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Search: a106483 -id:a106483
Displaying 1-10 of 25 results found. page 1 2 3
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A063440 Number of divisors of n-th triangular number. +10
18
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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)
REFERENCES
Steve Dinh, The Hard Mathematical Olympiad Problems And Their Solutions, AuthorHouse, 2011, Problem 2 of the British Mathematical Olympiad 2007, page 28.
LINKS
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.
FORMULA
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)
EXAMPLE
a(6) = 4 since 1+2+3+4+5+6 = 21 has four divisors {1,3,7,21}.
MAPLE
seq(numtheory:-tau(n*(n+1)/2), n=1..100); # Robert Israel, Oct 26 2015
MATHEMATICA
DivisorSigma[0, #]&/@Accumulate[Range[90]] (* Harvey P. Dale, Apr 15 2019 *)
PROG
(PARI) for (n=1, 10000, write("b063440.txt", n, " ", numdiv(n*(n + 1)/2)) ) \\ Harry J. Smith, Aug 21 2009
(PARI) a(n)=factorback(apply(numdiv, if(n%2, [n, (n+1)/2], [n/2, n+1]))) \\ Charles R Greathouse IV, Dec 27 2014
(PARI) vector(100, n, numdiv(n*(n+1)/2)) \\ Altug Alkan, Oct 26 2015
CROSSREFS
Cf. A000005, A000217.
Cf. A001108, A005383, A005384, A048161, A060778, A081978 (greedy inverse), A106483, A101755 (indices of records), A101756 (records).
KEYWORD
nonn,easy
AUTHOR
Henry Bottomley, Jul 24 2001
STATUS
approved
A092057 Primes of the form 2*p^2 - 1, where p is prime. +10
9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
MATHEMATICA
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 *)
PROG
(PARI) for (i=1, 300, if(isprime(2*prime(i)^2-1), print1(2*prime(i)^2-1, ", ")))
CROSSREFS
Cf. A092058.
Cf. A106483 (primes p such that 2p^2 - 1 is also prime).
KEYWORD
easy,nonn
AUTHOR
Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Feb 19 2004
STATUS
approved
A213078 Primes p such that 2p^2-1 and 3p^2-2 are also prime. +10
8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Subsequence of A106483: a(1)=A106483(19), a(2)=A106483(25),
a(3)=A106483(28).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
MATHEMATICA
Select[Prime[Range[2000]], PrimeQ[2 #^2 - 1] && PrimeQ[3 #^2 - 2] &] (* T. D. Noe, Jun 06 2012 *)
PROG
(Magma) [p: p in PrimesUpTo(17000) | IsPrime(2*p^2-1)and IsPrime(3*p^2-2)]; // Vincenzo Librandi, Apr 08 2013
CROSSREFS
Cf. A106483.
KEYWORD
nonn,easy
AUTHOR
Zak Seidov, Jun 04 2012
STATUS
approved
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. +10
8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
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
LINKS
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.
EXAMPLE
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.
MATHEMATICA
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}]
CROSSREFS
Cf. A000040, A066049, A106483, A219864, A230252, A230254, A230261.
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 16 2013
STATUS
approved
A213079 Primes p such that 2p^2-1, 3p^2-2 and 4p^2-3 are also prime. +10
7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Subsequence of A213078:
a(1) = 409 = A213078(4) = A106483(29) = A000040(80),
a(2) = 941 = A213078(7) = A106483(50) = A000040(160).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
MATHEMATICA
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 *)
PROG
(Magma) [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
CROSSREFS
Cf. A000040, A106483, A213078.
KEYWORD
nonn,easy
AUTHOR
Zak Seidov, Jun 04 2012
STATUS
approved
A092058 Numbers n such that 2*prime(n)^2 - 1 is prime. +10
6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
FORMULA
A106483(n) = prime(a(n)) . - R. J. Mathar, Aug 20 2019
EXAMPLE
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.
MATHEMATICA
Select[Range[500], PrimeQ[2Prime[#]^2-1]&] (* Harvey P. Dale, Dec 13 2010 *)
PROG
(PARI) for (i=1, 300, if(isprime(2*prime(i)^2-1), print1(i, ", ")))
(Magma) [n: n in [1..220]| IsPrime(2*NthPrime(n)^2-1)]; // Vincenzo Librandi, Jan 18 2013
CROSSREFS
Cf. A092057.
KEYWORD
easy,nonn
AUTHOR
mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Feb 19 2004
STATUS
approved
A182785 Primes p such that 2*p^4-1 is also prime. +10
6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
FORMULA
A000040 INTERSECT A182783.
MATHEMATICA
Select[Prime[Range[500]], PrimeQ[2 #^4 - 1]&] (* Vincenzo Librandi, Apr 17 2013 *)
PROG
(Magma) [p: p in PrimesUpTo(2600)| IsPrime(2*p^4 - 1)]; // Vincenzo Librandi, Apr 17 2013
CROSSREFS
Cf. A182783, A182784, A106483 (2p^2-1 prime), A177104 (2p^3-1 prime), A309855 (2p^5-1 prime).
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Dec 02 2010
STATUS
approved
A213107 Primes p such that 2p^2-1, 3p^2-2, 4p^2-3, and 5p^2-4 are also prime. +10
6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Subsequence of A213079: a(1) = 17569 = A213079(10) =A213078(44)= A106483(389) = A000040(2019).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..740
MATHEMATICA
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 *)
PROG
(Magma) [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
CROSSREFS
Cf. A000040, A106483, A213078, A213079.
KEYWORD
nonn,easy
AUTHOR
Zak Seidov, Jun 05 2012
STATUS
approved
A292989 Triangular numbers having exactly 6 divisors. +10
6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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).
LINKS
Jon E. Schoenfield, Table of n, a(n) for n = 1..10000
EXAMPLE
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}.
MATHEMATICA
Select[Array[PolygonalNumber, 10^5], DivisorSigma[0, #] == 6 &] (* Michael De Vlieger, Dec 09 2017 *)
CROSSREFS
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.
KEYWORD
nonn
AUTHOR
Jon E. Schoenfield, Dec 08 2017
STATUS
approved
A230493 Number of ways to write n = (2-(n mod 2))*p + q + r with p <= q <= r such that p, q, r, 2*p^2 - 1, 2*q^2 - 1, 2*r^2 - 1 are all prime. +10
5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,17
COMMENTS
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.
LINKS
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.
EXAMPLE
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.
MATHEMATICA
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}]
CROSSREFS
Cf. A000040, A068307, A106483, A230219, A230351, A230494, A230502.
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 20 2013
STATUS
approved
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Last modified August 30 00:57 EDT 2024. Contains 375520 sequences. (Running on oeis4.)