A046992 -id:A046992 - cp's OEIS frontend

A368611 a(n) = Sum_{k=1..n} pi(k) * floor(n/k).

0, 1, 3, 6, 9, 15, 19, 26, 32, 40, 45, 58, 64, 75, 86, 99, 106, 123, 131, 149, 163, 177, 186, 212, 224, 240, 255, 277, 287, 316, 327, 351, 369, 388, 406, 441, 453, 474, 494, 528, 541, 578, 592, 622, 651, 675, 690, 737, 756, 788, 812, 845, 861, 903, 927, 969, 995, 1022

Offset: 1

Wesley Ivan Hurt, Dec 31 2023

Cf. A000720 (pi), A013939, A046992, A368610, A368612, A368613.

Table[Sum[PrimePi[k] Floor[n/k], {k, n}], {n, 100}]

a(n) = A013939(n) + A368613(n).

A002815 a(n) = n + Sum_{k=1..n} pi(k), where pi() = A000720.

Original entry on oeis.org

0, 1, 3, 6, 9, 13, 17, 22, 27, 32, 37, 43, 49, 56, 63, 70, 77, 85, 93, 102, 111, 120, 129, 139, 149, 159, 169, 179, 189, 200, 211, 223, 235, 247, 259, 271, 283, 296, 309, 322, 335, 349, 363, 378, 393, 408, 423, 439, 455, 471

Offset: 0

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

H. Brocard, Reply to Query 1421, Nombres premiers dans une suite de différences, L'Intermédiaire des Mathématiciens, 7 (1900), 135-137.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A000720, A046992.

a002815 0 = 0
a002815 n = a046992 n + toInteger n  -- Reinhard Zumkeller, Feb 25 2012

Table[n + Sum[PrimePi[k], {k, 1, n}], {n, 0, 50}]
Module[{nn=50,pp},pp=Accumulate[PrimePi[Range[0,nn]]];Total/@ Thread[ {Range[ 0,nn],pp}]] (* This program is significantly faster than the program above. *) (* Harvey P. Dale, Jan 03 2013 *)

a(n) = my(p=primes([0,n])); n + (n+1)*#p - vecsum(p); \\ Ruud H.G. van Tol, Feb 16 2024

from sympy import primerange
def A002815(n): return n+(n+1)*len(p:=list(primerange(n+1)))-sum(p) # Chai Wah Wu, Jan 01 2024

a(n) = A046992(n) + n for n > 0. - Reinhard Zumkeller, Feb 25 2012

Conjectured g.f.: (Sum_{N>=1} x^A008578(N))/(1-x)^2 = (x + x^2 + x^3 + x^5 + x^7 + x^11 + x^13 + ...)/(1-x)^2. - L. Edson Jeffery, Nov 25 2013

A034957 Divide natural numbers in groups with prime(n) elements and add together.

Original entry on oeis.org

1, 9, 35, 91, 242, 442, 833, 1273, 2024, 3306, 4464, 6586, 8897, 11137, 14288, 18762, 24190, 28670, 35778, 42813, 49275, 59329, 69056, 81702, 98067, 112110, 124836, 140919, 155325, 173568, 210312, 233835, 262903, 285923, 327949, 355001, 393285

Offset: 1

Patrick De Geest, Oct 15 1998

nonn

Natural numbers starting from 0,1,2,3,...

			{0,1} #2 S=1;
{2,3,4} #3 S=9;
{5,6,7,8,9} #5 S=35;
{10,11,12,13,14,15,16} #7 S=91.

Hieronymus Fischer, Table of n, a(n) for n = 1..1000

Cf. A006003, A027441, A034956.

Cf. A046992, A034958, A034959, A034960.

