A004125 -id:A004125 - cp's OEIS frontend

A294629 Partial sums of A294628.

4, 16, 28, 56, 68, 120, 132, 192, 228, 296, 308, 440, 452, 536, 612, 736, 748, 920, 932, 1112, 1204, 1320, 1332, 1624, 1676, 1808, 1916, 2144, 2156, 2496, 2508, 2760, 2884, 3048, 3156, 3600, 3612, 3792, 3932, 4336, 4348, 4784, 4796, 5120, 5388, 5600, 5612, 6224, 6292, 6640, 6812, 7184, 7196, 7728, 7868, 8384

Offset: 1

Omar E. Pol, Nov 05 2017

nonn

a(n) is also the volume (and the number of cubes) in the n-th level (starting from the top) of the stepped pyramid described in A294630.

Number of terms less than 10^k, k=1,2,3,...: 1, 5, 19, 61, 195, 623, 1967, 6225, ... - Muniru A Asiru, Mar 04 2018

			Illustration of initial terms (n = 1..6):
.                                                  _ _ _ _ _ _
.                                _ _ _ _         _|     |     |_
.                _ _ _ _       _|   |   |_      |       |       |
.      _ _      |   |   |     |    _|_    |     |      _|_      |
.     |_|_|     |_ _|_ _|     |_ _|   |_ _|     |_ _ _|   |_ _ _|
.     |_|_|     |   |   |     |   |_ _|   |     |     |_ _|     |
.               |_ _|_ _|     |_    |    _|     |       |       |
.       4                       |_ _|_ _|       |_      |      _|
.                  16                             |_ _ _|_ _ _|
.                                  28
.                                                      56
.
.                                        _ _ _ _ _ _ _ _
.             _ _ _ _ _ _              _|       |       |_
.            |     |     |           _|         |         |_
.         _ _|     |     |_ _       |           |           |
.        |      _ _|_ _      |      |          _|_          |
.        |     |       |     |      |        _|   |_        |
.        |_ _ _|       |_ _ _|      |_ _ _ _|       |_ _ _ _|
.        |     |       |     |      |       |_     _|       |
.        |     |_ _ _ _|     |      |         |_ _|         |
.        |_ _      |      _ _|      |           |           |
.            |     |     |          |_          |          _|
.            |_ _ _|_ _ _|            |_        |        _|
.                                       |_ _ _ _|_ _ _ _|
.                 68
.                                              120
.
Note that for n >= 2 the structure has a hole (or hollow) in the center.
a(n) is the number of ON cells in the n-th diagram.

Iain Fox, Table of n, a(n) for n = 1..10000

For other related diagrams see A294630 (partial sums), A294016 and A237593.

Cf. A000203, A004125, A016742, A024916, A067436, A239050, A243980, A294015, A294017, A294628.

List([1..1000],n->Sum([1..n],k->8*(Sigma(k)-k+(1/2)))); # Muniru A Asiru, Mar 04 2018

with(numtheory): seq(sum(8*(sigma(k)-k+(1/2)),k=1..n),n=1..1000); # Muniru A Asiru, Mar 04 2018

f[n_] := 8 (DivisorSigma[1, n] - n) + 4; Accumulate@Array[f, 56] (* Robert G. Wilson v, Dec 12 2017 *)

a(n) = 4*(sum(k=1, n, n\k*k) - sum(k=2, n, n%k)) \\ Iain Fox, Dec 10 2017

first(n) = my(res = vector(n)); res[1] = 4; for(x=2, n, res[x] = res[x-1] + 8*(sigma(x) - x + (1/2))); res; \\ Iain Fox, Dec 10 2017

from math import isqrt
def A294629(n): return -(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))-n**2<<2 # Chai Wah Wu, Oct 22 2023

a(n) = 4*A294016(n).
a(n) = A016742(n) - 8*A004125(n).
a(n) = A016742(n) - 4*A067436(n).
a(n) = A243980(n) - 4*A004125(n).
a(n) = A243980(n) - 2*A067436(n).

A294630 Partial sums of A294629.

