A034953 -id:A034953 - cp's OEIS frontend

A082749 Difference between the sum of next prime(n) natural numbers and the sum of next n primes.

1, 4, 9, 10, 54, 71, 191, 236, 446, 1025, 1310, 2259, 3245, 3820, 5048, 7321, 10060, 11473, 15328, 18358, 20381, 25672, 30222, 36561, 46367, 53031, 58108, 65444, 70971, 78391, 104184, 116542, 133095, 142728, 169931, 181324, 203429, 226622

Offset: 1

Amarnath Murthy, Apr 17 2003

nonn

Group the natural numbers with prime(n) elements in each group. (1,2),(3,4,5),(6,7,8,9,10),(11,12,13,14,15,16,17),... The sum corresponding the groups is 3,12,40,98,... Group the prime numbers such that the n-th group contains n primes. (2),(3,5),(7,11,13),(17,19,23,29),... The sum corresponding the groups is 2,8,31,88,... The required difference is 1,4,9,10,...

The following sequences (allowing offset of 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 next prime(n) natural numbers and the sum of next n primes; A006254, numbers n such that 2n-1 is prime; A067076, 2n+3 is a prime. - Jeremy Gardiner, Sep 10 2004

Module[{nn=80,trms=40,c,nat,pr},c=(nn(nn+1))/2;nat=Total/@TakeList[Range[c],Prime[Range[trms]]];pr=Total/@TakeList[Prime[Range[c]], Range[trms]]; Differences/@ Thread[{pr,nat}]]//Flatten (* Harvey P. Dale, Apr 13 2025 *)

a(n) = ((A061802(n-1) + 1)*A000040(n))/2 - A007468(n). - Gionata Neri, May 17 2015

More terms from Ray Chandler, May 13 2003

A127920 1/6 of product of three numbers: n-th prime, previous and following number.

Original entry on oeis.org

1, 4, 20, 56, 220, 364, 816, 1140, 2024, 4060, 4960, 8436, 11480, 13244, 17296, 24804, 34220, 37820, 50116, 59640, 64824, 82160, 95284, 117480, 152096, 171700, 182104, 204156, 215820, 240464, 341376, 374660, 428536, 447580, 551300, 573800, 644956

Offset: 1

Artur Jasinski, Feb 06 2007

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

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

[(NthPrime(n) + 1)*NthPrime(n)*(NthPrime(n) - 1)/6: n in [1..40]]; // Vincenzo Librandi, Apr 09 2017

Table[(Prime[n] + 1) Prime[n](Prime[n] - 1)/6, {n, 1, 100}]
((#-1)#(#+1))/6&/@Prime[Range[40]] (* Harvey P. Dale, Dec 23 2019 *)

forprime(p=2,1e3,print1(binomial(p+1,3)", ")) \\ Charles R Greathouse IV, Jun 16 2011

from sympy import prime
print([(prime(n) - 1)*prime(n)*(prime(n) + 1)//6 for n in range(1, 101)]) # Indranil Ghosh, Apr 09 2017

a(n) = A127918(n)/3. - Michel Marcus, Apr 09 2017

A116995 Pentagonal numbers with prime indices.

Original entry on oeis.org

5, 12, 35, 70, 176, 247, 425, 532, 782, 1247, 1426, 2035, 2501, 2752, 3290, 4187, 5192, 5551, 6700, 7526, 7957, 9322, 10292, 11837, 14065, 15251, 15862, 17120, 17767, 19097, 24130, 25676, 28085, 28912, 33227, 34126, 36895, 39772, 41750, 44807

Offset: 1

Jonathan Vos Post, Apr 02 2006

See also: A001248 Squares of primes. A034953 Triangular numbers (A000217) with prime indices.

			a(1) = Pentagonal(prime(1)) = A000326(2) = 2*(3*2-1)/2 = 5.
a(2) = Pentagonal(prime(2)) = A000326(3) = 3*(3*3-1)/2 = 12.
a(3) = Pentagonal(prime(3)) = A000326(5) = 5*(3*5-1)/2 = 35.

Cf. A000040, A000326, A001248, A034953.

Table[Prime[n]*(3*Prime[n] - 1)/2, {n, 1, 60}] (* Stefan Steinerberger, Apr 04 2006 *)

a(n) = Prime(n)*(3*Prime(n)-1)/2. a(n) = A000326(A000040(n)).

A127922 1/24 of product of three numbers: n-th prime, previous and following number.