{1}~Join~Map[Abs@ Apply[Subtract, Map[PolygonalNumber, #]] &, Partition[Accumulate@ Prime@ Range@ 37 - 1, 2, 1]] (* Michael De Vlieger, Oct 06 2019 *)
Module[{nn=40,tprs},tprs=Total[Prime[Range[nn]]];Total/@TakeList[Range[0,tprs],Prime[Range[nn]]]] (* Harvey P. Dale, Apr 18 2025 *)

from itertools import islice
from sympy import nextprime
def A034957_gen(): # generator of terms
    a, p = 0, 2
    while True:
        yield p*((a<<1)+p-1)>>1
        a, p = a+p, nextprime(p)
A034957_list = list(islice(A034957_gen(),20)) # Chai Wah Wu, Mar 22 2023

From Hieronymus Fischer, Sep 27 2012: (Start)
a(n) = Sum_{k=A007504(n-1)+1..A007504(n)} (k-1), n > 1.
a(n) = (A007504(n) - A007504(n-1))*(A007504(n) + A007504(n-1) - 1)/2, n > 1.
a(n) = (A000217(A007504(n) - 1) - A000217(A007504(n-1) - 1)), n > 1.
If we define A007504(0):=0, then the formulas above are also true for n=1.
a(n) = A034959(n)/2.
a(n) = A034956(n) - A000040(n).
(End)

A230850 A054541 and A000012 interleaved.

Original entry on oeis.org

2, 1, 1, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 6, 1, 2, 1, 6, 1, 4, 1, 2, 1, 4, 1, 6, 1, 6, 1, 2, 1, 6, 1, 4, 1, 2, 1, 6, 1, 4, 1, 6, 1, 8, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 14, 1, 4, 1, 6, 1, 2, 1, 10, 1, 2, 1, 6, 1, 6, 1, 4, 1, 6, 1, 6, 1, 2, 1, 10, 1, 2, 1

Offset: 1

Omar E. Pol, Oct 31 2013

nonn

a(n) is also the length of the n-th edge of a staircase which represents the function pi(x) on the first quadrant of the square grid, see A000720.
a(2n-1) is the length of the n-th horizontal edge in the staircase.
a(2n) is the length of the n-th vertical edge in the staircase.
For another version see A230849.

			Illustration of initial terms, n = 1..22:
.
1                                                              _ _|
1                                                  _ _ _ _ _ _|
1                                          _ _ _ _|
1                                      _ _|
1                              _ _ _ _|
1                          _ _|
1                  _ _ _ _|
1              _ _|
1          _ _|
1        _|
1    _ _|
.
.      2 1   2   2       4   2       4   2       4           6   2
.
Drawing vertical line segments below the staircase (as shown below) we have that the number of cells in the vertical bars gives 0 together A000720.
Drawing horizontal line segments above the staircase we have that the number of cells in the k-th horizontal bar is A000040(k).
.    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
31  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
29  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
23  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | |
19  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | |
17  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | |
13  |_ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | |
11  |_ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | |
7   |_ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | |
5   |_ _ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | | | |
3   |_ _ _| | | | | | | | | | | | | | | | | | | | | | | | | | | | |
2   |_ _|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.
.    0 0 1 2 2 3 3 4 4 4 4 5 5 6 6 6 6 7 7 8 8 8 8 9 9 9 9 9 9 10 10
.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000012, A000040, A000720, A001223, A007504, A046992, A054541, A141042, A152535, A182986, A230849.

Riffle[Join[{2},Differences[Prime[Range[100]]]],1] (* Paolo Xausa, Oct 31 2023 *)

A230850(n) = if(1==n,2,if((n%2),prime((n+1)/2)-prime(((n+1)/2)-1),1)); \\ Antti Karttunen, Dec 23 2018

a(1) = 2; for n > 1, a(n) = A230849(n). - Antti Karttunen, Dec 23 2018

A249727 Start with a(1) = 1; then numbers 1 .. primepi(2), followed by numbers 1 .. primepi(3), and then numbers 1 .. primepi(4), ..., etc, where A000720 gives primepi.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

Can be used to construct the irregular table A249809.

This is a fractal sequence; i.e., the removal of the first occurrence of each term in A249727 leaves A249727, so that the sequence contains itself infinitely many times. The corresponding interspersion is A272616. - Clark Kimberling, May 12 2016

Antti Karttunen, Table of n, a(n) for n = 1..10016

Cf. A000720, A046992, A249728, A249809, A272616.

Flatten[Table[Range[PrimePi[n]], {n, 2, 100}]]
(* Clark Kimberling, May 14 2016 *)

(define (A249727 n) (if (= 1 n) 1 (- n (+ 1 (A046992 (- (A249728 n) 1))))))

A368612 a(n) = Sum_{k=2..n} pi(k-1) * ceiling(n/k).

Original entry on oeis.org

0, 0, 1, 4, 8, 13, 20, 27, 37, 46, 56, 65, 81, 92, 107, 122, 140, 153, 175, 190, 214, 234, 254, 271, 304, 324, 347, 370, 399, 418, 454, 475, 509, 536, 564, 591, 635, 658, 689, 719, 763, 788, 835, 862, 904, 945, 981, 1010, 1070, 1103, 1148, 1185, 1231, 1262, 1318, 1356

Offset: 1

Wesley Ivan Hurt, Dec 31 2023

Cf. A000720 (pi), A046992, A092494, A368610, A368611, A368613.

Table[Sum[PrimePi[k - 1] Ceiling[n/k], {k, 2, n}], {n, 100}]

a(n) = A368610(n) - A092494(n).

A368613 a(n) = Sum_{k=2..n} pi(k-1) * floor(n/k).

Original entry on oeis.org

0, 0, 1, 3, 5, 9, 12, 18, 23, 29, 33, 44, 49, 58, 67, 79, 85, 100, 107, 123, 135, 147, 155, 179, 190, 204, 218, 238, 247, 273, 283, 306, 322, 339, 355, 388, 399, 418, 436, 468, 480, 514, 527, 555, 582, 604, 618, 663, 681, 711, 733, 764, 779, 819, 841, 881, 905, 930

Offset: 1

Wesley Ivan Hurt, Dec 31 2023

Cf. A000720 (pi), A013939, A046992, A368610, A368611, A368612.

Table[Sum[PrimePi[k - 1] Floor[n/k], {k, 2, n}], {n, 100}]

a(n) = A368611(n) - A013939(n).

A147693 Irregular triangle read by rows: T(n, k) = n mod prime(k), n >= 2, 1 <= k <= PrimePi(n), where PrimePi(n) = A000720(n).

Original entry on oeis.org

0, 1, 0, 0, 1, 1, 2, 0, 0, 0, 1, 1, 1, 2, 0, 0, 2, 3, 1, 1, 0, 4, 2, 0, 1, 0, 3, 1, 2, 1, 4, 0, 0, 0, 2, 5, 1, 1, 1, 3, 6, 2, 0, 0, 2, 4, 0, 3, 1, 1, 0, 0, 1, 4, 2, 0, 1, 1, 2, 5, 3, 1, 2, 2, 3, 6, 4, 0, 0, 0, 3, 4, 7, 5, 1, 1, 1, 4, 5, 8, 6, 2, 0, 0, 2, 0, 6, 9, 7, 3, 1, 1, 0, 1, 0, 10, 8, 4, 2

Offset: 2

Reikku Kulon, Nov 10 2008

Equivalently, we define table, P, with columns numbered by the primes (2, 3, 5, ...) instead of 1, 2, 3, ... so that P(n, p) = n mod p.
P begins with P(2, 2).
A complex pattern emerges if values in the triangle are taken modulo 2.
Rows are unique. Row n has length A000720(n). - Jason Kimberley, Nov 2012

			Triangle P begins:
       2 3 5 7
     +---------
   2 | 0
   3 | 1 0
   4 | 0 1
   5 | 1 2 0
   6 | 0 0 1
   7 | 1 1 2 0
   8 | 0 2 3 1
   9 | 1 0 4 2
  10 | 0 1 0 3
  ...
Each row can be produced from the previous row by adding one to each number and resetting to zero any which would equal their column number. A row number n is prime, initiating a new column numbered n, iff P(n, p) is nonzero for all prime p < n; P(n, n) is then 0.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Exercise 3.7.22 on page 125.

Jason Kimberley, Rows n = 2..294 of irregular triangle, flattened
Eric Weisstein's World of Mathematics, Redheffer Matrix

Cf. A000720, A002321, A046992, A083058.

A147693 :=
func< n | [n mod p:p in PrimesUpTo(n)] >;
[A147693(n):n in[2..19]]; // Jason Kimberley, Nov 28 2012

row[n_]:=Table[Mod[n,Prime[i]], {i, PrimePi[n]}]; Array[row, 20, 2]//Flatten (* Stefano Spezia, Jul 17 2025 *)

a(A046992(n-1)+i) = T(n,i) = n mod A000040(i), where 1 <= i <= A000720(n). - Jason Kimberley, Nov 21 2012

Edited by Peter Munn, May 25 2025

A230849 A075526 and A000012 interleaved.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 6, 1, 2, 1, 6, 1, 4, 1, 2, 1, 4, 1, 6, 1, 6, 1, 2, 1, 6, 1, 4, 1, 2, 1, 6, 1, 4, 1, 6, 1, 8, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 14, 1, 4, 1, 6, 1, 2, 1, 10, 1, 2, 1, 6, 1, 6, 1, 4, 1, 6, 1, 6, 1, 2, 1, 10, 1, 2, 1

Offset: 1

Omar E. Pol, Nov 01 2013

nonn

a(n) is also the length of the n-th edge of a staircase which represents the function pi(x) on the first quadrant of the square grid, see A000720.
a(2n-1) is the length of the n-th horizontal edge in the staircase.
a(2n) is the length of the n-th vertical edge in the staircase.
For another version see A230850.

			Illustration of initial terms, n = 1..22:
.
1                                                            _ _|
1                                                _ _ _ _ _ _|
1                                        _ _ _ _|
1                                    _ _|
1                            _ _ _ _|
1                        _ _|
1                _ _ _ _|
1            _ _|
1        _ _|
1      _|
1    _|
.
.    1 1   2   2       4   2       4   2       4           6   2
.
Drawing vertical line segments below the staircase (as shown below) we have that the number of cells in the vertical bars gives A000720.
Drawing horizontal line segments above the staircase we have that the number of cells in the k-th horizontal bar is A006093(k).
.    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
30  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
28  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
22  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | |
18  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | |
16  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | |
12  |_ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | |
10  |_ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | |
6   |_ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | |
4   |_ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | | | |
2   |_ _| | | | | | | | | | | | | | | | | | | | | | | | | | | | |
1   |_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.
.    0 1 2 2 3 3 4 4 4 4 5 5 6 6 6 6 7 7 8 8 8 8 9 9 9 9 9 9 10 10
.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000012, A000040, A000720, A001223, A006093, A007504, A046992, A075526, A141042, A152535, A230850.

Riffle[Join[{1},Differences[Prime[Range[100]]]],1] (* Paolo Xausa, Oct 31 2023 *)

A230849(n) = if((n%2)&&(n>1),prime((n+1)/2)-prime(((n+1)/2)-1),1); \\ Antti Karttunen, Dec 23 2018

A249728 After a(1) = 1 each n appears A000720(n) times.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 20, 20, 20

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10016

Cf. A000720, A046992, A249727, A249809.

Join[{1},Table[Table[n,PrimePi[n]],{n,20}]//Flatten] (* Harvey P. Dale, Nov 24 2018 *)

cp's OEIS Frontend

A368611 a(n) = Sum_{k=1..n} pi(k) * floor(n/k).

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A002815 a(n) = n + Sum_{k=1..n} pi(k), where pi() = A000720.

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A034957 Divide natural numbers in groups with prime(n) elements and add together.

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A230850 A054541 and A000012 interleaved.

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A249727 Start with a(1) = 1; then numbers 1 .. primepi(2), followed by numbers 1 .. primepi(3), and then numbers 1 .. primepi(4), ..., etc, where A000720 gives primepi.

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A368612 a(n) = Sum_{k=2..n} pi(k-1) * ceiling(n/k).

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A368613 a(n) = Sum_{k=2..n} pi(k-1) * floor(n/k).

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A147693 Irregular triangle read by rows: T(n, k) = n mod prime(k), n >= 2, 1 <= k <= PrimePi(n), where PrimePi(n) = A000720(n).

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