A106483 -id:A106483 - cp's OEIS frontend

A063440 Number of divisors of n-th triangular number.

1, 2, 4, 4, 4, 4, 6, 9, 6, 4, 8, 8, 4, 8, 16, 8, 6, 6, 8, 16, 8, 4, 12, 18, 6, 8, 16, 8, 8, 8, 10, 20, 8, 8, 24, 12, 4, 8, 24, 12, 8, 8, 8, 24, 12, 4, 16, 24, 9, 12, 16, 8, 8, 16, 24, 24, 8, 4, 16, 16, 4, 12, 36, 24, 16, 8, 8, 16, 16, 8, 18, 18, 4, 12, 24, 16, 16, 8, 16, 40, 10, 4, 16

Offset: 1

Henry Bottomley, Jul 24 2001

a(n) = 4 iff either n is in A005383 or n/2 is in A005384.
a(n) is odd iff n is in A001108.
a(n) = 6 if either n = 18 or n = q^2 where q is in A048161 or n = 2 q^2 - 1 where q is in A106483. - Robert Israel, Oct 26 2015
From Bernard Schott, Aug 29 2020: (Start)
a(n-1) is the number of solutions in positive integers (x, y, z) to the simultaneous equations (x + y - z = n, x^2 + y^2 - z^2 = n) for n > 1. See the British Mathematical Olympiad link. In this case, one always has z > x and z > y.
For n = 12 as in the Olympiad problem, the a(11) = 8 solutions are (13,78,79), (14,45,47), (15,34,37), (18,23,29), (23,18,29), (34,15,37), (45,14,47), (78,13,79). (End)

			a(6) = 4 since 1+2+3+4+5+6 = 21 has four divisors {1,3,7,21}.

Steve Dinh, The Hard Mathematical Olympiad Problems And Their Solutions, AuthorHouse, 2011, Problem 2 of the British Mathematical Olympiad 2007, page 28.

Ray Chandler, Table of n, a(n) for n = 1..10000
British Mathematical Olympiad 2007/2008, Round 1, Problem 2.
Index to sequences related to Olympiads.

Cf. A000005, A000217.

Cf. A001108, A005383, A005384, A048161, A060778, A081978 (greedy inverse), A106483, A101755 (indices of records), A101756 (records).

seq(numtheory:-tau(n*(n+1)/2), n=1..100); # Robert Israel, Oct 26 2015

DivisorSigma[0,#]&/@Accumulate[Range[90]] (* Harvey P. Dale, Apr 15 2019 *)

for (n=1, 10000, write("b063440.txt", n, " ", numdiv(n*(n + 1)/2)) ) \\ Harry J. Smith, Aug 21 2009

a(n)=factorback(apply(numdiv,if(n%2,[n,(n+1)/2],[n/2,n+1]))) \\ Charles R Greathouse IV, Dec 27 2014

vector(100, n, numdiv(n*(n+1)/2)) \\ Altug Alkan, Oct 26 2015

a(n) = A000005(A000217(n)).
From Robert Israel, Oct 26 2015: (Start)
a(2k) = A000005(k)*A000005(2k+1).
a(2k+1) = A000005(2k+1)*A000005(k+1).
gcd(a(2k), a(2k+1)) = A000005(2k+1) * A060778(k). (End)

A092057 Primes of the form 2*p^2 - 1, where p is prime.

Original entry on oeis.org

7, 17, 97, 241, 337, 577, 3361, 3697, 6961, 10657, 23761, 25537, 32257, 37537, 49297, 64081, 65521, 77617, 79201, 89041, 126001, 138337, 153457, 171697, 193441, 249217, 269377, 287281, 334561, 351121, 374977, 474337, 633937, 652081, 665857

Offset: 1

Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Feb 19 2004

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A092058.

Cf. A106483 (primes p such that 2p^2 - 1 is also prime).

lst={};Do[p=Prime[n];If[PrimeQ[r=2*p^2-1],AppendTo[lst,r]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 20 2009 *)
Select[2#^2-1&/@Prime[Range[200]],PrimeQ] (* Harvey P. Dale, Jun 26 2017 *)

for (i=1,300,if(isprime(2*prime(i)^2-1),print1(2*prime(i)^2-1,",")))

A213078 Primes p such that 2p^2-1 and 3p^2-2 are also prime.

Original entry on oeis.org

199, 311, 379, 409, 419, 659, 941, 1009, 1439, 2351, 2789, 3079, 3221, 4421, 4999, 5351, 5531, 5839, 6299, 7129, 7321, 7349, 8819, 9029, 10091, 10151, 10391, 10459, 11131, 11551, 12251, 12391, 13049, 13759, 14281, 14669, 15091, 15329, 15581, 16381, 16811

Offset: 1

Zak Seidov, Jun 04 2012

Subsequence of A106483: a(1)=A106483(19), a(2)=A106483(25),

a(3)=A106483(28).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A106483.

[p: p in PrimesUpTo(17000) | IsPrime(2*p^2-1)and IsPrime(3*p^2-2)]; // Vincenzo Librandi, Apr 08 2013

Select[Prime[Range[2000]], PrimeQ[2 #^2 - 1] && PrimeQ[3 #^2 - 2] &] (* T. D. Noe, Jun 06 2012 *)

A230351 Number of ordered ways to write n = p + q (q > 0) with p, 2p^2 - 1 and 2q^2 - 1 all prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 16 2013

nonn

Conjecture: a(n) > 0 for all n > 3.
We have verified this for n up to 2*10^7.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Aug 07 2023

			a(7) = 1 since 7 = 3 + 4 with 3, 2*3^2 - 1 = 17, 2*4^2 - 1 = 31 all prime.
a(40) = 1 since 40 = 2 + 38, and 2, 2*2^2 - 1 = 7 , 2*38^2 - 1 = 2887 are all prime.
a(68) = 1 since 68 = 43 + 25, and all the three numbers 43, 2*43^2 - 1 = 3697 and 2*25^2 - 1 = 1249 are prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A000040, A066049, A106483, A219864, A230252, A230254, A230261.

a[n_]:=Sum[If[PrimeQ[2Prime[i]^2-1]&&PrimeQ[2(n-Prime[i])^2-1],1,0],{i,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]

A213079 Primes p such that 2p^2-1, 3p^2-2 and 4p^2-3 are also prime.

Original entry on oeis.org

409, 941, 6299, 10459, 11131, 11551, 15581, 16831, 17321, 17569, 25771, 25969, 26701, 31511, 36131, 40529, 43781, 50231, 52879, 54631, 54779, 56711, 60271, 61331, 70321, 71081, 83101, 83299, 85931, 100649, 110681, 116381, 118409, 118751, 120641, 130469

Offset: 1

Zak Seidov, Jun 04 2012

Subsequence of A213078:
a(1) = 409 = A213078(4) = A106483(29) = A000040(80),
a(2) = 941 = A213078(7) = A106483(50) = A000040(160).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A106483, A213078.

[p: p in PrimesUpTo(140000) | IsPrime(2*p^2-1) and IsPrime(3*p^2-2) and IsPrime(4*p^2-3)]; // Vincenzo Librandi, Apr 08 2013

Select[Prime[Range[20000]], PrimeQ[2 #^2 - 1] && PrimeQ[3 #^2 - 2] && PrimeQ[4 #^2 - 3] &] (* T. D. Noe, Jun 06 2012 *)
Select[Prime[Range[12500]],AllTrue[{2#^2-1,3#^2-2,4#^2-3},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 11 2015 *)

A092058 Numbers n such that 2*prime(n)^2 - 1 is prime.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 13, 14, 17, 21, 29, 30, 31, 33, 37, 41, 42, 45, 46, 47, 54, 56, 59, 62, 64, 71, 73, 75, 80, 81, 84, 93, 103, 105, 106, 113, 114, 120, 126, 131, 132, 134, 139, 141, 144, 145, 146, 148, 159, 160, 169, 175, 179, 183, 185, 186, 188, 192, 212, 217, 220

Offset: 1

mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Feb 19 2004

			2*prime(1)^2 - 1 = 7 is prime so a(1)=1;
2*prime(2)^2 - 1 = 17 is prime so a(2)=2;
2*prime(3)^2 - 1 = 97 is not prime;
2*prime(4)^2 - 1 = 241 is prime so a(3)=4.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A092057.