Original entry on oeis.org

1, 5, 14, 55, 91, 204, 285, 506, 1015, 1240, 2109, 2870, 3311, 4324, 6201, 8555, 9455, 12529, 14910, 16206, 20540, 23821, 29370, 38024, 42925, 45526, 51039, 53955, 60116, 85344, 93665, 107134, 111895, 137825, 143450, 161239, 180441, 194054

Offset: 2

Artur Jasinski, Feb 06 2007

nonn

The product of (n-1), n, and (n+1) = n^3 - n. - Harvey P. Dale, Jan 17 2011
For n > 2, a(n) = A001318(n-2) * A007310(n-1), if A007310(n-1) is prime. Also a(n) is a subsequence of A000330. - Richard R. Forberg, Dec 25 2013
If p is an odd prime it can always be the side length of a leg of a primitive Pythagorean triangle. However it constrains the other leg to have a side length of (p^2-1)/2 and the hypotenuse to have a side length of (p^2+1)/2. The resulting triangle has an area equal to (p-1)*p*(p+1)/4. a(n) is 1/6 the area of such triangles. - Frank M Jackson, Dec 06 2017

G. C. Greubel, Table of n, a(n) for n = 2..1000
César Aguilera, Two Prime Number Objects and The Velucchi Numbers, hal-02909691 [math.NT], 2020.

Cf. A000040, A000330, A011842.

Cf. A036689, A034953, A127917, A127918, A127919, A127920, A127921, A011842.

Table[(Prime[n] + 1) Prime[n](Prime[n] - 1)/24, {n, 1, 100}] (#^3-#)/ 24&/@ Prime[Range[2,40]] (* Harvey P. Dale, Jan 17 2011 *)
((#-1)#(#+1))/24&/@Prime[Range[2,40]] (* Harvey P. Dale, Jan 20 2023 *)

for(n=2,25, print1((prime(n)+1)*prime(n)*(prime(n)-1)/24, ", ")) \\ G. C. Greubel, Jun 19 2017

a(n) = A011842(A000040(n) + 1) = A000330((A000040(n) - 1) / 2).

A147846 Triangular numbers n*(n+1)/2 with n or n+1 prime.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 55, 66, 78, 91, 136, 153, 171, 190, 253, 276, 406, 435, 465, 496, 666, 703, 820, 861, 903, 946, 1081, 1128, 1378, 1431, 1711, 1770, 1830, 1891, 2211, 2278, 2485, 2556, 2628, 2701, 3081, 3160, 3403, 3486, 3916, 4005, 4656, 4753, 5050

Offset: 1

Giovanni Teofilatto, Nov 15 2008

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000217, A034953, A008837.

lista(nn) = {for (n=1, nn, if (isprime(n) || isprime(n+1), print1(n*(n+1)/2, ", ")););} \\ Michel Marcus, Jun 03 2013

print1(1);forprime(p=3,7,print1(", "p*(p-1)/2", "p*(p+1)/2)) \\ Charles R Greathouse IV, Jun 03 2013

a(n) ~ (n^2 log^2 n)/8. - Charles R Greathouse IV, Jun 03 2013

A034953 UNION A008837. - R. J. Mathar, Jun 13 2025

Missing terms 28=7*8/2, 91=13*14/2 etc. inserted by R. J. Mathar, Jan 30 2010

A117961 Hexagonal numbers with prime indices.

Original entry on oeis.org

6, 15, 45, 91, 231, 325, 561, 703, 1035, 1653, 1891, 2701, 3321, 3655, 4371, 5565, 6903, 7381, 8911, 10011, 10585, 12403, 13695, 15753, 18721, 20301, 21115, 22791, 23653, 25425, 32131, 34191, 37401, 38503, 44253, 45451, 49141, 52975, 55611

Offset: 1

Jonathan Vos Post, Apr 05 2006

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). 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.

Cf. A000005, A000040, A000384, A001021, A034953, A001248, A116995.

With[{hex=Table[n(2n-1),{n,250}]},Flatten[Table[Take[hex,{Prime[n]}],{n, 40}]]] (* Harvey P. Dale, Dec 04 2011 *)

a(n) = A000040(n)*(2*A000040(n)-1). a(n) = A000384(prime(n)). a(n) = number of divisors of 12^(prime(n)-1) = A000005(A001021(A000040(n)-1)).

A034954 Odd triangular numbers with prime indices.

Original entry on oeis.org

3, 15, 91, 153, 435, 703, 861, 1431, 1891, 2701, 4005, 4753, 5151, 5995, 6441, 9453, 11175, 12403, 15051, 16471, 18721, 19503, 26335, 27261, 29161, 33153, 36315, 38503, 39621, 43071, 49141, 50403, 56953, 61075, 62481, 69751, 75855, 79003

Offset: 1

Patrick De Geest, Oct 15 1998

nonn

For prime indices p such that p(p+1)/2 is 'odd' see A002313 (primes of form 4n+1).

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Triangular Number

Cf. A000217, A002313.

Cf. A014493, A034953, A034955.

Select[Table[Prime[n] (Prime[n] + 1)/2, {n, 500}], OddQ[#] &] (* G. C. Greubel, Nov 03 2017 *)
With[{nn=400},Select[Thread[{Accumulate[Range[nn]],Range[nn]}],OddQ[ #[[1]]] && PrimeQ[#[[2]]]&]][[All,1]] (* Harvey P. Dale, Mar 06 2019 *)

a(n) = A000217(A002313(n)). - Michel Marcus, Nov 04 2017

A034955 Even triangular numbers with prime indices.

Original entry on oeis.org

6, 28, 66, 190, 276, 496, 946, 1128, 1770, 2278, 2556, 3160, 3486, 5356, 5778, 8128, 8646, 9730, 11476, 13366, 14028, 16110, 18336, 19900, 22366, 24976, 25878, 28680, 31626, 34716, 36856, 40186, 47278, 48516, 54946, 60378, 64620, 67528, 72010, 73536, 87990

Offset: 1

Patrick De Geest, Oct 15 1998

nonn

For prime indices p such that p(p+1)/2 is even, see A002145 (Primes of form 4n+3).

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Triangular Number

Cf. A000217, A002145.

Cf. A014493, A034953, A034954.

Select[Table[Prime[n] (Prime[n] + 1)/2, {n, 500}], EvenQ[#] &] (* G. C. Greubel, Nov 03 2017 *)

from sympy import primerange
def aupto(lim):
    pitri = (p*(p+1)//2 for p in primerange(2, int((2*lim)**.5)+1))
    return [t for t in pitri if t%2 == 0]
print(aupto(90000)) # Michael S. Branicky, Jun 28 2021

a(n) = A000217(A002145(n)). - Michel Marcus, Nov 04 2017

A050482 Sum of remainders when n-th prime is divided by all preceding integers.

Original entry on oeis.org

0, 1, 4, 8, 22, 28, 51, 64, 98, 151, 167, 233, 297, 325, 403, 505, 635, 645, 790, 904, 923, 1113, 1244, 1422, 1654, 1800, 1888, 2056, 2098, 2256, 2849, 3066, 3326, 3450, 3969, 4045, 4329, 4696, 5014, 5325, 5767, 5759, 6499, 6565, 6898

Offset: 1

Brian Wallace (wallacebrianedward(AT)yahoo.co.uk), Dec 26 1999

nonn

a(n)/(n*log(n))^2 appears to approach a constant ~0.22... for large n. - Benedict W. J. Irwin, Dec 07 2016

Irwin's comment is incorrect. - Bill McEachen, Feb 04 2024. [Indeed, according to the first formula in A004125, a(n)/(n*log(n))^2 approaches a constant, which is not 0.22 but 1-Pi^2/12 = 0.1775... - Amiram Eldar, Feb 04 2024]

			a(4) = 8 because remainders when 7 is divided by 1..6 are 0,1,1,3,2,1, which add to 8.
a(2) = 3 mod (3-1) = 1.
a(3) = (5 mod (5-1)) + (5 mod (5-2)) + (5 mod (5-3)) = 2 + 1 + 1 = 4.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A004125, A034953.

A050482 := proc(n) local a,i; a := 0; for i from 1 to ithprime(n)-1 do a := a+(ithprime(n) mod i); od: end;

Table[Sum[Mod[Prime[n],k],{k,Prime[n]-1}],{n,45}] (* James C. McMahon, Feb 08 2024 *)

a(n)=my(p=prime(n));sum(k=2, p, p%k) \\ Charles R Greathouse IV, Jun 03 2013

from math import isqrt
from sympy import prime
def A050482(n): return (p:=prime(n))**2+((s:=isqrt(p))**2*(s+1)-sum((q:=p//k)*((k<<1)+q+1) for k in range(1,s+1))>>1) # Chai Wah Wu, Nov 01 2023

a(n) = A004125(A000040(n)). - R. J. Mathar, Jun 12 2009

A064366 a(n) = binomial(sigma(n), phi(n)).

Original entry on oeis.org

1, 3, 6, 21, 15, 66, 28, 1365, 1716, 3060, 66, 20475, 91, 134596, 735471, 7888725, 153, 3262623, 190, 118030185, 225792840, 254186856, 276, 2558620845, 84672315, 11058116888, 113380261800, 558383307300, 435, 11969016345, 496, 366395202809685, 16735679449896, 21094923659355

Offset: 1

Labos Elemer, Sep 27 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..200

Cf. A000010, A000203, A000217, A034953, A066090, A066449.

Array[Binomial[DivisorSigma[1, #], EulerPhi@ #] &, 31] (* Michael De Vlieger, Nov 03 2017 *)

a(n) = { binomial(sigma(n), eulerphi(n)) } \\ Harry J. Smith, Sep 12 2009

a(n) = binomial(A000203(n), A000010(n));

a(p) = A000217(p) for prime p.

cp's OEIS Frontend

A082749 Difference between the sum of next prime(n) natural numbers and the sum of next n primes.

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A127920 1/6 of product of three numbers: n-th prime, previous and following number.

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A116995 Pentagonal numbers with prime indices.

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A127922 1/24 of product of three numbers: n-th prime, previous and following number.

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A147846 Triangular numbers n*(n+1)/2 with n or n+1 prime.

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A117961 Hexagonal numbers with prime indices.

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A034954 Odd triangular numbers with prime indices.

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A034955 Even triangular numbers with prime indices.

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