Displaying 21-30 of 49 results found.
Primes p such that 29*p + 14 and 41*p + 20 are also prime.
+10
3
61, 103, 127, 271, 313, 331, 373, 457, 547, 571, 577, 613, 877, 967, 997, 1201, 1423, 1597, 2251, 2287, 2311, 2713, 2791, 2887, 3307, 3433, 3511, 3697, 3733, 3847, 4261, 4327, 4363, 4483, 4861, 4951, 5023, 5407, 5563, 5743, 6553, 6571, 6781, 6991, 7177, 7333
COMMENTS
p*(p + 1)/2 is the first number in a set of three triangular numbers with prime indices in arithmetic progression with difference 420*p*(p + 1) + 105. - Arkadiusz Wesolowski, Oct 29 2013
REFERENCES
Wacław Sierpiński, 200 zadan z elementarnej teorii liczb, Warsaw: PZWS, 1964, pp. 12, 61.
Wacław Sierpiński, 250 Problems in Elementary Number Theory. (Modern Analytic and Computational Methods in Science and Mathematics, No. 26), American Elsevier Publishing Co., Inc., New York; PWN Polish Scientific Publishers, Warsaw, 1970, pp. 7, 50.
MATHEMATICA
lst = {}; Do[p = Prime[n]; If[PrimeQ[29*p + 14] && PrimeQ[41*p + 20], AppendTo[lst, p]], {n, 10^3}]; lst
Select[Prime[Range[1000]], AllTrue[{29#+14, 41#+20}, PrimeQ]&] (* Harvey P. Dale, Oct 05 2022 *)
PROG
(Magma) [p : p in PrimesUpTo(7333) | IsPrime(29*p+14) and IsPrime(41*p+20)]; // Arkadiusz Wesolowski, Oct 29 2013
Sum of composite numbers less than n-th prime.
+10
2
0, 0, 4, 10, 37, 49, 94, 112, 175, 305, 335, 505, 622, 664, 799, 1049, 1329, 1389, 1709, 1916, 1988, 2368, 2611, 3041, 3692, 3989, 4091, 4406, 4514, 4847, 6407, 6794, 7464, 7602, 8898, 9048, 9818, 10618, 11113, 11963, 12843, 13023, 14697, 14889, 15474
FORMULA
a(n) = prime(n)*(prime(n)+1)/2 - sum_{1..n} prime(k) - 1.
Asymptotic expression: a(n) ~ n^2 * log(n)^2 / 2.
EXAMPLE
Prime(6) = 13, so a(6) = 4 + 6 + 8 + 9 + 1 0 + 12 = 49 = 13*14/2 - 13 - 11 - 7 - 5 - 3 - 2 - 1.
MAPLE
with(numtheory): A079725 := proc(n) local i: RETURN(ithprime(n)*(ithprime(n)+1)/2 add(ithprime(i), i=1..n)-1):
end;
MATHEMATICA
a[n_] := Block[{p = Prime[n], k}, k = p(p + 1)/2 - 1 - Sum[Prime[i], {i, 1, n}]]; Table[ a[n], {n, 1, 45}]
Prime numbers that are 2 less than a prime-indexed odd triangular number or 1 more than a prime-indexed even triangular number.
+10
2
7, 13, 29, 67, 89, 151, 191, 277, 433, 701, 859, 947, 1129, 1429, 1889, 2557, 2699, 4003, 4751, 5779, 8647, 11173, 12401, 13367, 14029, 16111, 18719, 19501, 22367, 24977, 27259, 31627, 33151, 36313, 36857, 38501, 39619, 47279, 49139, 56951
REFERENCES
David Wells, The Penguin Dictionary of Curious & Interesting Numbers. In the entry for 496 he remarks that 496 is the smallest counterexample to the conjecture that an even, prime-indexed triangular plus 1 equals a prime, since 497 is not prime.
FORMULA
Given the numbers of A034953, triangular numbers with prime indices, subtract 2 from the odd numbers on the list and add 1 to the even numbers on the list, then remove from the list the composite numbers.
EXAMPLE
a(2) = 13 because 15 is the 5th triangular number and since it is odd and we take 2 away from it, it yields the prime 13.
a(3) = 29 because 28 is the 7th triangular number and since it is even and we add 1 to it, it yields the prime 29.
497 is not on the list because although 496 is the 31st triangular number, but 496 + 1 = 7 * 71.
That sequence continues: 1771, 2279, 3161, 3487, 5149, 5357, 5993, 6439, 8129, 9451, 9731, ....
MATHEMATICA
tri[n_] := n(n + 1)/2; tp = Table[ tri[ Prime[n]], {n, 2, 70}]; f[n_] := If[ OddQ[n], n - 2, n + 1]; Select[f /@ tp, PrimeQ[ # ] &] (* Robert G. Wilson v, Aug 12 2004 *) Select[If[OddQ[#], #-2, #+1]&/@Table[(n(n+1))/2, {n, Prime[Range[ 100]]}], PrimeQ] (* Harvey P. Dale, Sep 19 2016 *)
Parity of p*(p+1)/2 for n-th prime p.
+10
2
1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0
COMMENTS
The following sequences (possibly with a different offset for first term) all appear to have the same parity: A034953 = triangular numbers with prime indices; A054269 = length of period of continued fraction for sqrt(p), p prime; A082749 = difference between the sum of the next prime(n) natural numbers and the sum of the next n primes; A006254 = numbers n such that 2n-1 is prime; A067076 = numbers n such that 2n+3 is a prime. See also A089253 = 2n-5 is a prime. For n > 1, if A000040(n) == 1 (mod 4), then a(n) = 1, otherwise a(n)=0, so (for n>1) also a(n) = number of representations of A000040(n) as a difference of hexagonal numbers ( A000384) (cf. [Nyblom, p. 262]). - L. Edson Jeffery, Feb 16 2013
FORMULA
a(n) = parity of p*(p+1)/2 for n-th prime p.
a(n) = 1 - A100672(n), n > 1. - Steven G. Johnson (stevenj(AT)math.mit.edu), Sep 18 2008 For n > 1, a(n) = (prime(n) mod 4) mod 3. - Gary Detlefs, Oct 27 2011
EXAMPLE
a(1) = parity of (2*(2+1)/2 = 3) = 1 (odd).
MAPLE
seq((ithprime(n) mod 4) mod 3, n = 2..105] # Gary Detlefs, Oct 27 2011
MATHEMATICA
Mod[(#(#+1))/2, 2]&/@Prime[Range[110]] (* Harvey P. Dale, Mar 29 2015 *)
Prime partial sums of triangular numbers with prime indices.
+10
2
3, 1759, 3323, 469303, 605113, 641969, 1110587, 1426669, 11148289, 18352349, 20473721, 21820391, 24710753, 30048589, 36690923, 40785301, 97060681, 155135369, 160593239, 168132247, 361391623, 377965069, 416572171, 645803201
EXAMPLE
a(1) = Sum_{i=1..1} prime(i)*(prime(i)+1)/2 = T(2) = 3.
a(2) = Sum_{i=1..11} prime(i)*(prime(i)+1)/2 = T(2)+T(3)+T(5)+T(7)+T(11)+T(13)+T(17)+T(19)+T(23)+T(29)+T(31) = 1759.
a(3) = Sum_{i=1..13} prime(i)*(prime(i)+1)/2 = 3323.
a(4) = Sum_{i=1..53} prime(i)*(prime(i)+1)/2 = T(2) + ... + T(241) = 469303.
a(5) = Sum_{i=1..57} prime(i)*(prime(i)+1)/2 = T(2) + ... + T(269) = 605113.
a(6) = Sum_{i=1..58} prime(i)*(prime(i)+1)/2 = T(2) + ... + T(271) = 641969.
a(7) = Sum_{i=1..68} prime(i)*(prime(i)+1)/2 = T(2) + ... + T(337) = 1110587.
MAPLE
T:=n->n*(n+1)/2: a:=proc(n): if isprime(sum(T(ithprime(j)), j=1..n))=true then sum(T(ithprime(j)), j=1..n) else fi end: seq(a(n), n=1..500); # Emeric Deutsch, Apr 06 2006
MATHEMATICA
Select[Accumulate[Table[(n(n+1))/2, {n, Prime[Range[500]]}]], PrimeQ] (* Harvey P. Dale, Jan 25 2015 *)
Partial sums of hexagonal numbers with prime indices.
+10
2
6, 21, 66, 157, 388, 713, 1274, 1977, 3012, 4665, 6556, 9257, 12578, 16233, 20604, 26169, 33072, 40453, 49364, 59375, 69960, 82363, 96058, 111811, 130532, 150833, 171948, 194739, 218392, 243817, 275948, 310139, 347540, 386043, 430296, 475747
COMMENTS
There are no prime hexagonal numbers. The n-th hexagonal number A000384(n) = n*(2*n-1) is semiprime iff both n and 2*n-1 are prime iff A000384(n) is an element of A001358 iff n is an element of A005382.
EXAMPLE
a(4) = hexagonal(2) + hexagonal(3) + hexagonal(5) + hexagonal(7) = 6 + 15 + 45 + 91 = 157 is prime.
a(12) = 6 + 15 + 45 + 91 + 231 + 325 + 561 + 703 + 1035 + 1653 + 1891 + 2701 = 9257 is prime.
a(26) = 150833 is prime.
MATHEMATICA
Accumulate[Table[n(2n-1), {n, Prime[Range[50]]}]] (* Harvey P. Dale, Jan 30 2014 *)
CROSSREFS
See also: A034953 Triangular numbers ( A000217) with prime indices. A001248 Squares of primes. A116995 Pentagonal numbers with prime indices. A000384 Hexagonal numbers: n(2n-1). A117961 Hexagonal numbers with prime indices. A117965 Prime partial sums of hexagonal numbers with prime indices.
1/12 of product of three numbers: n-th prime, previous and following number.
+10
2
2, 10, 28, 110, 182, 408, 570, 1012, 2030, 2480, 4218, 5740, 6622, 8648, 12402, 17110, 18910, 25058, 29820, 32412, 41080, 47642, 58740, 76048, 85850, 91052, 102078, 107910, 120232, 170688, 187330, 214268, 223790, 275650, 286900, 322478, 360882, 388108, 431462
COMMENTS
Summation of products of partitions into two parts of prime(n): a(6) = (1*12) + (2*11) + (3*10) + (4*9) + (5*8) + (6*7) = 182. - César Aguilera, Feb 20 2018
MAPLE
a:= n-> (p->p*(p^2-1)/12)(ithprime(n)):
MATHEMATICA
Table[(Prime[n] + 1) Prime[n](Prime[n] - 1)/12, {n, 2, 100}]
((#-1)#(#+1))/12&/@Prime[Range[2, 40]] (* Harvey P. Dale, Mar 08 2022 *)
PROG
(Magma) [(NthPrime(n) + 1)*NthPrime(n)*(NthPrime(n) - 1)/12: n in [2..50]]; // G. C. Greubel, Apr 30 2018
Fully multiplicative with a(p) = p*(p+1)/2 for prime p.
+10
2
1, 3, 6, 9, 15, 18, 28, 27, 36, 45, 66, 54, 91, 84, 90, 81, 153, 108, 190, 135, 168, 198, 276, 162, 225, 273, 216, 252, 435, 270, 496, 243, 396, 459, 420, 324, 703, 570, 546, 405, 861, 504, 946, 594, 540, 828, 1128, 486, 784, 675, 918, 819, 1431, 648, 990, 756
COMMENTS
There are analogs with the triangular numbers replaced by some other sequence, but this was chosen because of the parity coincidences of A034953.
