A061802 -id:A061802 - 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

A034960 Divide odd numbers into groups with prime(n) elements and add together.

Original entry on oeis.org

4, 21, 75, 189, 495, 897, 1683, 2565, 4071, 6641, 8959, 13209, 17835, 22317, 28623, 37577, 48439, 57401, 71623, 85697, 98623, 118737, 138195, 163493, 196231, 224321, 249775, 281945, 310759, 347249, 420751, 467801, 525943, 571985, 656047

Offset: 1

Patrick De Geest, Oct 15 1998

nonn

			{1,3} #2 S=4;
{5,7,9} #3 S=21;
{11,13,15,17,19} #5 S=75;
{21,23,25,27,29,31,33} #7 S=189.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A006003, A027441, A034959.
Cf. A007504.
Cf. A000040, A034956, A034957, A034958, A046992, A061802.

S:= n-> sum(ithprime(k), k=1..n): seq(S(n+1)^2-S(n)^2, n=0..40); # Gary Detlefs, Dec 20 2011

Accumulate[Join[{2}, ListConvolve[{1, 1}, #]]]*# & [Prime[Range[50]]] (* Paolo Xausa, Jun 23 2025 *)

a0(n) = vecsum(primes(n))^2 - vecsum(primes(n-1))^2; \\ Michel Marcus, Jun 16 2024

from itertools import islice
from sympy import nextprime
def A034960_gen(): # generator of terms
    a, p = 0, 2
    while True:
        yield p*((a<<1)+p)
        a, p = a+p, nextprime(p)
A034960_list = list(islice(A034960_gen(),20)) # Chai Wah Wu, Mar 22 2023

From Hieronymus Fischer, Sep 26 2012: (Start)
a(n) = Sum_{k=A007504(n-1)+1..A007504(n)} (2*k-1).
a(n) = A007504(n)^2 - A007504(n-1)^2.
a(n) = 2*A034957(n) + A000040(n).
a(n) = 2*A034956(n) - A000040(n).
a(n) = A034959(n) + A000040(n). (End)
a(n) = A061802(n)*A000040(n). - Marco Zárate, May 12 2023

A138143 Triangle read by rows in which row n lists p(1), p(2), ..., p(n), p(n-1), ..., p(1), where p(i) = i-th prime.

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 5, 3, 2, 2, 3, 5, 7, 5, 3, 2, 2, 3, 5, 7, 11, 7, 5, 3, 2, 2, 3, 5, 7, 11, 13, 11, 7, 5, 3, 2, 2, 3, 5, 7, 11, 13, 17, 13, 11, 7, 5, 3, 2, 2, 3, 5, 7, 11, 13, 17, 19, 17, 13, 11, 7, 5, 3, 2, 2, 3, 5, 7, 11, 13, 17, 19, 23, 19, 17, 13, 11, 7, 5, 3, 2, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 23, 19, 17, 13, 11, 7, 5, 3, 2

Offset: 1

Omar E. Pol, Mar 09 2008

Row n contains 2n-1 terms and each column lists the prime numbers A000040.

Triangle of primes mentioned in A061802.

			Triangle begins:
.............. 2
........... 2, 3, 2
........ 2, 3, 5, 3, 2
..... 2, 3, 5, 7, 5, 3, 2
.. 2, 3, 5, 7,11, 7, 5, 3, 2

Cf. A000040, A061802, A066622, A037126.

nn=10;Module[{pr=Prime[Range[nn]],e},Flatten[Table[e=Take[pr,n];Join[ e,Reverse[Most[e]]],{n,nn}]]] (* Harvey P. Dale, Mar 14 2015 *)

Edited by N. J. A. Sloane, Apr 07 2008, Nov 15 2008

A023662 Convolution of odd numbers and primes.

Original entry on oeis.org

2, 9, 24, 51, 96, 165, 264, 399, 576, 805, 1094, 1451, 1886, 2405, 3014, 3723, 4544, 5485, 6554, 7761, 9112, 10615, 12280, 14117, 16140, 18361, 20786, 23421, 26272, 29345, 32658, 36229, 40068, 44183, 48586, 53289, 58300, 63631, 69292

Offset: 1

Clark Kimberling

nonn

Cf. A000040, A005408, A061802 (first differences).

Cf. A167214, A062020, A316322, A014148, A007504.

A023662 := proc(n)
    add( ithprime(n-i)*(2*i+1),i=0..n-1) ;
end proc: # R. J. Mathar, Nov 29 2015

Table[Sum[Prime[n - k + 1] (2 k - 1), {k, n}], {n, 39}] (* Michael De Vlieger, Nov 29 2015 *)

a(n) = sum(i=1, n, prime(n-i+1)*(2*i-1)); \\ Michel Marcus, Sep 30 2013

a(n) = Sum_{i=0..n-1} A000040(n-i)*A005408(i). - R. J. Mathar, Nov 29 2015
a(n) = Sum_{i=0..n-1} A061802(i). - Marco Zárate, Jun 09 2024
From Ridouane Oudra, Feb 19 2025: (Start)
a(n) = Sum_{i=1..n} Sum_{j=1..n} min(prime(i), prime(j)).
a(n) = A167214(n) - A062020(n).
a(n) = 2*A167214(n) - A316322(n).
a(n) = A014148(n) + A014148(n-1).
a(n) = A007504(n) + 2*A014148(n-1). (End)

A073612 Group the positive integers as (1, 2), (3, 4, 5), (6, 7, 8, 9, 10), (11, 12, 13, 14, 15, 16, 17), ... the n-th group containing prime(n) elements. Except the first, all groups contain an odd number of elements and hence have a middle term. Sequence gives the middle terms starting from group 2.

Original entry on oeis.org

4, 8, 14, 23, 35, 50, 68, 89, 115, 145, 179, 218, 260, 305, 355, 411, 471, 535, 604, 676, 752, 833, 919, 1012, 1111, 1213, 1318, 1426, 1537, 1657, 1786, 1920, 2058, 2202, 2352, 2506, 2666, 2831, 3001, 3177, 3357, 3543, 3735, 3930, 4128, 4333, 4550, 4775

Offset: 2

Amarnath Murthy, Aug 05 2002

nonn

Christian Krause, LODA program

Cf. A007504, A034956, A061802.

Table[ Sum[ Prime[i], {i, 1, n}] - Floor[ Prime[n]/2], {n, 2, 50}]
For[lst={}; n1=3; n=2, n<=100, n++, n2=n1+Prime[n]; AppendTo[lst, (n2+n1-1)/2]; n1=n2]; lst
Module[{nn=50,no,pr},no=Total[Prime[Range[2,nn+1]]];pr=Prime[Range[2,nn]]; #[[ (Length[ #]+1)/2]]&/@TakeList[Range[3,no],pr]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Sep 20 2017 *)

Difference of the triangular numbers corresponding to the sum of first (n+1) primes and that of first n primes/prime(n) for n > 1.
a(n) = (A061802(n-1) + 1)/2. - Hugo Pfoertner, Apr 30 2021
a(n) = A007504(n) - (prime(n)-1)/2. - Andrew Howroyd, Apr 30 2021
a(n) = (Sum_{i=2..n-1} A001043(i)) / 2 + 4. - Christian Krause, May 06 2021

Edited by Robert G. Wilson v and T. D. Noe, Aug 08 2002

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Programs

Mathematica

Formula

Extensions

A034960 Divide odd numbers into groups with prime(n) elements and add together.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A138143 Triangle read by rows in which row n lists p(1), p(2), ..., p(n), p(n-1), ..., p(1), where p(i) = i-th prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A023662 Convolution of odd numbers and primes.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions