cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A237878 Composite numbers n such that n | A072514 (n).

Original entry on oeis.org

4, 16, 22, 34, 108, 121, 187, 232, 289, 300, 372, 2230, 4322, 8882, 23771, 36737, 48851, 50946, 67650, 75497, 424689, 482998, 581765, 4669921, 7374225, 9597001, 19722481, 110714642, 416101557, 1478286411, 1773582750
Offset: 1

Views

Author

Paolo P. Lava, Feb 14 2014

Keywords

Comments

a(24) > 3000000. - Giovanni Resta, Feb 17 2014
a(32) > 4 * 10^9. - Hiroaki Yamanouchi, Oct 01 2014

Examples

			The unrelated numbers to 16 are 6, 10, 12 and 14.
The remainders when 16 is divided by these numbers are 4, 6, 4, 2.
Their sum is 16 and 16 / 16 = 1.

Crossrefs

Cf. A072514.

Programs

  • Maple
    P:=proc(q) local i,n,t; for n from 2 to q do if not isprime(n) then t:=0;
for i from 1 to n-1 do if not gcd(i,n)=1 then if not type(n/i,integer) then t:=t+(n mod i); fi; fi; od;
if type(t/n,integer) then print(n); fi; fi; od; end: P(10^6);

Extensions

a(16)-a(23) from Giovanni Resta, Feb 17 2014
a(24)-a(28) from Hiroaki Yamanouchi, Oct 01 2014
a(29)-a(31) from Hiroaki Yamanouchi, Sep 27 2015

A004125 Sum of remainders of n mod k, for k = 1, 2, 3, ..., n.

Original entry on oeis.org

0, 0, 1, 1, 4, 3, 8, 8, 12, 13, 22, 17, 28, 31, 36, 36, 51, 47, 64, 61, 70, 77, 98, 85, 103, 112, 125, 124, 151, 138, 167, 167, 184, 197, 218, 198, 233, 248, 269, 258, 297, 284, 325, 328, 339, 358, 403, 374, 414, 420, 449, 454, 505, 492, 529, 520, 553, 578, 635, 586, 645, 672
Offset: 1

Views

Author

N. J. A. Sloane, Jeffrey Shallit

Keywords

Comments

Row sums of A051778, A048158. Antidiagonal sums of A051127. - L. Edson Jeffery, Mar 03 2012
Let u_m(n) = Sum_{k=1..n} (n^m mod k^m) with m integer. As n-->+oo, u_m(n) ~ (n^(m+1))*(1-(1/(m+1))*Zeta(1+1/m)). Proof: using Riemann sums, we have u_m(n) ~ (n^(m+1))*int(((1/x)[nonascii character here])*(1-floor(x^m)/(x^m)),x=1..+oo) and the result follows. - Yalcin Aktar, Jul 30 2008 [x is the real variable of integration. The nonascii character (which was illegible in the original message) is probably some form of multiplication sign. I suggest that we leave it the way it is for now. - N. J. A. Sloane, Dec 07 2014]
Also the alternating row sums of A236112. - Omar E. Pol, Jan 26 2014
If n is prime then a(n) = a(n-1) + n - 2. - Omar E. Pol, Mar 19 2014
If n is a power of 2 greater than 1, then a(n) = a(n-1). - David Morales Marciel, Oct 21 2015
It appears that if n is an even perfect number, then a(n) = a(n-1) - 1. - Omar E. Pol, Oct 21 2015
Partial sums of A235796. - Omar E. Pol, Jun 26 2016
Aside from a(n) = a(n-1) for n = 2^m, the only values appearing more than once among the first 6*10^8 terms are those at n = 38184 +- 1, 458010 +- 1, 776112 +- 1, 65675408 +- 1, and 113393280 +- 2. - Trevor Cappallo, Jun 07 2021
The off-by-1 terms in the comment above are the terms of A068077. Proof: If a(n-1) = a(n+1), then (n-1)^2 - Sum_{k=1..n-1} sigma(k) = (n+1)^2 - Sum_{k=1..n+1} sigma(k) via the formula; rearranging terms gives sigma(n)+sigma(n+1)=4n. - Lewis Chen, Sep 24 2021

Examples

			a(5) = 4. The remainder when 5 is divided by 2,3,4 respectively is 1,2,1 and their sum = 4.

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000290, A006218, A023196, A048158, A050482, A051778, A120444 (first differences).
Cf. A067439, A072514, A111490.

