A034953 -id:A034953 - cp's OEIS frontend

A098996 a(n) = p(p + 1)(2*p + 1) where p is the n-th prime.

30, 84, 330, 840, 3036, 4914, 10710, 14820, 25944, 51330, 62496, 105450, 142926, 164604, 214320, 306234, 421260, 465186, 615060, 731016, 794094, 1004880, 1164324, 1433790, 1853670, 2091306, 2217384, 2484540, 2625810, 2924214, 4145280, 4547796, 5199150, 5429340

Offset: 1

Parthasarathy Nambi, Nov 05 2004

nonn

The unique primitive Pythagorean triple whose inradius is the n-th prime p and whose short leg is an odd number is (2*p+1, 2*p*(p+1), 2*p*(p+1)+1) and its area is a(n) = p*(p+1)*(2*p+1). - Miguel-Ángel Pérez García-Ortega, Mar 16 2025

			a(4) = 7*(7+1)*(2*7+1) = 840.

Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2025.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A034953, A367573, A382097, A382070.

[(p*(p+1)*(2*p+1)): p in PrimesUpTo(200)]; // Vincenzo Librandi, Feb 04 2011

#(#+1)(2#+1)&/@Prime[Range[30]] (* Harvey P. Dale, Jun 23 2020 *)

a(n) = prime(n)*(prime(n)+1)*(2*prime(n)+1).

More terms from Klaus Brockhaus, Nov 09 2004

A127921 1/12 of product of three numbers: n-th prime, previous and following number.

Original entry on oeis.org

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

Offset: 2

Artur Jasinski, Feb 06 2007

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

G. C. Greubel, Table of n, a(n) for n = 2..10000

Cf. A000040, A036689, A034953, A127917, A127918, A127919, A127920.

[(NthPrime(n) + 1)*NthPrime(n)*(NthPrime(n) - 1)/12: n in [2..50]]; // G. C. Greubel, Apr 30 2018

a:= n-> (p->p*(p^2-1)/12)(ithprime(n)):
seq(a(n), n=2..40);  # Alois P. Heinz, Mar 08 2022

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 *)

a(n,p=prime(n))=binomial(p+1,3)/2 \\ Charles R Greathouse IV, Feb 28 2018

a(n) ~ (n log n)^3/12. - Charles R Greathouse IV, Feb 28 2018

A138383 If prime(i) = i-th prime, a(n) = prime(n)+1 + prime(n)+2 + ... + prime(n+1). a(0) = 3 by convention.

Original entry on oeis.org

3, 3, 9, 13, 38, 25, 62, 37, 86, 159, 61, 207, 158, 85, 182, 303, 339, 121, 387, 278, 145, 459, 326, 519, 748, 398, 205, 422, 217, 446, 1687, 518, 807, 277, 1445, 301, 927, 963, 662, 1023, 1059, 361, 1865, 385, 782, 397, 2466, 2610, 902, 457, 926, 1419, 481, 2465, 1527

Offset: 0

Odimar Fabeny, May 08 2008

First differences of A034953 for n > 0. - Gionata Neri, May 17 2015

			3 = 1 + 2;
3 = 3;
9 = 4 + 5;
13 = 6 + 7;
38 = 8 + 9 + 10 + 11;
...

Cf. A000040.

[3] cat [(NthPrime(n+1) - NthPrime(n))*(NthPrime(n+1) + NthPrime(n)+1)/2: n in [1..60]]; // Vincenzo Librandi, May 18 2015

3, seq((ithprime(n+1)-ithprime(n))*(ithprime(n+1)+ithprime(n)+1)/2, n=1..100); # Robert Israel, May 17 2015

Join[{3}, Table[(Prime[n+1] - Prime[n]) (Prime[n+1] + Prime[n] + 1)/2, {n, 60}]] (* Vincenzo Librandi, May 18 2015 *)
Join[{3},(#[[2]]-#[[1]]) (Total[#]+1)/2&/@Partition[Prime[Range[ 60]],2,1]] (* Harvey P. Dale, Oct 27 2020 *)

from sympy import prime, nextprime
def A138383(n):
    if n == 0: return 3
    q = nextprime(p:=prime(n))
    return (q-p)*(p+q+1)>>1 # Chai Wah Wu, Jun 01 2024

a(n) = (prime(n+1)-prime(n))*(prime(n+1)+prime(n)+1)/2 for n >= 1. - N. J. A. Sloane, May 08 2008

A203263 Primes p such that 29p + 14 and 41p + 20 are also prime.

Original entry on oeis.org

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

Offset: 1

Arkadiusz Wesolowski, Jan 13 2012

nonn

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

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.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Cf. A034953.

[p : p in PrimesUpTo(7333) | IsPrime(29*p+14) and IsPrime(41*p+20)]; // Arkadiusz Wesolowski, Oct 29 2013

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 *)

A382070 Semiperimeter of the unique primitive Pythagorean triple whose inradius is the n-th prime and whose short leg is an odd number.

Original entry on oeis.org

15, 28, 66, 120, 276, 378, 630, 780, 1128, 1770, 2016, 2850, 3486, 3828, 4560, 5778, 7140, 7626, 9180, 10296, 10878, 12720, 14028, 16110, 19110, 20706, 21528, 23220, 24090, 25878, 32640, 34716, 37950, 39060, 44850, 46056, 49770, 53628, 56280, 60378

Offset: 1

Miguel-Ángel Pérez García-Ortega, Mar 15 2025

			For n=2, the short leg is A367573(2,1) = 7, the long leg is A367573(2,2) = 24 and the hypotenuse is A367573(2,3) = 25 so the semiperimeter is then a(2) = (7 + 24 + 25)/2 = 28.

Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2025.

Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas

Cf. A034953, A367573, A382097, A098996.

a=Table[Prime[n],{n,1,40}];Apply[Join,Map[{(#+1)(2#+1)}&,a]]

a(n) = (prime(n)+1) * (2*prime(n)+1).

a(n) = (A367573(n,1) + A367573(n,2) + A367573(n,3))/2.

A382097 Sum of the legs of the unique primitive Pythagorean triple whose inradius is the n-th prime and whose short leg is an odd number.

Original entry on oeis.org

17, 31, 71, 127, 287, 391, 647, 799, 1151, 1799, 2047, 2887, 3527, 3871, 4607, 5831, 7199, 7687, 9247, 10367, 10951, 12799, 14111, 16199, 19207, 20807, 21631, 23327, 24199, 25991, 32767, 34847, 38087, 39199, 44999, 46207, 49927, 53791, 56447, 60551

Offset: 1

Miguel-Ángel Pérez García-Ortega, Mar 15 2025

			For n=2, the short leg is A367573(2,1) = 7 and the long leg is A367573(2,2) = 24 so the sum of the legs is then a(2) = 7 + 24 = 31.

Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2025.

Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas

Cf. A034953, A367573, A382070, A098996.

a=Table[Prime[n],{n,1,40}];Apply[Join,Map[{2#^2+4#+1}&,a]]

a(n) = 2*(prime(n))^2 + 4*prime(n) + 1.

a(n) = A367573(n,1) + A367573(n,2).

A079725 Sum of composite numbers less than n-th prime.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Feb 18 2003

			Prime(6) = 13, so a(6) = 4 + 6 + 8 + 9 + 1 0 + 12 = 49 = 13*14/2 - 13 - 11 - 7 - 5 - 3 - 2 - 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Equals A000217(Prime_n) - A007504(n) - 1 = A034953 - A007504 - A000012.

Partial sums of A054265.

with(numtheory): A079725 := proc(n) local i:
RETURN(ithprime(n)*(ithprime(n)+1)/2 add(ithprime(i),i=1..n)-1):
end;

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}]

a(n) = prime(n)*(prime(n)+1)/2 - sum_{1..n} prime(k) - 1.

Asymptotic expression: a(n) ~ n^2 * log(n)^2 / 2.

Edited and extended by Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Robert G. Wilson v and T. D. Noe, Feb 18 2003

A096333 Prime numbers that are 2 less than a prime-indexed odd triangular number or 1 more than a prime-indexed even triangular number.

Original entry on oeis.org

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

Offset: 1

Alonso del Arte, Aug 02 2004

nonn

			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, ....

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.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A034953.

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 *)

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.

Edited and extended by Robert G. Wilson v, Aug 12 2004

A098033 Parity of p*(p+1)/2 for n-th prime p.

Original entry on oeis.org

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

Offset: 1

Jeremy Gardiner, Sep 10 2004

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.
Analogous to the prime race (mod 3). - Robert G. Wilson v, Sep 17 2004
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

			a(1) = parity of (2*(2+1)/2 = 3) = 1 (odd).

Amiram Eldar, Table of n, a(n) for n = 1..10000
M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263.

Cf. A034953, A054269, A082749, A006254, A067076, A100672.

seq((ithprime(n) mod 4) mod 3, n = 2..105] # Gary Detlefs, Oct 27 2011

Table[ Mod[ Prime[n](Prime[n] + 1)/2, 2], {n, 105}] (* Robert G. Wilson v, Sep 17 2004 *)
Mod[(#(#+1))/2,2]&/@Prime[Range[110]] (* Harvey P. Dale, Mar 29 2015 *)

a(n)=prime(n)%4<3 \\ Charles R Greathouse IV, Oct 27 2011

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

More terms from Robert G. Wilson v, Sep 17 2004

A116994 Prime partial sums of triangular numbers with prime indices.

Original entry on oeis.org

3, 1759, 3323, 469303, 605113, 641969, 1110587, 1426669, 11148289, 18352349, 20473721, 21820391, 24710753, 30048589, 36690923, 40785301, 97060681, 155135369, 160593239, 168132247, 361391623, 377965069, 416572171, 645803201

Offset: 1

Jonathan Vos Post, Apr 02 2006

			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.

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A000040, A000217, A034953, A085739.

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

Select[Accumulate[Table[(n(n+1))/2,{n,Prime[Range[500]]}]],PrimeQ] (* Harvey P. Dale, Jan 25 2015 *)

A000040 INTERSECTION A085739. Primes in A085739.

More terms from Emeric Deutsch, Apr 06 2006

cp's OEIS Frontend

A098996 a(n) = p*(p + 1)*(2*p + 1) where p is the n-th prime.

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A127921 1/12 of product of three numbers: n-th prime, previous and following number.

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A138383 If prime(i) = i-th prime, a(n) = prime(n)+1 + prime(n)+2 + ... + prime(n+1). a(0) = 3 by convention.

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A203263 Primes p such that 29*p + 14 and 41*p + 20 are also prime.

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A382070 Semiperimeter of the unique primitive Pythagorean triple whose inradius is the n-th prime and whose short leg is an odd number.

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A382097 Sum of the legs of the unique primitive Pythagorean triple whose inradius is the n-th prime and whose short leg is an odd number.

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A079725 Sum of composite numbers less than n-th prime.

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A098996 a(n) = p(p + 1)(2*p + 1) where p is the n-th prime.

A203263 Primes p such that 29p + 14 and 41p + 20 are also prime.