A004125 -id:A004125 - cp's OEIS frontend

1, 3, 6, 8, 13, 16, 23, 25, 30, 35, 46, 46, 59, 66, 75, 74, 91, 91, 110, 112, 125, 136, 159, 152, 169, 182, 195, 199, 228, 223, 254, 253, 274, 291, 316, 297, 334, 353, 378, 373, 414, 409, 452, 460, 475, 498, 545, 520, 557, 565, 598, 608, 661, 652, 693, 690

Offset: 1

Hieronymus Fischer, Jul 05 2007, Jul 15 2007, Jan 07 2009

Sums of rows of the triangle in A138530. - Reinhard Zumkeller, Mar 26 2008

			5 = 11111(base 1) = 101(base 2) = 12(base 3) = 11(base 4) = 10(base 5). Thus a(5) = ds_1(5)+ds_2(5)+ds_3(5)+ds_4(5)+ds_5(5) = 5+2+3+2+1 = 13.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digit Sum

Cf. A131384, A007953.

Table[n + Total@ Map[Total@ IntegerDigits[n, #] &, Range[2, n]], {n, 56}] (* Michael De Vlieger, Jan 03 2017 *)

a(n)=sum(i=2,n+1,vecsum(digits(n,i))); \\ R. J. Cano, Jan 03 2017

a(n) = n^2-sum{k>0, sum{2<=p<=n, (p-1)*floor(n/p^k)}}.

a(n) = n^2-sum{2<=p<=n, (p-1)*sum{0

a(n) = n^2-A024916(n)+A006218(n)-sum{k>1, sum{2<=p<=n, (p-1)*floor(n/p^k)}}.

a(n) = A004125(n)+A006218(n)-sum{k>1, sum{2<=p<=n, (p-1)*floor(n/p^k)}}.

Asymptotic behavior: a(n) = (1-Pi^2/12)*n^2 + O(n*log(n)) = A004125(n) + A006218(n) + O(n*log(n)).

Lim a(n)/n^2 = 1 - Pi^2/12 for n-->oo.

G.f.: (1/(1-x))*(x(1+x)/(1-x)^2-sum{k>0,sum{j>1,(j-1)*x^(j^k)/(1-x^(j^k))}= }).

Also: (1/(1-x))*(x(1+x)/(1-x)^2-sum{m>1, sum{10,j^(1/k) is an integer, j^(1/k)-1}}*x^m}).

a(n) = n^2-sum{10,sum{1

Recurrence: a(n)=a(n-1)-b(n)+2n-1, where b(n)=sum{1

a(n) = sum{1<=p<=n, ds_p(n)} where ds_p = digital sum base p.

a(n) = A043306(n) + n (that sequence ignores unary) = A014837(n) + n + 1 (that sequence ignores unary and base n in which n is "10"). - Alonso del Arte, Mar 26 2009

A154585 a(n) = abs(Sum_{k=1..n} (-1)^k * (n-k+1 mod k)).

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 4, 0, 3, 4, 5, 4, 1, 10, 4, 3, 1, 2, 9, 2, 11, 12, 17, 11, 0, 13, 0, 1, 6, 7, 23, 8, 7, 20, 10, 9, 8, 25, 14, 13, 4, 3, 20, 13, 34, 35, 34, 26, 8, 13, 6, 5, 8, 25, 24, 1, 26, 27, 34, 33, 4, 37, 25, 6, 11, 12, 11, 16, 37, 38, 60, 59, 24, 25, 0, 19, 40, 41, 54, 14, 25, 26, 51

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jan 12 2009

			a(5) = -(5 mod 1)+(4 mod 2)-(3 mod 3)+(2 mod 4)-(1 mod 5) = -0+0-0+2-1 = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A004125, A051127, A154586.

P:=proc(i) local a,n; for n from 1 by 1 to i do a:=abs(sum('(-1)^k*((n-k+1) mod k)','k'=1..n)); print(a); od; end: P(100);

a[n_] := Abs @ Sum[(-1)^k * Mod[n - k + 1, k], {k, 1, n}]; Array[a, 100] (* Amiram Eldar, Sep 18 2021 *)

a(n) = abs(sum(k=1, n, (-1)^k * lift(Mod(n-k+1, k)))); \\ Michel Marcus, Sep 18 2021

A244970 Total number of regions after n-th stage in the diagram of the symmetric representation of sigma on the four quadrants.

Original entry on oeis.org

1, 2, 6, 7, 11, 12, 16, 17, 25, 29, 33, 34, 38, 42, 50, 51, 55, 56, 60, 61, 73, 77, 81, 82, 90, 94, 106, 107, 111, 112, 116, 117, 129, 133, 141, 142, 146, 150, 162, 163, 167, 168, 172, 176, 184, 188, 192, 193, 201, 209, 221, 225, 229, 230, 242, 243, 255, 259, 263, 264

Offset: 1

Omar E. Pol, Jul 08 2014

nonn

Partial sums of A244971.
If we use toothpicks of length 1/2, so the area of the central square is equal to 1. The total area of the structure after n-th stage is equal to A024916(n), the sum of all divisors of all positive integers <= n, hence the total area of the n-th set of symmetric regions added at n-th stage is equal to sigma(n) = A000203(n), the sum of divisors of n.
If we use toothpicks of length 1, so the number of cells (and the area) of the central square is equal to 4. The number of cells (and the total area) of the structure after n-th stage is equal to 4*A024916(n) = A243980(n), hence the number of cells (and the total area) of the n-th set of symmetric regions added at n-th stage is equal to 4*A000203(n) = A239050(n).
a(n) is also the total number of terraces of the stepped pyramid with n levels described in A244050. - Omar E. Pol, Apr 20 2016

			Illustration of the structure after 15 stages (contains 50 regions):
.
.                   _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.                  |  _ _ _ _ _ _ _ _ _ _ _ _ _ _  |
.               _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _
.             _|  _| |  _ _ _ _ _ _ _ _ _ _ _ _  | |_  |_
.           _|   |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _|   |_
.          |  _ _|     |  _ _ _ _ _ _ _ _ _ _  |     |_ _  |
.     _ _ _|_| |      _| |_ _ _ _ _ _ _ _ _ _| |_      | |_|_ _ _
.    | |  _ _ _|    _|_ _|  _ _ _ _ _ _ _ _  |_ _|_    |_ _ _  | |
.    | | | |  _ _ _| |  _| |_ _ _ _ _ _ _ _| |_  | |_ _ _  | | | |
.    | | | | | |  _ _|_|  _|  _ _ _ _ _ _  |_  |_|_ _  | | | | | |
.    | | | | | | | |  _ _|   |_ _ _ _ _ _|   |_ _  | | | | | | | |
.    | | | | | | | | | |  _ _|  _ _ _ _  |_ _  | | | | | | | | | |
.    | | | | | | | | | | | |  _|_ _ _ _|_  | | | | | | | | | | | |
.    | | | | | | | | | | | | | |  _ _  | | | | | | | | | | | | | |
.    | | | | | | | | | | | | | | |   | | | | | | | | | | | | | | |
.    | | | | | | | | | | | | | | |_ _| | | | | | | | | | | | | | |
.    | | | | | | | | | | | | |_|_ _ _ _|_| | | | | | | | | | | | |
.    | | | | | | | | | | |_|_  |_ _ _ _|  _|_| | | | | | | | | | |
.    | | | | | | | | |_|_    |_ _ _ _ _ _|    _|_| | | | | | | | |
.    | | | | | | |_|_ _  |_  |_ _ _ _ _ _|  _|  _ _|_| | | | | | |
.    | | | | |_|_ _  | |_  |_ _ _ _ _ _ _ _|  _| |  _ _|_| | | | |
.    | | |_|_ _    |_|_ _| |_ _ _ _ _ _ _ _| |_ _|_|    _ _|_| | |
.    |_|_ _ _  |     |_  |_ _ _ _ _ _ _ _ _ _|  _|     |  _ _ _|_|
.          | |_|_      | |_ _ _ _ _ _ _ _ _ _| |      _|_| |
.          |_    |_ _  |_ _ _ _ _ _ _ _ _ _ _ _|  _ _|    _|
.            |_  |_  | |_ _ _ _ _ _ _ _ _ _ _ _| |  _|  _|
.              |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _|
.                  | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
The diagram is also the top view of the stepped pyramid with 15 levels described in A244050. - _Omar E. Pol_, Apr 20 2016

Cf. A000203, A004125, A024916, A196020, A235791, A236104, A237270, A237271, A237590, A237591, A237593, A239050, A239660, A239931-A239934, A243980, A244050, A244360-A244363, A244370, A244371, A244971, A245092.

A072514 Sum of n mod k for k in {1...n} with gcd(k,n) > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 2, 3, 8, 0, 9, 0, 16, 17, 16, 0, 27, 0, 31, 31, 44, 0, 52, 20, 56, 36, 69, 0, 92, 0, 72, 77, 102, 68, 127, 0, 128, 97, 152, 0, 192, 0, 165, 153, 196, 0, 230, 56, 231, 170, 237, 0, 319, 154, 290, 211, 302, 0, 421, 0, 334, 273, 334, 192, 478, 0, 411, 317

Offset: 1

Amarnath Murthy, Jul 30 2002

nonn

Previous name was: Sum of the remainders when n is divided by unrelated numbers (numbers which are neither coprime to n nor divide n).

			a(18) = 27: the unrelated numbers to 18 are 4, 8, 10, 12, 14, 15 and 16. The remainders when 18 is divided by these numbers are 2, 2, 8, 6, 4, 3 and 2 whose sum is 27.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A004125, A067439, A008683, A111490.

seq(add(n mod k, k=select(j->gcd(j,n)>1, [$1..n])),n=1..69); # Peter Luschny, Oct 01 2014

snmk[n_]:=Total[Mod[n,Complement[Complement[Range[n],Divisors[n]], Select[ Range[n], CoprimeQ[ #,n]&]]]]; Array[snmk,70] (* Harvey P. Dale, Dec 09 2018 *)

lista(nn) = vector(nn, n, sum(i=1, n, (n % i)*(gcd(n, i) != 1))); \\ Michel Marcus, Oct 01 2014

From Ridouane Oudra, May 14 2025: (Start)
a(n) = A004125(n) - A067439(n).
a(n) = Sum_{d|n, d>1} d*A067439(n/d).
a(p) = 0, for p prime.
a(p*q) = p*A067439(q) + q*A067439(p), for p and q two distinct prime numbers.
a(p^k) = p*A004125(p^(k-1)), for p a prime and k >= 0. (End)

Corrected and extended by David Garber, Oct 22 2002

New name from Robert Israel, Oct 01 2014

A103289 Numbers k such that both sigma(k) >= 2k-1 and sigma(k+1) >= 2(k+1)-1.

Original entry on oeis.org

1, 4095, 5775, 5984, 7424, 11024, 21735, 21944, 26144, 27404, 39375, 43064, 49664, 56924, 58695, 61424, 69615, 70784, 76544, 77175, 79695, 81080, 81675, 82004, 84524, 84644, 89775, 91664, 98175, 103455, 104895, 106784, 109395, 111824

Offset: 1

Max Alekseyev, Jan 28 2005

nonn

A004125(n) is less than A004125(n-1) and A004125(n+1) iff n belongs to A023196 but not to the current sequence.
Numbers k that both k and k+1 are in A103288.
Union of sequences A096399 and {2^m-1} for m in A103291.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2500 from Muniru A Asiru)

Cf. A103288, A096399, A103291, A004125, A023196.

Filtered([1..120000],n->Sigma(n)>=2*n-1 and Sigma(n+1)>2*(n+1)-1); # Muniru A Asiru, Jun 26 2018

with(numtheory): a:=`if`(sigma(n)>=2*n-1 and sigma(n+1)>=2*(n+1)-1,n,NULL): seq(a(n),n=1..120000); # Muniru A Asiru, Jun 26 2018

Select[Range[120000], And[DivisorSigma[1, #] >= 2 #1 - 1, DivisorSigma[1, # + 1] >= 2 (#1 + 1) - 1] &] (* Michael De Vlieger, Jun 29 2018 *)

for(i=1,1000000,if(sigma(i)>=2*i-1 && sigma(i+1)>=2*i+1, print1(i, ", ")));