Programs

  • GAP
    List([1..70],n->n^2-Sum([1..n],k->Sigma(k))); # Muniru A Asiru, Mar 28 2018
  • Haskell
    a004125 n = sum $ map (mod n) [1..n]
-- Reinhard Zumkeller, Jan 28 2011
  • Magma
    [&+[n mod r: r in [1..n]]: n in [1..70]]; // Bruno Berselli, Jul 06 2014
  • Maple
    A004125 := n -> add( modp(n,k), k=2..n); /* much faster and unambiguous; "a mod b" may be mods(a,b) */ # M. F. Hasler, Nov 22 2007
  • Mathematica
    Table[Sum[Mod[n,k],{k,2,n-1}],{n,70}] (* Harvey P. Dale, Nov 23 2011 *)
Accumulate[Table[2n-1-DivisorSigma[1,n],{n,70}]] (* Harvey P. Dale, Jul 11 2014 *)
  • PARI
    A004125(n)=sum(k=2,n,n%k) \\ M. F. Hasler, Nov 22 2007
  • Python
    def a(n): return sum(n%k for k in range(1, n))
print([a(n) for n in range(1, 63)]) # Michael S. Branicky, Jun 08 2021
  • Python
    from math import isqrt
def A004125(n): return 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 21 2023
  • SageMath
    def a(n): return sum(n.mod(k) for k in (1..n))
print([a(n) for n in (1..62)])  # Peter Luschny, May 12 2025

Formula

a(n) = n^2 - Sum_{k=1..n} sigma(k) = A000290(n) - A024916(n), hence asymptotically a(n) = n^2*(1-Pi^2/12) + O(n*log(n)^(2/3)). - Benoit Cloitre, Apr 28 2002. Asymptotics corrected/improved by Charles R Greathouse IV, Feb 22 2015
a(n) = A008805(n-3) + A049798(n-1), for n > 2. - Carl Najafi, Jan 31 2013
a(n) = A000217(n-1) - A153485(n). - Omar E. Pol, Jan 28 2014
G.f.: x^2/(1-x)^3 - (1-x)^(-1) * Sum_{k>=1} k*x^(2*k)/(1-x^k). - Robert Israel, Aug 13 2015
a(n) = Sum_{i=1..n} (n mod i). - Wesley Ivan Hurt, Sep 15 2017
From Ridouane Oudra, May 12 2025: (Start)
a(n) = A067439(n) + A072514(n).
a(n) = Sum_{d|n} d*A067439(n/d).
a(p) = A067439(p), for p prime.
a(p^k) = A072514(p^(k+1))/p, for p prime and k >= 0. (End)
a(n) = A111490(n) - n. - Peter Luschny, May 12 2025

Extensions

Edited by M. F. Hasler, Apr 18 2015

A067439 a(n) = sum of all the remainders when n is divided by positive integers less than and coprime to n.

Original entry on oeis.org

0, 0, 1, 1, 4, 1, 8, 6, 9, 5, 22, 8, 28, 15, 19, 20, 51, 20, 64, 30, 39, 33, 98, 33, 83, 56, 89, 55, 151, 46, 167, 95, 107, 95, 150, 71, 233, 120, 172, 106, 297, 92, 325, 163, 186, 162, 403, 144, 358, 189, 279, 217, 505, 173, 375, 230, 342, 276, 635, 165, 645, 338
Offset: 1

Views

Author

Amarnath Murthy, Jan 29 2002

Keywords

Examples

			a(8) = 6. The remainders when 8 is divided by the coprime numbers 1, 3, 5 and 7 are 0, 2, 3 and 1, whose sum = 6.

Links

Crossrefs

Cf. A004125, A067435, A067436, A024934, A033955, A111490.
Cf. A072514, A008683.

Programs

  • Maple
    a := n -> add(ifelse(igcd(n, i) = 1, irem(n, i), 0), i = 1..n-1):
seq(a(n), n = 1..62);  # Peter Luschny, May 14 2025
  • Mathematica
    a[n_] := Sum[If[GCD[i, n]>1, 0, Mod[n, i]], {i, 1, n-1}]
Table[Total[Mod[n,#]&/@Select[Range[n-1],CoprimeQ[#,n]&]],{n,70}] (* Harvey P. Dale, May 22 2012 *)
  • PARI
    a(n)=sum(i=1,n-1,if(gcd(n,i)==1,n%i)) \\ Charles R Greathouse IV, Jul 17 2012

Formula

From Ridouane Oudra, May 14 2025: (Start)
a(n) = A004125(n) - A072514(n).
a(n) = Sum_{d|n} d*mu(d)*A004125(n/d).
a(n) = Sum_{d|n} mu(d)*f(n,d), where f(n,d) = Sum_{i=1..n/d} (n mod d*i).
a(p) = A004125(p), for p prime.
a(p^k) = A004125(p^k) - p*A004125(p^(k-1)), for p prime and k >= 0.
a(p^k) = A072514(p^(k+1))/p - A072514(p^k), for p prime and k >= 0. (End)

Extensions

Edited by Dean Hickerson, Feb 15 2002
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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