[n: n in [1..220]| IsPrime(2*NthPrime(n)^2-1)]; // Vincenzo Librandi, Jan 18 2013

Select[Range[500],PrimeQ[2Prime[#]^2-1]&] (* Harvey P. Dale, Dec 13 2010 *)

for (i=1,300,if(isprime(2*prime(i)^2-1),print1(i,",")))

A106483(n) = prime(a(n)) . - R. J. Mathar, Aug 20 2019

A182785 Primes p such that 2*p^4-1 is also prime.

Original entry on oeis.org

2, 5, 7, 47, 79, 103, 131, 139, 149, 173, 197, 229, 307, 313, 331, 373, 439, 541, 547, 593, 659, 743, 761, 797, 853, 859, 863, 883, 919, 937, 1051, 1093, 1097, 1163, 1171, 1301, 1303, 1451, 1471, 1549, 1601, 1657, 1721, 1861, 1973, 2039, 2081, 2087, 2099, 2129, 2161, 2239, 2269, 2393, 2417, 2437, 2473, 2521

Offset: 1

Vincenzo Librandi, Dec 02 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A182783, A182784, A106483 (2p^2-1 prime), A177104 (2p^3-1 prime), A309855 (2p^5-1 prime).

[p: p in PrimesUpTo(2600)| IsPrime(2*p^4 - 1)]; // Vincenzo Librandi, Apr 17 2013

Select[Prime[Range[500]], PrimeQ[2 #^4 - 1]&] (* Vincenzo Librandi, Apr 17 2013 *)

A000040 INTERSECT A182783.

A213107 Primes p such that 2p^2-1, 3p^2-2, 4p^2-3, and 5p^2-4 are also prime.

Original entry on oeis.org

17569, 43781, 70321, 229561, 251231, 426131, 426551, 453289, 635051, 727201, 729791, 741709, 944689, 981091, 1015309, 1078081, 1128761, 1228429, 1231229, 1282961, 1289149, 1302349, 1351099, 1723481, 1763159, 1823779, 2078339, 2260889, 2336519, 2357879

Offset: 1

Zak Seidov, Jun 05 2012

Subsequence of A213079: a(1) = 17569 = A213079(10) =A213078(44)= A106483(389) = A000040(2019).

Vincenzo Librandi, Table of n, a(n) for n = 1..740

Cf. A000040, A106483, A213078, A213079.

[p: p in PrimesUpTo(2500000) | forall{i*p^2-i+1: i in [2..5] | IsPrime(i*p^2-i+1)}]; // Vincenzo Librandi, Apr 08 2013

Select[Prime[Range[200000]], PrimeQ[2 #^2 - 1] && PrimeQ[3 #^2 - 2] && PrimeQ[4 #^2 - 3] && PrimeQ[5 #^2 - 4] &] (* T. D. Noe, Jun 06 2012 *)

A292989 Triangular numbers having exactly 6 divisors.

Original entry on oeis.org

28, 45, 153, 171, 325, 4753, 7381, 29161, 56953, 65341, 166753, 354061, 5649841, 6060421, 6835753, 6924781, 12708361, 19478161, 24231241, 52035301, 56791153, 147258541, 186660181, 282304441, 326081953, 520273153, 536657941, 704531953, 784139401, 1215121753

Offset: 1

Jon E. Schoenfield, Dec 08 2017

nonn

Intersection of A000217 (triangular numbers) and A030515 (numbers with exactly 6 divisors).
This sequence is also the union of
(1) numbers of the form p*(2p-1) where p is prime and 2p-1 is the square of a prime (this sequence begins 45, 325, 7381, 65341, 354061, ...),
(2) numbers of the form p^2*(2p^2 - 1) where both p and 2p^2 - 1 are prime (this sequence begins 28, 153, 4753, 29161, ...), and
(3) numbers of the form p^2*(2p^2 + 1) where both p and 2p^2 + 1 are prime (the only such number is 171).