A129365 a(n) = A092287(n)/A129364(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 6, 48, 48, 48, 48, 1536, 207360, 207360, 207360, 1105920, 1105920, 17694720, 30098718720, 15410543984640, 15410543984640, 481579499520, 60197437440000, 123284351877120000, 29958097506140160000

Offset: 1

Peter Bala, Apr 13 2007

Conjectures:
A) a(n) is always an integer.
B) If p is a prime then p|a(n) if and only if p <= n/3. Let ordp(n,p) denote the exponent of the largest power of p which divides n. For example, ordp(48,2) = 4 since 48 = 3*(2^4). The precise decomposition of a(n) into primes would follow from the following two conjectures:
C) For each positive integer n and prime p, ordp(a(n*p),p) = ordp(a(n*p+1),p) = ordp(a(n*p+2),p) = . . . = ordp(a(n*p+p-1),p).
D) Let b(n) = A004125(n). Then ordp(a(n*p),p) = b(n) + b(floor(n/p)) + b(floor(n/p^2)) + b(floor(n/p^3)) + .... This is reminiscent of de Polignac's formula (also due to Legendre) for the prime factorization of n! (see the link).

Wikipedia, De Polignac's formula.

Cf. A004125, A092287, A129364.

a(n) = ( Product_{j = 1..n} Product_{k = 1..n} gcd(j,k) ) / ( Product_{j = 1..n} Product_{d|j} d^(j/d) ).

a(n) = ( Product_{j = 1..n} Product_{k = 1..n} gcd(j,k) ) / ( Product_{k = 1..n} (floor(n/k)!)^k ).

A163553 First differences of A024816.

Original entry on oeis.org

0, 2, 1, 6, 0, 11, 1, 11, 5, 17, -4, 27, 4, 15, 9, 30, -3, 38, -2, 31, 18, 35, -12, 54, 15, 29, 12, 55, -12, 71, 1, 48, 28, 41, -7, 90, 16, 43, 6, 89, -12, 95, 4, 51, 52, 71, -28, 116, 14, 72, 26, 97, -12, 103, 8, 97, 48, 89, -48, 167, 28, 55, 41, 108, 6, 143, 10, 99, 22, 143

Offset: 1

John W. Layman, Jul 30 2009

sign

A024816(n) is the sum of the non-divisors k of n for k=2,3,...,n-1.

It appears that (1) a(n) = A120444(n)+1 if and only if n is a prime, (2) if a(n)<0 then A120444(n)<0, and (3) a(n)<=0 whenever n is of the form 6k-1. Are these conjectures easy to prove/disprove? (A120444 is the first difference of A004125 Sum of remainders of n mod k, for k = 1,2,3,...,n).

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A004125, A024816, A120444.

Differences[Table[Total[Complement[Range[n],Divisors[n]]],{n,80}]] (* Harvey P. Dale, Mar 05 2013 *)

a(n) = n + 1 + sigma(n) - sigma(n+1); \\ Michel Marcus, Jul 29 2017

A224923 a(n) = Sum_{i=0..n} Sum_{j=0..n} (i XOR j), where XOR is the binary logical exclusive-or operator.

Original entry on oeis.org

0, 2, 12, 24, 68, 114, 168, 224, 408, 594, 788, 984, 1212, 1442, 1680, 1920, 2672, 3426, 4188, 4952, 5748, 6546, 7352, 8160, 9096, 10034, 10980, 11928, 12908, 13890, 14880, 15872, 18912, 21954, 25004, 28056, 31140, 34226, 37320, 40416, 43640, 46866, 50100, 53336, 56604

Offset: 0

Alex Ratushnyak, Apr 19 2013

Alois P. Heinz, Table of n, a(n) for n = 0..1023
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 42-43.

Cf. A004125, A222423, A224915, A224924.

Array[Sum[BitXor[i, j], {i, 0, #}, {j, 0, #}] &, 45, 0] (* Michael De Vlieger, Nov 03 2022 *)

for n in range(99):
    s = 0
    for i in range(n+1):
        for j in range(n+1):
            s += i ^ j
    print(s, end=",") # Alex Ratushnyak, Apr 19 2013

# O(log(n)) version, whereas program above is O(n^2)
def countPots2Until(n):
    nbPots = {1:n>>1}
    lftMask = ~3
    rgtMask = 1
    digit = 2
    while True:
        lft = (n & lftMask) >> 1
        rgt = n & rgtMask
        nbDigs = lft
        if n & digit:
            nbDigs |= rgt
        if nbDigs == 0:
            return nbPots
        nbPots[digit] = nbDigs
        rgtMask |= digit
        digit <<= 1
        lftMask = lftMask ^ digit
def sumXorSquare(n):
    """Returns sum(i^j for i, j <= n)"""
    n += 1
    nbPots = countPots2Until(n)
    return 2 * sum(pot * freq * (n - freq) for pot, freq in nbPots.items())
print([sumXorSquare(n) for n in range(100)])  # Miguel Garcia Diaz, Nov 19 2014

# O(log(n)) version, same as previous, but simpler and about 3x faster.
def xor_square(n: int) -> int:
    return sum((((n + 1 >> i) ** 2 >> 1 << i) +
               ((n + 1) & ((1 << i) - 1)) * (n + 1 + (1 << i) >> i + 1 << 1)
               << 2 * i) for i in range(n.bit_length()))
print([xor_square(n) for n in range(100)]) # Gabriel F. Ushijima, Feb 24 2024

A224924 Sum_{i=0..n} Sum_{j=0..n} (i AND j), where AND is the binary logical AND operator.

Original entry on oeis.org

0, 1, 3, 12, 16, 33, 63, 112, 120, 153, 211, 300, 408, 553, 735, 960, 976, 1041, 1155, 1324, 1536, 1809, 2143, 2544, 2952, 3433, 3987, 4620, 5320, 6105, 6975, 7936, 7968, 8097, 8323, 8652, 9072, 9601, 10239, 10992, 11800, 12729, 13779, 14956, 16248, 17673, 19231, 20928

Offset: 0

Alex Ratushnyak, Apr 19 2013

For n>0, a(2^n)-A000217(2^n)=a(2^n-1)-A000217(2^n-1) [See links]. - R. J. Cano, Aug 21 2013

Enrique Pérez Herrero, Table of n, a(n) for n = 0..1000
R. J. Cano, Additional information
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 42-43.

Cf. A004125, A222423, A224915, A224923, A000217.

read("transforms") :
A224924 := proc(n)
    local a,i,j ;
    a := 0 ;
    for i from 0 to n do
    for j from 0 to n do
        a := a+ANDnos(i,j) ;
    end do:
    end do:
    a ;
end proc: # R. J. Mathar, Aug 22 2013

a[n_] := Sum[BitAnd[i, j], {i, 0, n}, {j, 0, n}];
Table[a[n], {n, 0, 20}]
(* Enrique Pérez Herrero, May 30 2015 *)

a(n)=sum(i=0,n,sum(j=0,n,bitand(i,j))); \\ R. J. Cano, Aug 21 2013

for n in range(99):
    s = 0
    for i in range(n+1):
      for j in range(n+1):
        s += i & j
    print(s, end=',')

a(2^n) = a(2^n - 1) + 2^n.

a(n) -a(n-1) = 2*A222423(n) -n. - R. J. Mathar, Aug 22 2013

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cp's OEIS Frontend

A067435 a(n) is the sum of all the remainders when n-th odd number is divided by odd numbers < 2n-1.

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A131383 Total digital sum of n: sum of the digital sums of n for all the bases 1 to n (a 'digital sumorial').

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A154585 a(n) = abs(Sum_{k=1..n} (-1)^k * (n-k+1 mod k)).

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A244970 Total number of regions after n-th stage in the diagram of the symmetric representation of sigma on the four quadrants.

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A072514 Sum of n mod k for k in {1...n} with gcd(k,n) > 1.

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A103289 Numbers k such that both sigma(k) >= 2*k-1 and sigma(k+1) >= 2*(k+1)-1.

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A129365 a(n) = A092287(n)/A129364(n).

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A163553 First differences of A024816.

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A103289 Numbers k such that both sigma(k) >= 2k-1 and sigma(k+1) >= 2(k+1)-1.