, 1|2}][[All, 1]] (* _Jean-François Alcover

A361502 Index of n-th prime in A359804.

2, 3, 4, 8, 13, 42, 347, 3466, 49012, 528231, 717126, 63056215, 1375559400, 7038527851

Offset: 1

N. J. A. Sloane, Mar 18 2023, based on a comment made by Michael De Vlieger in A359804 in which he gave the values of a(1) to a(12)

Theorem: Every prime appears in A359804. For proof see A359804.

It appears that the primes in A359804 appear in order.

Rémy Sigrist, C++ program

Cf. A359804, A361504.

nn = 2^20; c[] = False; q[] = 1;
 i = 1; j = 2; c[1] = c[2] = True; u = 3;
 {2}~Join~Reap[Monitor[Do[
      (k = q[#]; While[c[k #], k++]; k *= #;
         While[c[# q[#]], q[#]++]) &[(p = 2;
        While[Divisible[i j, p], p = NextPrime[p]]; p)];
      If[PrimeQ[k], Sow[n]; Print[n]];
      Set[{c[k], i, j}, {True, j, k}];
If[k == u, While[c[u], u++]], {n, 3, nn}], n]][[-1, -1]] (* Michael De Vlieger, Mar 19 2023 *)

a(13)-a(14) from Rémy Sigrist, Mar 19 2023

A354168 Let M_p = 2^p-1 be a Mersenne prime, where p is an odd prime. Sequence lists p such that b_{p-2} == -2^((p+1)/2) mod M_p, where {b_k} is defined in the Comments.

Original entry on oeis.org

7, 17, 19, 89, 107, 521, 607, 1279, 2281, 3217, 4423, 9689, 11213, 21701, 44497, 216091, 859433, 1257787, 24036583, 30402457, 32582657, 42643801, 57885161, 74207281, 82589933

Offset: 1

N. J. A. Sloane, Jun 02 2022, based on Section 16.1 of Cosgrave (2022)

Let M_p = 2^p-1 (not necessarily a prime) where p is an odd prime, and define b_1 = 4; b_k = b_{k-1}^2 - 2 (mod M_p) for k >= 2.
The Lucas-Lehmer theorem says that M_p is a prime iff b_{p-1} == 0 (mod M_p).
Furthermore, if M_p is a prime, then b_{p-2} is congruent to +- 2^((p+1)/2) (mod M_p).
This partitions the Mersenne prime exponents A000043 into two classes, listed here and in A354167.

J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 16.1.

Mersenneforum, data for all known Mersenne penultimate residues (up to M#51)
Wikipedia, Lucas-Lehmer primality test. Sign of penultimate term

Cf. A000043, A000668, A354167.

Cf. A123271 (sign of the penultimate term of the Lucas-Lehmer sequence).

Thanks to Chai Wah Wu for several corrections. - N. J. A. Sloane, Jun 02 2022
a(16) from Chai Wah Wu, Jun 03 2022
a(17)-a(18) from Chai Wah Wu, Jun 04 2022
a(19)-a(25) from Serge Batalov, Jun 11 2022

A354167 Let M_p = 2^p-1 be a Mersenne prime, where p is an odd prime. Sequence lists p such that b_{p-2} == 2^((p+1)/2) mod M_p, where {b_k} is defined in the Comments.

Original entry on oeis.org

3, 5, 13, 31, 61, 127, 2203, 4253, 9941, 19937, 23209, 86243, 110503, 132049, 756839, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 25964951, 37156667, 43112609, 77232917

Offset: 1

N. J. A. Sloane, Jun 02 2022, based on Section 16.1 of Cosgrave (2022)

Let M_p = 2^p-1 (not necessarily a prime) where p is an odd prime, and define b_1 = 4; b_k = b_{k-1}^2 - 2 (mod M_p) for k >= 2.
The Lucas-Lehmer theorem says that M_p is a prime iff b_{p-1} == 0 (mod M_p).
Furthermore, if M_p is a prime, then b_{p-2} is congruent to +- 2^((p+1)/2) (mod M_p).
This partitions the Mersenne prime exponents A000043 into two classes, listed here and in A354168.

J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 16.1.

Mersenneforum, data for all known Mersenne penultimate residues (up to M#51)
Wikipedia, Lucas-Lehmer primality test. Sign of penultimate term

Cf. A000043, A000668, A354168.

Cf. A123271 (sign of the penultimate term of the Lucas-Lehmer sequence).

Thanks to Chai Wah Wu for several corrections. - N. J. A. Sloane, Jun 02 2022
a(15) from Chai Wah Wu, Jun 04 2022
a(16)-a(25) from Serge Batalov, Jun 11 2022

A292766 Numbers n whose trajectory under iteration of the map k -> (sigma(k)+phi(k))/2 consists only of integers and is unbounded, excluding numbers n whose trajectory merges with the trajectory of a smaller number.

Original entry on oeis.org

270, 440, 496, 702, 737, 813, 828, 897, 905, 1027, 1066, 1099, 1240, 1241, 1260, 1331, 1353, 1368, 1371, 1422, 1507, 1537, 1754, 1760, 1834, 1848, 2002, 2016, 2282

Offset: 1

N. J. A. Sloane, Sep 27 2017, based on emails from Sean A. Irvine, Sep 14 2017, who computed a(1)-a(9), and Hans Havermann, same date, who computed a(10)-a(29). Hugo Pfoertner also computed many of these terms

nonn

These are the "seeds" in A291790, that is, every number which blows up under iteration of the map k -> (sigma(k)+phi(k))/2 belongs to one of these trajectories. AT PRESENT ALL TERMS ARE CONJECTURAL.
The trajectories of these numbers are pairwise disjoint for the first 400 steps.
This is unsatisfactory because it is possible that, at some later step, these trajectories may merge, reach a prime (a fixed point), or reach a fraction (and die). However, this seems unlikely on probabilistic grounds - see the remarks of Andrew R. Booker in A292108.
Normally such a sequence would not be included in the OEIS, but exceptions have been made for this and A291790 because a number of people have worked on them, and also in the hope that this will encourage resolution of some of the open questions.
Needs a b-file.

Sean A. Irvine, Showing how the initial portions of some of these trajectories merge

Cf. A291790, A292108.

A281432 E.g.f. C(x) satisfies: C(x) = cosh( Integral C(x)^5 dx ).

Original entry on oeis.org

1, 1, 21, 1341, 173961, 38032281, 12572222301, 5853055718421, 3649908317290641, 2936992590376253361, 2962874009751302380581, 3662113951884448455886701, 5442785335874229886957949721, 9576772316393556595041389524041, 19688717121629473508791582116311661, 46766278604566476892923500929537926981, 127098490344228968075529350992858163636001

Offset: 0

Paul D. Hanna, Jan 21 2017

nonn

terms = 20; max = 2terms; se = Series[(1/8)*((x*(5+3x^2))/(1+x^2)^2 + 3* ArcTan[x]), {x, 0, max}]; s[x_] = InverseSeries[se, x] // Normal; coes = CoefficientList[Sqrt[1+s[x]^2]+O[x]^(max+1), x]*Range[0, max]!; Partition[ coes, 2][[All, 1]] (* Jean-François Alcover, Mar 01 2017 *)

{a(n) = my(S=x, C=1); for(i=0, n, S = intformal( C^6 +x*O(x^(2*n))); C = 1 + intformal( S*C^5 ) ); (2*n)!*polcoeff(C, 2*n)}
for(n=0, 30, print1(a(n), ", "))

C(x)^2 - S(x)^2 = 1 and C(x) = 1 + Integral C(x)^5*S(x) dx, where S(x) is described by A281431.

A260036 Number of configurations of the general monomer-dimer model for an 8 X 2n square lattice.

Original entry on oeis.org

1, 7573, 211351945, 6158217253688, 179788343101980135, 5249581929453966097649, 153282416794739031814924079, 4475693398084103779028604877129, 130685784089846859975392644067590259, 3815894599555857747953551843550514992237

Offset: 0

N. J. A. Sloane, Jul 19 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..225
N. Allegra, Exact solution of the 2d dimer model: Corner free energy, correlation functions and combinatorics, arXiv:1410.4131 [cond-mat.stat-mech], 2014. See Table 5.

Bisection (even part) of A033511.

A033511 = Cases[Import["https://oeis.org/A033511/b033511.txt", "Table"], {, }][[All, 2]];
Partition[A033511, 2][[All, 1]] (* Jean-François Alcover, Feb 29 2020 *)

a(0), a(5)-a(9) from Alois P. Heinz, Mar 08 2016

A251411 Numbers k such that A098550(k) = k.

Original entry on oeis.org

1, 2, 3, 4, 12, 50, 86

Offset: 1

N. J. A. Sloane, Dec 02 2014

There is a strong conjecture that there are no further terms. See the discussion in the comments in A098550.

L. Edson Jeffery, Posting to Sequence Fans Mailing List, Nov 30 2014.

Hans Havermann, Loops and unresolved chains for map n -> A098550(n) trajectories
David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669, 2015 and J. Int. Seq. 18 (2015) 15.6.7.

Cf. A098550, A251412, A251556.

max = 100;
f[lst_] := Block[{k = 4}, While[GCD[lst[[-2]], k] == 1 || GCD[lst[[-1]], k] > 1 || MemberQ[lst, k], k++]; Append[lst, k]];
A098550 = Nest[f, {1, 2, 3}, max - 3];
Select[Transpose[{Range[max], A098550}], #[[1]] == #[[2]]&][[All, 1]] (* Jean-François Alcover, Sep 05 2018, after Robert G. Wilson v in A098550 *)

from math import gcd
A251411_list, l1, l2, s, b = [1,2,3], 3, 2, 4, {}
for n in range(4,10**4):
    i = s
    while True:
        if not i in b and gcd(i,l1) == 1 and gcd(i,l2) > 1:
            l2, l1, b[i] = l1, i, 1
            while s in b:
                b.pop(s)
                s += 1
            if i == n:
                A251411_list.append(n)
            break
        i += 1 # Chai Wah Wu, Dec 03 2014

A210493 Transits of Venus since the invention of the telescope by Julian Date (rounded).

Original entry on oeis.org

2317111, 2320030, 2364409, 2367328, 2405867, 2408786, 2453165, 2456085, 2458099, 2497542, 2541921, 2544841, 2583379, 2586298

Offset: 1

Fred Espenak (fred.espenak-1(AT)nasa.gov) or (info01(AT)MrEclipse.com) and Robert G. Wilson v, Jan 23 2013

nonn

"Transits of Venus are among the rarest of predictable astronomical phenomena. They occur in a pattern that generally repeats every 243 years, with pairs of transits eight years apart separated by long gaps of 121.5 years and 105.5 years. The periodicity is a reflection of the fact that the orbital periods of Earth and Venus are close to 8:13 and 243:395 commensurabilities." - Wikipedia

a(n) is approximately 365.25 * A171467(n+46). - Charles R Greathouse IV, Jan 24 2013

			05:19 07 Dec 1631 = 2317110.721528
18:25 04 Dec 1639 = 2320030.267361
05:19 06 Jun 1761 = 2364408.721528
22:25 03 Jun 1769 = 2367328.434028
04:05 09 Dec 1874 = 2405866.670139
17:06 06 Dec 1882 = 2408786.212500
08:19 08 Jun 2004 = 2453164.846528
01:28 06 Jun 2012 = 2456084.561111
02:48 11 Dec 2117 = 2458098.616667
16:01 08 Dec 2125 = 2497542.167361
11:30 11 Jun 2247 = 2541920.979167
04:36 09 Jun 2255 = 2544840.691667
01:40 13 Dec 2360 = 2583378.569444
14:43 10 Dec 2368 = 2586298.113194

Jean Meeus, Transits, Willmann-Bell, 1989.
Jean Meeus, Astronomical Algorithms, Second Ed., 1999.

Fred Espenak, The Ultimate Resource for Eclipse Photography, 2001.
NASA Science, Saturn's Transit of Venus on Dec. 21, 2012
NASA Eclipse Web Site, Planetary Transits Across the Sun
NASA Eclipse Web Site, Eclipses and Transits of 2012
NASA Eclipse Web Site, 2004 and 2012 Transits of Venus
NASA Eclipse Web Site, Six Millennium Catalog of Venus Transits: 2000 BCE to 4000 CE
NASA / Goddard Space Flight Center, Calendars, by L. E. Doggett
The United States Naval Observatory (USNO), Astronomical Applications Department, Julian Date Converter
Wikipedia, 2012 transit of Venus

Cf. A171467.

A210996 Number of free polyominoes with 2n cells.

Original entry on oeis.org

1, 1, 5, 35, 369, 4655, 63600, 901971, 13079255, 192622052, 2870671950, 43191857688, 654999700403, 9999088822075, 153511100594603, 2368347037571252, 36695016991712879, 570694242129491412, 8905339105809603405, 139377733711832678648, 2187263896664830239467, 34408176607279501779592

Offset: 0

Omar E. Pol, Sep 15 2012

nonn

It appears that we can write A216492(n) < A216583(n) < A056785(n) < A056786(n) < a(n) < A210988(n) < A210986(n), if n >= 3. - Omar E. Pol, Sep 16 2012

			For n = 1 there is only one free domino. For n = 2 there are 5 free tetrominoes. For n = 3 there are 35 free hexominoes. For n = 4 there are 369 free octominoes (see link section).

John Mason, Table of n, a(n) for n = 0..25
Herman Tulleken, Polyominoes 2.2: How they fit together, (2019).
Eric Weisstein's World of Mathematics, Polyomino
Wikipedia, All 5 free tetrominoes, Illustration of a(2) = 5.
Wikipedia, All 35 free hexominoes, Illustration of a(3) = 35.
Wikipedia, All 369 free octominoes, Illustration of a(4) = 369.
Wikipedia, Polyomino

Bisection of A000105.

Cf. A000988, A056785, A056786, A210988, A210997, A216492, A216583.

A000105 = Cases[Import["https://oeis.org/A000105/b000105.txt", "Table"], {, }][[All, 2]];
Partition[A000105, 2][[All, 1]] (* Jean-François Alcover, Jan 03 2020 *)

a(n) = A000105(2n).

a(n) = A213376(n) + A056785(n). - R. J. Mathar, Feb 08 2023

More terms from John Mason, Apr 15 2023

A198388 Square root of first term of a triple of squares in arithmetic progression.

Original entry on oeis.org

1, 2, 7, 3, 7, 4, 17, 14, 5, 1, 6, 14, 23, 7, 31, 21, 8, 34, 9, 28, 21, 10, 49, 17, 11, 47, 2, 12, 35, 23, 13, 28, 51, 46, 71, 14, 62, 42, 7, 15, 16, 41, 35, 17, 41, 49, 79, 3, 68, 18, 97, 19, 56, 7, 42, 20, 69, 98, 34, 21, 93, 31, 63, 22, 85, 94, 23, 49, 73

Offset: 1

Reinhard Zumkeller, Oct 24 2011

nonn

There is a connection to |x-y| of Pythagorean triangles (x,y,z). See a comment on the primitive Pythagorean triangle case under A198441 which applies mutatis mutandis. - Wolfdieter Lang, May 23 2013

			Connection to Pythagorean triangles: a(2) = 2 because (in the notation of the Zumkeller link) (u,v,w) = 2*(1,5,7) and the corresponding Pythagorean triangle is 2*((7-1)/2,(1+7)/2,5) = 2*(3,4,5) with |x-y| = 2*(4-3) = 2. - _Wolfdieter Lang_, May 23 2013

Ray Chandler, Table of n, a(n) for n = 1..10000
Keith Conrad, Arithmetic progressions of three squares
Reinhard Zumkeller, Table of initial values

Cf. A198384, A198409, A198439.

a198388 n = a198388_list !! (n-1)
a198388_list = map (\(x,,) -> x) ts where
   ts = [(u,v,w) | w <- [1..], v <- [1..w-1], u <- [1..v-1],
                   w^2 - v^2 == v^2 - u^2]

wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u, v, w}]]]]][[2]];
Flatten[DeleteCases[triples /@ Range[wmax], {}], 2][[All, 1]] (* Jean-François Alcover, Oct 20 2021 *)

A198384(n) = a(n)^2.

A198439(n) = a(A198409(n)).

cp's OEIS Frontend

User: , 1|2}][[All, 1]] (* _Jean-François Alcover

A361502 Index of n-th prime in A359804.

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A354168 Let M_p = 2^p-1 be a Mersenne prime, where p is an odd prime. Sequence lists p such that b_{p-2} == -2^((p+1)/2) mod M_p, where {b_k} is defined in the Comments.

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A354167 Let M_p = 2^p-1 be a Mersenne prime, where p is an odd prime. Sequence lists p such that b_{p-2} == 2^((p+1)/2) mod M_p, where {b_k} is defined in the Comments.

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A292766 Numbers n whose trajectory under iteration of the map k -> (sigma(k)+phi(k))/2 consists only of integers and is unbounded, excluding numbers n whose trajectory merges with the trajectory of a smaller number.

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Comments

Links

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A281432 E.g.f. C(x) satisfies: C(x) = cosh( Integral C(x)^5 dx ).

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Programs

Mathematica

PARI

Formula

A260036 Number of configurations of the general monomer-dimer model for an 8 X 2n square lattice.

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Mathematica

Extensions

A251411 Numbers k such that A098550(k) = k.

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Mathematica

Python

A210493 Transits of Venus since the invention of the telescope by Julian Date (rounded).

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A210996 Number of free polyominoes with 2n cells.

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