Original entry on oeis.org

4, 20, 48, 104, 172, 292, 424, 616, 844, 1140, 1448, 1888, 2340, 2876, 3488, 4224, 4972, 5892, 6824, 7936, 9140, 10460, 11792, 13416, 15092, 16900, 18816, 20960, 23116, 25612, 28120, 30880, 33764, 36812, 39968, 43568, 47180, 50972, 54904, 59240, 63588, 68372, 73168, 78288, 83676, 89276, 94888, 101112

Offset: 1

Omar E. Pol, Nov 05 2017

nonn

a(n) is also the volume of a stepped pyramid with n levels which is another version of the stepped pyramid described in A244050. Both pyramids have the same top view and the same front view, that is to say externally both pyramids are equal, but this pyramid with n levels contains a central chamber whose volume is 4*A072481(n). For more information about the central chamber see the diagrams in A294629.

a(n) is the number of unit cubes of the pyramid with n levels.

			Illustration of the top view of the pyramid with 16 levels and 4224 unit cubes:
.                 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                |  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.             _ _| |  _ _ _ _ _ _ _ _ _ _ _ _ _ _  | |_ _
.           _|  _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _  |_
.         _|  _|  _| |  _ _ _ _ _ _ _ _ _ _ _ _  | |_  |_  |_
.        |  _|   |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _|   |_  |
.   _ _ _| |  _ _|     |  _ _ _ _ _ _ _ _ _ _  |     |_ _  | |_ _ _
.  |  _ _ _|_| |      _| |_ _ _ _ _ _ _ _ _ _| |_      | |_|_ _ _  |
.  | | |  _ _ _|    _|_ _|  _ _ _ _ _ _ _ _  |_ _|_    |_ _ _  | | |
.  | | | | |  _ _ _| |  _| |_ _ _ _ _ _ _ _| |_  | |_ _ _  | | | | |
.  | | | | | | |  _ _|_|  _|  _ _ _ _ _ _  |_  |_|_ _  | | | | | | |
.  | | | | | | | | |  _ _|   |_ _ _ _ _ _|   |_ _  | | | | | | | | |
.  | | | | | | | | | | |  _ _|  _ _ _ _  |_ _  | | | | | | | | | | |
.  | | | | | | | | | | | | |  _|_ _ _ _|_  | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | |  _ _  | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |   | | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |_ _| | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | |_|_ _ _ _|_| | | | | | | | | | | | | |
.  | | | | | | | | | | | |_|_  |_ _ _ _|  _|_| | | | | | | | | | | |
.  | | | | | | | | | |_|_    |_ _ _ _ _ _|    _|_| | | | | | | | | |
.  | | | | | | | |_|_ _  |_  |_ _ _ _ _ _|  _|  _ _|_| | | | | | | |
.  | | | | | |_|_ _  | |_  |_ _ _ _ _ _ _ _|  _| |  _ _|_| | | | | |
.  | | | |_|_ _    |_|_ _| |_ _ _ _ _ _ _ _| |_ _|_|    _ _|_| | | |
.  | |_|_ _ _  |     |_  |_ _ _ _ _ _ _ _ _ _|  _|     |  _ _ _|_| |
.  |_ _ _  | |_|_      | |_ _ _ _ _ _ _ _ _ _| |      _|_| |  _ _ _|
.        | |_    |_ _  |_ _ _ _ _ _ _ _ _ _ _ _|  _ _|    _| |
.        |_  |_  |_  | |_ _ _ _ _ _ _ _ _ _ _ _| |  _|  _|  _|
.          |_  |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _|  _|
.            |_ _  | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |  _ _|
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
Note that the above diagram contains a hidden pattern, simpler, which emerges from the front view of every corner of the stepped pyramid. For more information about the hidden pattern see A237593 and A245092.

Muniru A Asiru, Table of n, a(n) for n = 1..2000

For other related diagrams see A294016 and A294629.

Cf. A000203, A002492, A004125, A072481, A237593, A239050, A243980, A244050, A245092, A294015, A294017, A294628.