FORMULA
a((p_1)^e_1)*(p_2)^e_2)*...*(p_k)^e_k)) = (T((p_1))^e_1)*T((p_2))^e_2)*...*T((p_k))^e_k, where T(i) = A000217(i). a(p_i) = A034953(i). Sum_{n>=1} 1/a(n) = Product_{p prime} (1 - 2/(p*(p+1))^(-1) = 2.12007865309570462566... . - Amiram Eldar, Dec 24 2022 Dirichlet g.f.: Product_{p prime} (1 + (p^2 + p) / (2*p^s - p^2 - p)). - Vaclav Kotesovec, Apr 05 2023
MATHEMATICA
f[p_, e_] := (p*(p + 1)/2)^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Dec 24 2022 *)
PROG
(PARI) a(n)=my(f=factor(n)); prod(i=1, #f[, 1], binomial(f[i, 1]+1, 2)^f[i, 2]) /* Charles R Greathouse IV, Sep 09 2010 */ (PARI) for(n=1, 100, print1(direuler(p=2, n, 1 + (p^2 + p) / (2/X - p^2 - p))[n], ", ")) \\ Vaclav Kotesovec, Apr 05 2023
Triangular numbers p*(p+1)/2 with p prime such that 1+(number of prime factors of p+1) is prime.
+10
2
3, 6, 15, 91, 276, 703, 1431, 1770, 1891, 2701, 3486, 4005, 5356, 8646, 9730, 11175, 11476, 12403, 18721, 19503, 24976, 25878, 27261, 28680, 38503, 43071, 47278, 49141, 60378, 61075, 64620, 72010, 75855, 79003, 88831, 98346, 104653, 106491
COMMENTS
Triangular numbers n(n+1)/2 such that n and A073093(n+1) are both prime.
EXAMPLE
3 has one prime factor; 1+1 = 2 is prime, hence 2*3/2 = 3 is in the sequence.
14 = 2*7 has two prime factors; 1+2 = 3 is prime, hence 13*14/2 = 91 is in the sequence.
24 = 2*2*2*3 has four prime factors; 1+4 = 5 is prime, hence 23*24/2 = 276 is in the sequence.
MATHEMATICA
aQ[n_] := PrimeQ[n] && PrimeQ[PrimeOmega[n + 1] + 1]; p = Select[Range[470], aQ]; p*(p + 1)/2 (* Amiram Eldar, Aug 31 2019 *)
PROG
(Magma) [ p*(p+1)/2: p in PrimesUpTo(490) | IsPrime(1 + &+[ f[2]: f in Factorization(p+1) ]) ];
EXTENSIONS
Edited, corrected (3 inserted) and extended beyond a(16) by Klaus Brockhaus, Jan 05 2009
Octagonal numbers with prime indices.
+10
2
8, 21, 65, 133, 341, 481, 833, 1045, 1541, 2465, 2821, 4033, 4961, 5461, 6533, 8321, 10325, 11041, 13333, 14981, 15841, 18565, 20501, 23585, 28033, 30401, 31621, 34133, 35425, 38081, 48133, 51221, 56033, 57685, 66305, 68101, 73633, 79381, 83333, 89441, 95765
MATHEMATICA
Table[Prime[n] (3 Prime[n] - 2), {n, 1, 45}]
PROG
(PARI) lista(nn) = forprime(p=2, nn, print1(p*(3*p-2), ", ")); \\ Altug Alkan, Jan 11 2016 (Magma) [NthPrime(n)*(3*NthPrime(n)-2): n in [1..50]]; // Vincenzo Librandi, Jan 12 2016 (Python)
from sympy import prime
def a(n): p = prime(n); return p*(3*p-2)
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