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A272075 Primes of the form k^4 + 29*k^2 + 101.

101, 131, 233, 443, 821, 1451, 2441, 3923, 6053, 9011, 13001, 18251, 25013, 33563, 44201, 57251, 73061, 92003, 114473, 140891, 207371, 295283, 476681, 951491, 1078373, 1369961, 1536251, 1913963, 3472523, 3804341, 4159451, 4943843, 5834531, 7972043, 9925541

Offset: 1

Robert Price, Apr 19 2016

			233 is prime and it is in this sequence since 233 = 2^4 + 29*2^2 + 101.

Robert Price, Table of n, a(n) for n = 1..2264
Eric Weisstein's World of Mathematics, Prime-Generating Polynomials

Cf. A050268, A050267, A005846, A007641, A007635, A048988, A050265, A050266.

Cf. A271980, A272074.

n = Range[0, 100]; Select[#^4 + 29#^2 + 101, PrimeQ[#] &]

lista(nn) = for(n=0, nn, if(ispseudoprime(p=n^4+29*n^2+101), print1(p, ", "))); \\ Altug Alkan, Apr 19 2016

A030567 Triangle T(n,k): Write n in base 6 and reverse order of digits to get row n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 0, 4, 1, 4, 2, 4, 3, 4, 4, 4, 5, 4, 0, 5, 1, 5, 2, 5, 3, 5, 4, 5, 5, 5, 0, 0, 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 4, 0, 1, 5, 0, 1, 0, 1, 1, 1, 1, 1, 2

Offset: 0

Clark Kimberling

If columns are numbered starting with k=0, then T(n,k) contains the coefficient of 6^k in n's base-6 expansion. - M. F. Hasler, Jul 21 2013

R. J. Mathar, Table of n, a(n) for n = 0..5950

See A030548 for a quite complete list of crossreferences.
Cf. A030568 - A030573 for positions of a given digit.
Cf. A030575 - A030580 for run lengths, A030581 - A030585 for more.
Row sums (same as those of A030548) are in A053827.
Cf. A030308, A030341, A030386, A031235, A031007, A031045, A031087, A031298 for the base-2 to base-10 analogs.

Flatten[Table[Reverse[IntegerDigits[n,6]],{n,0,50}]] (* Harvey P. Dale, Sep 27 2015 *)

A030567(n,k=-1)=/*k<0&&error("Flattened sequence not yet implemented.")*/n\6^k%6 \\ Assuming that columns start with k=0, cf. comment. TO DO: implement flattened sequence, such that A030567(n)=a(n). - M. F. Hasler, Jul 21 2013

Initial 0 and better name from Philippe Deléham, Oct 20 2011

Edited and crossrefs added by M. F. Hasler, Jul 21 2013

A046601 First numerator and then denominator of the 1/5-Pascal triangle (by row). To get a 1/5-Pascal triangle, replace "2" in the third row of the Pascal triangle with "1/5" and calculate all other rows as in the Pascal triangle.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 6, 5, 6, 5, 1, 1, 1, 1, 11, 5, 12, 5, 11, 5, 1, 1, 1, 1, 16, 5, 23, 5, 23, 5, 16, 5, 1, 1, 1, 1, 21, 5, 39, 5, 46, 5, 39, 5, 21, 5, 1, 1, 1, 1, 26, 5, 12, 1, 17, 1, 17, 1, 12, 1, 26, 5, 1, 1, 1, 1, 31, 5, 86, 5, 29, 1, 34, 1, 29, 1, 86, 5

Offset: 1

Mohammad K. Azarian

			1/1;
1/1  1/1;
1/1  1/5  1/1;
1/1  6/5  6/5  1/1;
1/1 11/5 12/5 11/5  1/1;
1/1 16/5 23/5 23/5 16/5  1/1;
1/1 21/5 39/5 46/5 39/5 21/5  1/1;
1/1 26/5 12/1 17/1 17/1 12/1 26/5 1/1; ...

A031235 Triangle T(n,k): write n in base 5, reverse order of digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 0, 4, 1, 4, 2, 4, 3, 4, 4, 4, 0, 0, 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 4, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 4, 1, 1, 0, 2, 1, 1, 2, 1, 2, 2, 1, 3, 2, 1, 4, 2, 1, 0

Offset: 0

Clark Kimberling

Reinhard Zumkeller, Rows n = 0..1000 of triangle, flattened

Cf. A030308, A030341, A030386, A030567, A031007, A031045, A031087, A031298 for the base-2 to base-10 analogs.

Cf. A007091.

a031235 n k = a031235_tabf !! n !! k
a031235_row n = a031235_tabf !! n
a031235_tabf = iterate succ [0] where
   succ []     = [1]
   succ (4:ts) = 0 : succ ts
   succ (t:ts) = (t + 1) : ts
-- Reinhard Zumkeller, Sep 18 2015

Reverse[IntegerDigits[#,5]]&/@Range[0,40]//Flatten (* Harvey P. Dale, Aug 02 2016 *)

A031235(n, k=-1)=/*k<0&&error("Flattened sequence not yet implemented.")*/n\5^k%5 \\ Assuming that columns are numbered starting with k=0 as in A030308, A030341, ... - M. F. Hasler, Jul 21 2013

Initial 0 and better name by Philippe Deléham, Oct 20 2011

A030386 Triangle T(n,k): write n in base 4, reverse order of digits.

Original entry on oeis.org

0, 1, 2, 3, 0, 1, 1, 1, 2, 1, 3, 1, 0, 2, 1, 2, 2, 2, 3, 2, 0, 3, 1, 3, 2, 3, 3, 3, 0, 0, 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 0, 2, 1, 1, 2, 1, 2, 2, 1, 3, 2, 1, 0, 3, 1, 1, 3, 1, 2, 3, 1, 3, 3, 1, 0, 0, 2, 1, 0, 2, 2, 0, 2, 3, 0, 2, 0, 1, 2

Offset: 0

Clark Kimberling

			Triangle begins:
0
1
2
3
0, 1
1, 1
2, 1
3, 1
0, 2
1, 2
2, 2
3, 2
0, 3
1, 3
2, 3
3, 3
0, 0, 1
1, 0, 1 ... - _Philippe Deléham_, Oct 20 2011

Reinhard Zumkeller, Rows n = 0..1000 of triangle, flattened

Cf. A030308, A030341, A031235, A030567, A031007, A031045, A031087, A031298 for the base-2 to base-10 analogs.

Cf. A007090.

a030386 n k = a030386_tabf !! n !! k
a030386_row n = a030386_tabf !! n
a030386_tabf = iterate succ [0] where
   succ []     = [1]
   succ (3:ts) = 0 : succ ts
   succ (t:ts) = (t + 1) : ts
-- Reinhard Zumkeller, Sep 18 2015

A030386_row := n -> op(convert(n, base, 4)):
seq(A030386_row(n), n=0..36); # Peter Luschny, Nov 28 2017

Flatten[Table[Reverse[IntegerDigits[n,4]],{n,0,50}]] (* Harvey P. Dale, Oct 13 2012 *)

A030386(n, k=-1)=/*k<0&&error("Flattened sequence not yet implemented.")*/n\4^k%4 \\ Assuming that columns are numbered starting with k=0 as in A030308, A030341, ... \\ M. F. Hasler, Jul 21 2013

Initial 0 and better name by Philippe Deléham, Oct 20 2011

A031087 Triangle T(n,k): write n in base 9, reverse order of digits.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

Reinhard Zumkeller, Rows n = 0..1000 of triangle, flattened

Cf. A030308, A030341, A030386, A031235, A030567, A031007, A031045, A031298 for the base-2 to base-10 analogs.

Cf. A031076, A007095.

a031087 n k = a031087_row n !! (k-1)
a031087_row n | n < 9     = [n]
              | otherwise = m : a031087_row n' where (n',m) = divMod n 9
a031087_tabf = map a031087_row [0..]
-- Reinhard Zumkeller, Jul 07 2015

A031087(n, k=-1)=/*k<0&&error("Flattened sequence not yet implemented.")*/n\9^k%9 \\ Assuming that columns are numbered starting with k=0 as in A030308, A030567 and others. - M. F. Hasler, Jul 21 2013

Initial 0 and better name by Philippe Deléham, Oct 20 2011

A046213 First numerator and then denominator of 1/2-Pascal triangle (by row). To get a 1/2-Pascal triangle, replace "2" in third row of Pascal triangle with "1/2" and calculate all other rows as in Pascal triangle.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 3, 2, 1, 1, 1, 1, 5, 2, 3, 1, 5, 2, 1, 1, 1, 1, 7, 2, 11, 2, 11, 2, 7, 2, 1, 1, 1, 1, 9, 2, 9, 1, 11, 1, 9, 1, 9, 2, 1, 1, 1, 1, 11, 2, 27, 2, 20, 1, 20, 1, 27, 2, 11, 2, 1, 1, 1, 1, 13, 2, 19, 1, 67, 2, 40, 1, 67, 2, 19, 1, 13, 2, 1, 1, 1, 1, 15, 2

Offset: 1

Mohammad K. Azarian

			1/1;
1/1  1/1;
1/1  1/2  1/1;
1/1  3/2  3/2  1/1;
1/1  5/2  3/1  5/2  1/1;
1/1  7/2 11/2 11/2  7/2  1/1;
1/1  9/2  9/1 11/1  9/1  9/2  1/1;
1/1 11/2 27/2 20/1 20/1 27/2 11/2 1/1; ...

Peter J. C. Moses, Table of n, a(n) for n = 1..10000

fractionalPascal[1,] = {1}; fractionalPascal[2,] = {1,1}; fractionalPascal[3,frac_] = {1,frac,1}; fractionalPascal[n_,frac_] := fractionalPascal[n,frac] = Join[{1}, Map[Total, Partition[fractionalPascal[n-1,frac],2,1]],{1}]; Flatten[Map[Transpose,Transpose[{Numerator[#], Denominator[#]}]&[Map[fractionalPascal[#,1/2]&, Range[15]]]]] (* Peter J. C. Moses, Apr 04 2013 *)

A031214 Initial term of sequence An.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2, 1, 0, 1, 2, 0, 1, 0, 2, 2, 2, 1, 2, 1, 1, 2, 1, 0, 1, 1, 1, 0, 1, 2, 8, 14, 4, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 0, 4, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1

Offset: 1

Jeff Burch

This ignores the offset and gives the first term of the actual entry.
Since the sequences in the OEIS occasionally change their initial terms (for editorial reasons), this is an especially ill-defined sequence! - N. J. A. Sloane, Jan 01 2005
Sequences like this are deprecated. - Joerg Arndt, Apr 16 2020

			A000001 begins 0,1,1,1,2,1,2,1,5,2,... so a(1) = 0 = a(31214).

PrimeFan, Esoteric Integer Sequences
PrimeFan, Esoteric Integer Sequences [Cached copy]
N. J. A. Sloane, Online Encyclopedia of Integer Sequences
Index entries for sequences whose definition involves A_n (or An).

Cf. A039928, A051070, A091967, A107357, A102288; also A000001, A000002, A000003, A000004, A000005, etc.

Data updated by Sean A. Irvine, Apr 16 2020

A144396 The odd numbers greater than 1.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133

Offset: 1

Paul Curtz, Oct 03 2008

Last number of the n-th row of the triangle described in A142717.
If negated, these are also the values at local minima of the sequence A141620.
a(n) is the shortest leg of the n-th Pythagorean triple with consecutive longer leg and hypotenuse. The n-th such triple is given by (2n+1,2n^2+2n, 2n^2+2n+1), so that the longer legs are A046092(n) and the hypotenuses are A099776(n). - Ant King, Feb 10 2011
Numbers k such that the symmetric representation of sigma(k) has a pair of bars as its ends (cf. A237593). - Omar E. Pol, Sep 28 2018
Numbers k such that there is a prime knot with k crossings and braid index 2. (IS this true with "prime" removed?) - Charles R Greathouse IV, Feb 14 2023

Muniru A Asiru, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1)

Complement of A004275 and of A004277.
Cf. A000217, A002378, A005408, A007395, A046092, A099776, A237593, A254858.
Essentially the same as A140139, A130773, A062545, A020735, A005818.

List([3,5..200],n->n); # Muniru A Asiru, Sep 28 2018

a144396 = (+ 1) . (* 2)
a144396_list = [3, 5 ..] -- Reinhard Zumkeller, Feb 09 2015

seq(n,n=3..200,2); # Muniru A Asiru, Sep 28 2018

Range[3, 200, 2] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2012 *)

a(n)=2*n+1 \\ Charles R Greathouse IV, Feb 14 2023

a(n) = A005408(n+1) = A000290(n+1) - A000290(n).
G.f.: x*(3-x)/(1-x)^2. - Jaume Oliver Lafont, Aug 30 2009
a(n) = A254858(n-1,2). - Reinhard Zumkeller, Feb 09 2015

Edited by R. J. Mathar, May 21 2009

A147307 Numbers A of the constrained search for ABC records described in A147306.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 5, 19, 41, 125, 23, 1, 1, 1, 95

Offset: 1

Artur Jasinski, Nov 09 2008

The sequences A147305, a(n) and A147307 are steered by searching for records in the ABC conjecture along increasing C confined as described in A147306, the main entry for these three sequences.

Cf. A085152, A085153, A147298-A147307.

a(n)+A147305(n) = A147306(n). gcd(a(n),A147305(n))=1.

Edited by R. J. Mathar, Aug 24 2009

cp's OEIS Frontend

A272075 Primes of the form k^4 + 29*k^2 + 101.

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PARI

A030567 Triangle T(n,k): Write n in base 6 and reverse order of digits to get row n.

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A046601 First numerator and then denominator of the 1/5-Pascal triangle (by row). To get a 1/5-Pascal triangle, replace "2" in the third row of the Pascal triangle with "1/5" and calculate all other rows as in the Pascal triangle.

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A031235 Triangle T(n,k): write n in base 5, reverse order of digits.

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Haskell

Mathematica

PARI

Extensions

A030386 Triangle T(n,k): write n in base 4, reverse order of digits.

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Haskell

Maple

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A031087 Triangle T(n,k): write n in base 9, reverse order of digits.

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Haskell

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Extensions

A046213 First numerator and then denominator of 1/2-Pascal triangle (by row). To get a 1/2-Pascal triangle, replace "2" in third row of Pascal triangle with "1/2" and calculate all other rows as in Pascal triangle.

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Mathematica

A031214 Initial term of sequence An.

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A144396 The odd numbers greater than 1.

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