List([1..50],n->Sum([1..n],m->Sum([1..m],k->8*(Sigma(k)-k+(1/2))))); # Muniru A Asiru, Mar 04 2018

with(numtheory): seq(sum(sum(8*(sigma(j)-j+(1/2)),j=1..k),k=1..n),n=1..50); # Muniru A Asiru, Mar 04 2018

f[n_] := 8 (DivisorSigma[1, n] - n) + 4; Accumulate@ Accumulate@ Array[f, 48] (* Robert G. Wilson v, Dec 12 2017 *)

from math import isqrt
def A294630(n): return ((((s:=isqrt(n))**2*(s+1)*((s+1)*((s<<1)+1)-6*(n+1))>>1) + sum((q:=n//k)*(-k*(q+1)*(3*k+(q<<1)+1)+3*(n+1)*((k<<1)+q+1)) for k in range(1, s+1))<<2)-(n*(n+1)*((n<<1)+1)<<1))//3 # Chai Wah Wu, Nov 01 2023

a(n) = 4*A294017(n).
a(n) = A002492(n) - 8*A072481(n).
a(n) = A244050(n) - 4*A072481(n).

A354238 Decimal expansion of 1 - Pi^2/12.

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, May 20 2022

Ratio of area between the polygon that is adjacent in the same plane to the base of the stepped pyramid with an infinite number of levels described in A245092 and the circumscribed square (see the first formula).

			0.177532966575886781763792416676987405390525049396600781132220885314996264798...

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013.

Ovidiu Furdui, Problem A, Nieuw Archief voor Wiskunde, Vol. 9, No. 1 (2008), p. 86; "Problem 2008/1-A, Solution to Problem A by Noud Aldenhoven and Daan Wanrooy, ibid., Vol. 9, No. 3 (2008), p. 303.
Ovidiu Furdui, Problem 1930, Mathematics Magazine, Vol. 86, No. 4 (2013), p. 289; A zeta series, Solution to Problem 1930 by Omran Kouba, ibid., Vol. 87, No. 4 (2014), pp. 296-298.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (C7.4).
Index entries for transcendental numbers.

Cf. A000290, A004125, A013661, A024916, A072691, A152416, A237593, A245092, A353908.

RealDigits[1 - Pi^2/12, 10, 100][[1]] (* Amiram Eldar, May 20 2022 *)

PARI
```
1-Pi^2/12
```
PARI
```
1-zeta(2)/2
```

Equals lim_{n->infinity} A004125(n)/(n^2).
Equals 1 - A013661/2.
Equals 1 - A072691.
Equals A152416/2.
Equals Sum_{k>=1} 1/(2*k*(k+1)^2). - Amiram Eldar, May 20 2022
Equals -1/4 + Sum_{k>=2} (-1)^k * k * (k - Sum_{i=2..k} zeta(i)) (Furdui, 2013 problem). - Amiram Eldar, Jun 09 2022
Equals Integral_{x>=1} {x}/x^3 dx where {.} is the fractional part. [Nahin]. R. J. Mathar, May 22 2024
From Amiram Eldar, Jul 31 2025: (Start)
Equals Integral_{x=0..1} {1/x} * x dx (Furdui, 2013 book, section 2.21, page 103).
Equals Integral_{x=0..1} Integral_{y=0..1} {x/y}*{y/x} dx dy, where {} denotes fractional part (Furdui, 2008 and 2013 book, section 2.36, page 105). (End)

A161517 Sum of remainders of c mod k where k = 1, 2, 3, ..., c and c is the n-th composite number.

Original entry on oeis.org

1, 3, 8, 12, 13, 17, 31, 36, 36, 47, 61, 70, 77, 85, 103, 112, 125, 124, 138, 167, 184, 197, 218, 198, 248, 269, 258, 284, 328, 339, 358, 374, 414, 420, 449, 454, 492, 529, 520, 553, 578, 586, 672, 693, 693, 738, 725, 799, 840, 835, 852, 956, 981, 992, 1049, 1036

Offset: 1

Juri-Stepan Gerasimov, Jun 12 2009

			a(1) = 1 (= (4 mod 3) + 0);
a(2) = 3 (= (6 mod 5) + (6 mod 4) + 0 + 0);
a(3) = 8 (= (8 mod 7) + (8 mod 6) + (8 mod 5) + 0 + (8 mod 3) + 0), etc.

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

Cf. A002808, A004125.

A002808 := proc(n) local a; if n = 1 then 4; else for a from procname(n-1)+1 do if not isprime(a) then RETURN(a) ; end if; end do; end if; end proc: A004125 := proc(n) add( n mod k, k=1..n) ; end: A161517 := proc(n) local c; A004125( A002808(n)) ; end: seq(A161517(n),n=1..80) ; # R. J. Mathar, Aug 03 2009

With[{cmps=Select[Range[200],CompositeQ]},Table[Total[Mod[n,Range[n-1]]],{n,cmps}]] (* Harvey P. Dale, Apr 09 2023 *)

a(n)=my(c=n+n\log(n+1));for(i=0,n-c+primepi(c),if(isprime(c++),i--));sum(k=2,c,c%k) \\ Charles R Greathouse IV, Oct 12 2009

a(n) = (c mod (c-1)) + (c mod (c-2)) + ... + (c mod 3) + (c mod 2).

Corrected and extended by R. J. Mathar, Aug 03 2009

A163180 a(n) = tau(n) + Sum_{k=1..n} (n mod k).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 10, 12, 15, 17, 24, 23, 30, 35, 40, 41, 53, 53, 66, 67, 74, 81, 100, 93, 106, 116, 129, 130, 153, 146, 169, 173, 188, 201, 222, 207, 235, 252, 273, 266, 299, 292, 327, 334, 345, 362, 405, 384, 417, 426, 453, 460, 507, 500, 533, 528, 557, 582, 637, 598, 647

Offset: 1

Juri-Stepan Gerasimov, Jul 22 2009

nonn

Number of divisors of n plus the sum of all the remainders modulo the numbers below n.

			a(1) = 1 + 0 = 1;
a(2) = 2 + 0 = 2;
a(3) = 2 + 1 = 3;
a(4) = 3 + 1 = 4;
a(5) = 2 + 4 = 6.

Cf. A000005, A004125.

A004125 := proc(n) add( modp(n,k),k=2..n) ; end: A163180 := proc(n) numtheory[tau](n)+A004125(n) ; end: seq(A163180(n),n=1..80) ; # R. J. Mathar, Jul 27 2009

Table[DivisorSigma[0,n]+Sum[Mod[n,k],{k,n}],{n,70}] (* Harvey P. Dale, Feb 11 2015 *)

from math import isqrt
from sympy import divisor_count
def A163180(n): return divisor_count(n)+n**2+((s:=isqrt(n))**2*(s+1)-sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1) # Chai Wah Wu, Oct 22 2023

a(n) = A000005(n) + A004125(n).

169 inserted by R. J. Mathar, Jul 27 2009

A173392 Product of nonzero remainders of n mod k, for k = 1,2,3,...,n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 6, 12, 24, 48, 720, 240, 2160, 23040, 45360, 40320, 2419200, 1935360, 65318400, 69672960, 163296000, 2786918400, 754427520000, 22992076800, 201180672000, 14714929152000, 211843247616000, 114776447385600, 32953394073600000, 2410305395097600

Offset: 1

Jaroslav Krizek, Nov 22 2010

nonn

			For n = 7; 7 mod k, for k = 1,2,3,...,7: (0,1,1,3,2,1,0). Product of nonzero remainders = 6. a(7) = 6.

Cf. A004125 (sum of remainders of n mod k).

Table[Times@@DeleteCases[Mod[n, Range[n]], 0], {n, 30}]

a(n) = prod(k=1, n, if (m = n % k, m, 1)); \\ Michel Marcus, May 23 2018

Extended by T. D. Noe, Nov 22 2010

A180492 Product of remainders of prime(n) mod k, for k = 2,3,4,...,prime(n)-1.

Original entry on oeis.org

1, 1, 2, 6, 720, 2160, 2419200, 65318400, 754427520000, 32953394073600000, 311409573995520000, 37269497815783833600000, 7890485108998805913600000000, 1096106738916569123487744000000, 4067286739206415827555188736000000000, 7924734685010508814047938347008000000000000

Offset: 1

Carl R. White, Sep 08 2010

nonn

Nonzero entries in A180491. Note that this sequence, while increasing in general, is not strictly increasing.

a(n) is divisible by (n-1)!. - Robert G. Wilson v, Sep 09 2010

			Since prime(4) = 7, a(4) = (7 mod 2) * (7 mod 3) * (7 mod 4) * (7 mod 5) * (7 mod 6) = 1 * 1 * 3 * 2 * 1 = 6.

Cf. A034386, A000142, A004125, A180491, A180493.