			28 = 2^2 * 7, so it has 6 divisors: {1, 2, 4, 7, 14, 28};
45 = 3^2 * 5, so it has 6 divisors: {1, 3, 5, 9, 15, 45};
171 = 3^2 * 19, so it has 6 divisors: {1, 3, 9, 19, 57, 171}.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Cf. A000217 (triangular numbers), A030515 (numbers with exactly 6 divisors).
Cf. A067756 (primes p such that 2p-1 is the square of a prime), A106483 (primes p such that 2p^2 - 1 is prime).
Cf. A263951.

Select[Array[PolygonalNumber, 10^5], DivisorSigma[0, #] == 6 &] (* Michael De Vlieger, Dec 09 2017 *)

A230493 Number of ways to write n = (2-(n mod 2))p + q + r with p <= q <= r such that p, q, r, 2p^2 - 1, 2q^2 - 1, 2r^2 - 1 are all prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 20 2013

nonn

Conjecture:  a(n) > 0 for all n > 6.
This is stronger than Goldbach's weak conjecture which was finally proved by H. Helfgott in 2013. It also implies that there are infinitely many primes p with 2*p^2 - 1 also prime.
We have verified the conjecture for n up to 10^6.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Sep 22 2023
See also A230351, A230494 and A230502 for similar conjectures.

			a(14) = 1 since 14 = 2*2 + 3 + 7 with 2, 3, 7, 2*2^2 - 1 = 7, 2*3^2 - 1 = 17, 2*7^2 - 1 = 97 all prime.
a(19) = 1 since 19 = 3 + 3 + 13, and 3, 13, 2*3^2 - 1 = 17 and 2*13^2 - 1 = 337 are all prime.
a(53) = 1 since 53 = 3 + 7 + 43, and all the six numbers 3, 7, 43, 2*3^2 - 1 = 17, 2*7^2 - 1 = 97, 2*43^2 - 1 = 3697 are prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A000040, A068307, A106483, A230219, A230351, A230494, A230502.

pp[n_]:=PrimeQ[2n^2-1]
pq[n_]:=PrimeQ[n]&&pp[n]
a[n_]:=Sum[If[pp[Prime[i]]&&pp[Prime[j]]&&pq[n-(2-Mod[n,2])Prime[i]-Prime[j]],1,0],{i,1,PrimePi[n/(4-Mod[n,2])]},{j,i,PrimePi[(n-(2-Mod[n,2])Prime[i])/2]}]
Table[a[n],{n,1,100}]

cp's OEIS Frontend

A063440 Number of divisors of n-th triangular number.

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A092057 Primes of the form 2*p^2 - 1, where p is prime.

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Mathematica

PARI

A213078 Primes p such that 2p^2-1 and 3p^2-2 are also prime.

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Magma

Mathematica

A230351 Number of ordered ways to write n = p + q (q > 0) with p, 2*p^2 - 1 and 2*q^2 - 1 all prime.

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Mathematica

A213079 Primes p such that 2p^2-1, 3p^2-2 and 4p^2-3 are also prime.

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Magma

Mathematica

A092058 Numbers n such that 2*prime(n)^2 - 1 is prime.

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Programs

Magma

Mathematica

PARI

Formula

A182785 Primes p such that 2*p^4-1 is also prime.

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Magma

Mathematica

Formula

A213107 Primes p such that 2p^2-1, 3p^2-2, 4p^2-3, and 5p^2-4 are also prime.

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A230351 Number of ordered ways to write n = p + q (q > 0) with p, 2p^2 - 1 and 2q^2 - 1 all prime.

A230493 Number of ways to write n = (2-(n mod 2))p + q + r with p <= q <= r such that p, q, r, 2p^2 - 1, 2q^2 - 1, 2r^2 - 1 are all prime.