1,1

fQ[n_]:=PrimeQ[n^2-48]&&PrimeQ[n^2+48]; lst={}; Do[If[fQ@Prime[n], AppendTo[lst, Prime[n]]], {n, 7!}]; lst

Cf. A153116, A153119, A153120

Sequence in context: A129745 A038964 A019401 this_sequence A098333 A162862 A043338

Adjacent sequences: A153317 A153318 A153319 this_sequence A153321 A153322 A153323

nonn

Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 23 2008

2,5

From Table 2, p.11, of Sikiric. Number of 2-self-hedrites with 4 <= n <= 40 and and 2 <= i <= 4. An i-hedrite is a 4-regular plane graph with faces of size 2, 3 and 4. We do a short survey of their known properties and explain some new algorithms that allow their efficient enumeration. Using this we give the symmetry groups of all i-hedrites and the minimal representative for each. We also review the link of 4-hedrites with knot theory and the classification of 4-hedrites with simple central circuits. An i-self-hedrite is a self-dual plane graph with faces and vertices of size/degree 2, 3 and 4. We give a new efficient algorithm for enumerating them based on i-hedrites. We give a classification of their possible symmetry groups and a classification of 4-self-hedrites of symmetry T, Td in terms of the Goldberg-Coxeter construction. Then we give a method for enumerating 4-self-hedrites with simple zigzags.

Mathieu Dutour Sikiric, Michel Deza, 4-regular and self-dual analogs of fullerenes, Oct 28, 2009.

Cf. A167156-A167160, A167227, A167228.

Sequence in context: A156097 A039597 A000937 this_sequence A068902 A077569 A073935

Adjacent sequences: A167226 A167227 A167228 this_sequence A167230 A167231 A167232

nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 30 2009

1,1

The root structure is nearly all semi-unique algebriac irrationals: NSolve[CharacteristicPolynomial[M, x] == 0, x] {{x -> -2.1893697030273365`}, { x -> -1.8207023936551185`}, {x -> -1.2128832881490639` - 0.35413663859184796`I}, {x -> -1.2128832881490639` + 0.35413663859184796` I}, {x -> -0.09433091961414544`}, {x -> 0.223526912678973`}, {x -> 1.2182640154457587`}, {x -> 1.9477296343029344`}, {x -> 3.140649030167062`}} The game value of the matrix is: Det[M]/(Sum[Sum[If[i == j, M[[i, j]], 0], {i, 1, 9}], {j, 1, 9}] - Sum[Sum[If[i == j, 0, M[[i, j]]], {i, 1, 9}], {j, 1, 9}]) 1/(3^3+1)=1/28

M = {{0, 1, 0, 1, 0, 0, 1, 0, 0}, {0, 0, 1, 0, 1, 0, 0, 1, 0}, {1, 1, 0, 0, 0, 1, 0, 0, 1}, {1, 0, 0, 0, 1, 1, 1, 0, 0}, {0, 1, 0, 1, 0, 0, 0, 1, 0}, {0, 0, 1, 0, 1, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 0, 0, 0, 1}, {0, 1, 0, 0, 1, 0, 0, 1, 0}, {0, 0, 1, 0, 0, 1, 1, 0, -1}}; v[0] = {1, 0, 0, 0, 1, 0, 0, 0, 1}; v[n_] := v[n] = M.v[n - 1]; a(n) = Sum[v[n][[i]],{i,1,9}]

M = {{0, 1, 0, 1, 0, 0, 1, 0, 0}, {0, 0, 1, 0, 1, 0, 0, 1, 0}, {1, 1, 0, 0, 0, 1, 0, 0, 1}, {1, 0, 0, 0, 1, 1, 1, 0, 0}, {0, 1, 0, 1, 0, 0, 0, 1, 0}, {0, 0, 1, 0, 1, 0, 0, 0, 1}, {1, 0, 0, 1, 0, 0, 0, 0, 1}, {0, 1, 0, 0, 1, 0, 0, 1, 0}, {0, 0, 1, 0, 0, 1, 1, 0, -1}}; v[0] = {1, 0, 0, 0, 1, 0, 0, 0, 1}; v[n_] := v[n] = M.v[n - 1]; a = Table[Apply[Plus, v[n]], {n, 0, 50}]

Sequence in context: A058145 A161590 A018361 this_sequence A123947 A135142 A098589

Adjacent sequences: A134322 A134323 A134324 this_sequence A134326 A134327 A134328

nonn,uned

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 16 2008

1,1

Sequence in context: A067378 A071886 A087939 this_sequence A139924 A155572 A107145

Adjacent sequences: A142408 A142409 A142410 this_sequence A142412 A142413 A142414

nonn

N. J. A. Sloane (njas(AT)research.att.com), Jul 11 2008

1,1

Discriminant=-132.

Consider the quadratic form f(x,y)=ax^2+bxy+cy^2. When the discriminant d=b^2-4ac is -4 times an idoneal number (A000926), there is exactly one class for each genus. As a result, the primes generated by f(x,y) are the same as the primes congruent to S (mod -d), where S is a set of numbers less than -d. The table on page 60 of Cox shows that there are exactly 331 quadratic forms having this property. The 217 sequences starting with this one complete the collection in OEIS.

When a=1 and b=0, f(x,y) is a quadratic form whose congruences are discussed in A139642. Let N be an idoneal number. Then there are 2^r reduced quadratic forms whose discriminant is -4N, where r=1,2,3, or 4. By collecting the residuals p (mod 4N) for primes p generated by the i-th reduced quadratic form, we can empirically find a set Si. To show that the 2^r sets Si are complete, we only need to show that the union of the Si is equal to the set of numbers k such the Jacobi symbol (-k/4N)=1.

David A. Cox, Primes of Form x^2 + n y^2, Wiley, 1989.

The primes are congruent to {2, 17, 29, 41, 65, 101} (mod 132).

f[x_, y_]:=2*x^2+2*x*y+17*y^2; lst={}; Do[Do[p=f[x, y]; If[PrimeQ[p], AppendTo[lst, p]], {y, -4!, 3*4!}], {x, -4!, 3*4!}]; Take[Union[lst], 90] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 30 2009]

QuadPrimes[2, -2, 17, 10000] (* see A106856 *)

Cf. A002313 (d=-4) and the following:

A033200 (d=-8),

A007645 (d=-12),

A002144 (d=-16),

A033205, A106865 (d=-20),

A033199, A084865 (d=-24),

A033207 (d=-28),

A007519, A007520 (d=-32),

A068228, A040117 (d=-36),

A033201, A106889 (d=-40),

A068228, A068229 (d=-48),

A033210, A106906 (d=-52),

A033212, A106859 (d=-60),

A007519, A007521 (d=-64),

A106950, A106949 (d=-72),

A033215, A102271, A102273, A106972 (d=-84),

A033216, A106984 (d=-88),

A107008, A107003, A107006, A107007 (d=-96),

A033205, A122487 (d=-100),

A107134, A107133 (d=-112),

A033220, A107135, A107136, A107137 (d=-120),

A033222, A107138, A139827, A139828 (d=-132),

A033225, A007639 (d=-148),

A107145, A107144, A139829, A139830 (d=-160),

A033229, A107146, A107147, A107148 (d=-168),

A107152, A107151, A139831, A139832 (d=-180),

A107008, A107006, A107154, A139530 (d=-192),

A033236, A107165, A139833, A139834 (d=-228),

A033237, A107166 (d=-232),

A107152, A107167, A107168, A107169 (d=-240),

A033245, A107178, A107179, A107180 (d=-280),

A107008, A107007, A107154, A107181 (d=-288),

A033250, A107188, A107189, A107190 (d=-312),

A033254, A107199, A139835, A139836 (d=-340),

A107202, A107201, A139837, A139838 (d=-352),

A033202, A107210, A139839, A139840 (d=-372),

A139643, A139841-A139843 (d=-408),

A139644, A139844-A139850 (d=-420),

A139645, A139851-A139853 (d=-448),

A139502, A139854-A139860 (d=-480),

A139646, A139861-A139863 (d=-520),

A139647, A139864-A139866 (d=-532),

A139648, A139867-A139873 (d=-660),

A139506, A139874-A139880 (d=-672),

A139649, A139881-A139883 (d=-708),

A139650, A139884-A139886 (d=-760),

A139651, A139887-A139893 (d=-840),

A139652, A139894-A139896 (d=-928),

A139502, A139855, A139857, A139858, A139897-A139899, A139902 (d=-960),

A139653, A139904-A139906 (d=-1012),

A139654, A139907-A139913 (d=-1092),

A139655, A139914-A139920 (d=-1120),

A139656, A139921-A139927 (d=-1248),

A139657, A139928-A139934 (d=-1320),

A139658, A139935-A139941 (d=-1380),

A139659, A139942-A139948 (d=-1428),

A139660, A139949-A139955 (d=-1540),

A139661, A139956-A139962 (d=-1632),

A139662, A139963-A139969 (d=-1848),

A139663, A139970-A139976 (d=-2080),

A139664, A139977-A139983 (d=-3040),

A139665, A139984-A139998 (d=-3360),

A139666, A139999-A140013 (d=-5280),

A139667, A140014-A140028 (d=-5460),

A139668, A140029-A140043 (d=-7392).

Sequence in context: A171605 A018759 A132146 this_sequence A063118 A141068 A162622

Adjacent sequences: A139824 A139825 A139826 this_sequence A139828 A139829 A139830

nonn,easy

T. D. Noe (noe(AT)sspectra.com), May 02 2008, May 07 2008

1,1

Some a(n) are equal to 1 (n=2,5,6,14,21,25,38,40,42,47,48,53,59,69,70..). All other a(n) are primes that belong to A038907 (33 is a square mod p).

a(n) = Abs[numerator[Det[DiagonalMatrix[Table[(i+1)/(i+2)-1,{i,1,n}]]+1]]].

Abs[Numerator[Table[Det[DiagonalMatrix[Table[(i+1)/(i+2)-1, {i, 1, n}]]+1], {n, 1, 70}]]]

Cf. A118679, A038907.

Sequence in context: A055459 A080958 A138351 this_sequence A063624 A101851 A111724

Adjacent sequences: A120290 A120291 A120292 this_sequence A120294 A120295 A120296

frac,nonn

Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 08 2006

1,2

R. K. Guy reports, Apr 14 2005: In Math. Mag. 48 (1975) 301 one finds "C. W. Trigg, C. C. Oursler and R. Cormier & J. L. Selfridge have sent calculations on Problem 886 [Nov 1973] for which we had received only partial results [Jan 1975]. Cormier and Selfridge sent the following results: There appear to be five sequences beginning with integers less than 1000 which do not merge. These sequences were carried out to 10^8 or more." The five sequences are A003508, A105210-A105213.

This suggests that there may be infinitely many different (non-merging) sequences obtained by choosing different starting values.

All terms of these five sequences are distinct up to least 10^30. - T. D. Noe, Oct 19 2007

Problem 886, Math. Mag., 48 (1975), 57-58.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..2000

a(6)=8, so a(7) = 8 + 1 + 2 = 11.

a[1] = 1; a[n_] := a[n] = a[n - 1] + 1 + Plus @@ Select[ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[ a[n - 1]]], # < a[n - 1] &]; Table[ a[n], {n, 54}] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 13 2005)

Cf. A096460, A105221, A105233

Sequence in context: A089190 A065294 A026808 this_sequence A078662 A050048 A122456

Adjacent sequences: A003505 A003506 A003507 this_sequence A003509 A003510 A003511

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Henry Bottomley (se16(AT)btinternet.com), May 09 2000

0,6

Also, number of partitions of n where there are fewer than k parts equal to k for all k. - Jon Perry (perry(AT)globalnet.co.uk) and Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 04 2004. E.g. a(8)=5 because we have 8=6+2=5+3=4+4=3+3+

G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004. page 48.

James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.

G.f.: Product_{m>0} (1-x^(m^2))/(1-x^m). - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 21 2003

a(n) = (1/n)*Sum_{k=1..n} (A000203(k)-A035316(k))*a(n-k), a(0)=1. - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 21 2003

G.f.: product(i=1, oo, sum(j=0, i-1, x^(i*j) )). - Jon Perry (perry(AT)globalnet.co.uk), Jul 26 2004

n=7: 2+5 = 2+2+3 = 7: a(7)=3;

n=8: 2+6 = 2+2+2+2 = 2+3+3 = 3+5 = 8: a(8)=5;

n=9: 2+7 = 2+2+5 = 2+2+2+3 = 3+3+3 = 3+6: a(9)=5.

g:=product((1-x^(i^2))/(1-x^i), i=1..70):gser:=series(g, x=0, 60):seq(coeff(gser, x^n), n=1..53); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 09 2006

Drop[ CoefficientList[ Series[ Product[ Sum[x^(i*j), {j, 0, i - 1}], {i, 1, 54}], {x, 0, 54}], x], 1] (from Robert G. Wilson v Aug 05 2004)

Cf. A087154, A001156, A000009, A000037, A052335 (<=k parts of k).

A115584, A172151. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 26 2010]

nonn,new

Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 21 2003

Zero term added by Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jan 25 2010

1,1

Discriminant = 85. Class = 2. Binary quadratic forms a*x^2+b*x*y+c*y^2 have discriminant d=b^2-4ac and gcd(a,b,c)=1

Borevich and Shafaewich, Number Theory.

D. B. Zagier, Zetafunktionen und quadratische Koerper.

a(1)=3 because we can write 3= 3*1^2+5*1*0-5*0^2 (or 3=7*0^2+13*0*1+3*1^2

Cf. A141773 (d=85). See also A038872 (d=5). A141131 (d=8). A141122, A141123 (d=12). A038883 (d=13). A038889 (d=17). A141158 (d=20). A141159, A141160 (d=21). A141170, A141171 (d=24). A141172, A141173 (d=28). A141174, A141175 (d=32). A141176, A141177 (d=33). A141178 (d=37). A141179, A141180 (d=40). A141181 (d=41). A141182, A141183 (d=44). A141184, A141185 (d=45). A141122, A141187 (d=48). A141188 (d=52). A141189 (d=53). A141190, A141191 (d=56). A141192, A141193 (d=57). A141301, A141302, A141303, A141304 (d=60). A141215 (d=61). A141111, A141112 (d=65). A141750 (d=73). A141161, A141162, A141163 (d=148). A141164, A141165, A141166 (d=229). A141167, A141168, A141169 (d=257).

Sequence in context: A137258 A053341 A086086 this_sequence A032496 A002092 A057476

Adjacent sequences: A141769 A141770 A141771 this_sequence A141773 A141774 A141775

nonn

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (sergarmor(AT)yahoo.es), Jul 04 2008

0,1

The least prime which has n prime prime residues.

f[n_] := Length[ Select[ Mod[ Prime[n], Prime[ Range[n]]], PrimeQ[ # ] &]]; g[n_] := Block[{k = 1}, While[ f[ k] != n, k++ ]; Prime[k]]; Table[ g[n], {n, 0, 48}]

Cf. A102351.

Sequence in context: A087952 A124255 A079936 this_sequence A156009 A164620 A156013

Adjacent sequences: A102851 A102852 A102853 this_sequence A102855 A102856 A102857

nonn

Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 28 2005