Cf. A173392, A000040.

a:= n-> (p-> mul(irem(p, k), k=2..p-1))(ithprime(n)):
seq(a(n), n=1..17);  # Alois P. Heinz, Jul 16 2022

f[n_]:=Times@@(Mod[n,# ]&/@ Range[2,n-1]); Table[f[Prime[i]],{i,20}] (* Harvey P. Dale, Sep 18 2010 *)
f[n_] := Times @@ Mod[n, Range[2, n - 1]]; Table[ f@ Prime@ n, {n, 10}] (* Robert G. Wilson v, Sep 09 2010 *)

a(n) = A173392(A000040(n)) = A180491(A000040(n)). - Ridouane Oudra, Nov 01 2024

A180493 Numbers n such that prime(n)! is divisible by A180491(prime(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 29, 30, 35, 38, 39, 40, 53

Offset: 1

Carl R. White, Sep 08 2010

Also numbers n such that prime(n)! is divisible by A180492(n).
Is this sequence finite?
From Robert G. Wilson v, Sep 09 2010: (Start)
Checked to 10000 for more terms.
Conjecture: This sequence is finite and all its terms are present. (End)

Cf. A034386, A000142, A004125, A180491, A180492.

f[n_] := Times @@ Mod[n, Range[2, n - 1]]; k = 1; lst = {}; While[k < 10001, If[ Divisible[ Prime@k!, f@Prime@k], AppendTo[lst, k]; Print@k]; k++ ]; lst (* Robert G. Wilson v, Sep 09 2010 *)

A233131 Sum of remainders of n modulo all smaller composite numbers.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 4, 2, 5, 9, 14, 9, 15, 21, 28, 24, 33, 27, 37, 33, 44, 56, 69, 52, 66, 81, 88, 87, 105, 92, 111, 102, 122, 143, 165, 139, 163, 187, 212, 196, 223, 209, 237, 239, 244, 274, 305, 266, 298, 296, 330, 335, 371, 347, 384, 368, 407, 447, 488, 432, 474, 516, 529, 513, 558, 543, 590, 599, 647, 637, 687, 620

Offset: 0

Max Alekseyev, Dec 07 2013

nonn

Cf. A143853, A233344

Table[Total[Mod[n,Select[Range[n-1],CompositeQ]]],{n,0,80}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 01 2018 *)

a(n) = A004125(n) - A024934(n).

A283565 Numbers n such that n = Sum_{k=1..m} (n mod k) for some m.

Original entry on oeis.org

0, 1, 2, 7, 8, 9, 10, 13, 15, 19, 22, 23, 25, 31, 37, 49, 51, 52, 57, 72, 95, 98, 100, 133, 140, 146, 152, 158, 168, 189, 196, 212, 315, 348, 376, 383, 396, 407, 416, 451, 452, 497, 521, 541, 548, 551, 568, 583, 586, 592, 593, 657, 663, 683, 729, 780, 784, 794

Offset: 1

Rémy Sigrist, Mar 11 2017

nonn

A283593 gives the least m > 0 as described in the name.

Numbers t such that t and 2*t are both in this sequence are 0, 1, 49, 98, 1249, 2599, 3784, 9565, 10933, ... - Altug Alkan, Mar 11 2017

			(7 mod 1) + (7 mod 2) + (7 mod 3) + (7 mod 4) + (7 mod 5) = 0 + 1 + 1 + 3 + 2 = 7, hence 7 appears in this sequence.
(4 mod 1) + (4 mod 2) + (4 mod 3) + (4 mod 4) = 0 + 0 + 1 + 0 = 1, and (4 mod 1) + (4 mod 2) + (4 mod 3) + (4 mod 4) + (4 mod 5) = 0 + 0 + 1 + 0 + 4 = 5, hence 4 does not appear in this sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A004125, A283593.

isok(n) = my (s=0); my (k=1); while (s

cp's OEIS Frontend

A294629 Partial sums of A294628.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

PARI

Python

Formula

A294630 Partial sums of A294629.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

Python

Formula

A354238 Decimal expansion of 1 - Pi^2/12.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A161517 Sum of remainders of c mod k where k = 1, 2, 3, ..., c and c is the n-th composite number.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A163180 a(n) = tau(n) + Sum_{k=1..n} (n mod k).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Python

Formula

Extensions

A173392 Product of nonzero remainders of n mod k, for k = 1,2,3,...,n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI