) -3*a(n-8) +a(n-12). [Bruno Berselli, Jul 01 2013]
a(4n) = -A002939(n), a(4n+1) = A054569(n+1), a(4n+2) = A054554(n+2), a(4n+3) = A068377(n+2). [Bruno Berselli, Jul 02 2013]

More terms from Bruno Berselli, Jul 01 2013

A244679 The spiral of Champernowne read by the Northeast ray.

Original entry on oeis.org

1, 3, 1, 0, 3, 0, 1, 6, 1, 1, 3, 1, 2, 3, 2, 3, 7, 4, 4, 5, 5, 6, 7, 7, 7, 3, 9, 9, 7, 1, 2, 2, 5, 3, 6, 4, 5, 6, 2, 7, 7, 9, 0, 1, 1, 2, 0, 4, 7, 6, 2, 8, 5, 0, 6, 2, 5, 4, 2, 7, 7, 9, 0, 2, 1, 4, 0, 7, 7, 0, 2, 2, 5, 5, 6, 8, 5, 1, 2, 4, 7, 7, 0, 1, 1, 4, 0, 8, 7, 1, 2, 5, 5, 8, 6, 2, 5, 6, 2, 0, 0, 4, 0, 0, 8

Offset: 1

Robert G. Wilson v, Jul 04 2014

			See A244677 for the spiral of David Gawen Champernowne.

Cf. A033307, A054554, A244677 - A244688, A033952, A244690 - A244692.

almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]]; f[n_] := 4n^2 - 10n + 7 (* see A244677 formula section *); Array[ almostNatural[ f@#, 10] &, 105]

See A244677 formula section.

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A163586 a(1) = 3. For n > 1, Ulam's spiral is started with a(n-1), and the primes p on the NE spoke are considered. a(n) is the minimal p that is the lesser of a twin prime pair.

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A226940 a(0)=0; if a(n-1) is odd, a(n) = n + a(n-1), otherwise a(n) = n - a(n-1).

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A244679 The spiral of Champernowne read by the Northeast ray.

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