A033879 -id:A033879 - cp's OEIS frontend

A286385 a(n) = A003961(n) - A000203(n).

0, 0, 1, 2, 1, 3, 3, 12, 12, 3, 1, 17, 3, 9, 11, 50, 1, 36, 3, 21, 23, 3, 5, 75, 18, 9, 85, 43, 1, 33, 5, 180, 17, 3, 29, 134, 3, 9, 29, 99, 1, 69, 3, 33, 97, 15, 5, 281, 64, 54, 23, 55, 5, 255, 19, 177, 35, 3, 1, 147, 5, 15, 171, 602, 35, 51, 3, 45, 49, 87, 1, 480, 5, 9, 121, 67, 47, 87, 3, 381, 504, 3, 5, 271, 25, 9, 35, 171, 7, 291, 75, 93, 57, 15, 41, 963

Offset: 1

Antti Karttunen, May 09 2017

nonn

Are all terms nonnegative? This question is equivalent to the question posed in A285705.
From Antti Karttunen, Aug 05 2020: (Start)
The answer to the above question is yes. Because both A000203 and A003961 are multiplicative sequences, it suffices to prove that for any prime p, and e >= 1, q^e >= sigma(p^e) = ((p^(1+e))-1) / (p-1), where q = A151800(p), i.e., the next larger prime after p. If p is a lesser twin prime, then q = p+2 (and this difference can't be less than 2, apart from case p=2), and it is easy to see that (n+2)^e > ((n^(e+1)) - 1) / (n-1), for all n >= 2, e >= 1.
See comments in A326042.
(End)
This is the inverse Möbius transform of A337549, from which it is even easier to see that all terms are nonnegative. - Antti Karttunen, Sep 22 2020

Antti Karttunen, Table of n, a(n) for n = 1..16383

Cf. A000203, A001359, A003961, A001065, A031924, A033879, A048673, A151800, A285705, A326042, A336702, A336851, A336852, A337549 (Möbius transform).

Cf. A326057 [= gcd(a(n), A252748(n))].

Array[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] - DivisorSigma[1, #] &, 96] (* Michael De Vlieger, Oct 05 2020 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A286385(n) = (A003961(n) - sigma(n));
for(n=1, 16384, write("b286385.txt", n, " ", A286385(n)));

from sympy import factorint, nextprime, divisor_sigma as D
from operator import mul
def a048673(n):
    f = factorint(n)
    return 1 if n==1 else (1 + reduce(mul, [nextprime(i)**f[i] for i in f]))/2
def a(n): return 2*a048673(n) - D(n) - 1 # Indranil Ghosh, May 12 2017

(define (A286385 n) (- (A003961 n) (A000203 n)))

a(n) = A285705(A048673(n)) - 1 = 2*A048673(n) - A000203(n) - 1.
a(n) = A336852(n) - A336851(n). - Antti Karttunen, Aug 05 2020
a(n) = Sum_{d|n} A337549(d). - Antti Karttunen, Sep 22 2020
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-q(p))) - Pi^2/12 = 1.24152934..., where q(p) = nextprime(p) (A151800). - Amiram Eldar, Dec 21 2023

A357976 Numbers with a divisor having the same sum of prime indices as their quotient.

Original entry on oeis.org

1, 4, 9, 12, 16, 25, 30, 36, 40, 48, 49, 63, 64, 70, 81, 84, 90, 100, 108, 112, 120, 121, 144, 154, 160, 165, 169, 192, 196, 198, 210, 220, 225, 252, 256, 264, 270, 273, 280, 286, 289, 300, 324, 325, 336, 351, 352, 360, 361, 364, 390, 400, 432, 441, 442, 448

Offset: 1

Gus Wiseman, Oct 26 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
   1: {}
   4: {1,1}
   9: {2,2}
  12: {1,1,2}
  16: {1,1,1,1}
  25: {3,3}
  30: {1,2,3}
  36: {1,1,2,2}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  49: {4,4}
For example, 40 has factorization 8*5, and both factors have the same sum of prime indices 3, so 40 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

The partitions with these Heinz numbers are counted by A002219.
A subset of A300061.
The squarefree case is A357854, counted by A237258.
Positions of nonzero terms in A357879.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A033879, A033880, A064914, A181819, A213086, A235130, A237194, A276107, A300273, A321144, A357975.

filter:= proc(n) local F,s,t,i,R;
  F:= ifactors(n)[2];
  F:= map(t -> [numtheory:-pi(t[1]),t[2]], F);
  s:= add(t[1]*t[2],t=F)/2;
  if not s::integer then return false fi;
  try
  R:= Optimization:-Maximize(0, [add(F[i][1]*x[i],i=1..nops(F)) = s, seq(x[i]<= F[i][2],i=1..nops(F))], assume=nonnegint, depthlimit=20);
  catch "no feasible integer point found; use feasibilitytolerance option to adjust tolerance": return false;
  end try;
  true
end proc:
filter(1):= true:
select(filter, [$1..1000]); # Robert Israel, Oct 26 2023

sumprix[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]];
Select[Range[100],MemberQ[sumprix/@Divisors[#],sumprix[#]/2]&]

A153881 1 followed by -1, -1, -1, ... .

Original entry on oeis.org

1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1

Offset: 1

Mats Granvik, Jan 03 2009

Dirichlet inverse of A074206.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1).

If prefixed by initial 0, we get A134824.
Cf. A074206 (Dirichlet inverse).
Cf. A033879, A055615, A057427, A063524, A344587, A346234.

[1 - 2*Sign(n-1) : n in [1..100]]; // Wesley Ivan Hurt, Jun 20 2014

A153881:=n->`if`(n=1, 1, -1); seq(A153881(n), n=1..100); # Wesley Ivan Hurt, Jun 20 2014

Table[1 - 2 Sign[n - 1], {n, 100}] (* Wesley Ivan Hurt, Jun 20 2014 *)

a(n)=if(n>1,-1,1) \\ Charles R Greathouse IV, Sep 24 2015

G.f: x*(1-2*x)/(1-x). - Mats Granvik, Mar 09 2009, rewritten R. J. Mathar, Mar 31 2010
a(n) = (-1)^A000040(n). - Juri-Stepan Gerasimov, Sep 10 2009
G.f.: x / (1 + x / (1 - 2*x)). - Michael Somos, Apr 02 2012
From Wesley Ivan Hurt, Jun 20 2014: (Start)
a(1) = 1; a(n) = -1, n > 1.
a(n) = 1 - 2*sign(n-1) = 1 - 2*A057427(n-1).
a(n) = (-1)^sign(1-n) = (-1)^A057427(1-n).
a(n) = 2*floor(1/n)-1 = 2*A063524(n)-1. (End)
Dirichlet g.f.: 2 - zeta(s). - Álvar Ibeas, Dec 30 2018
a(n) = Sum_{d|n} A033879(d)*A055615(n/d) = Sum_{d|n} A344587(d)*A346234(n/d). - Antti Karttunen, Nov 22 2024

Edited by Charles R Greathouse IV, Mar 18 2010

More terms from Antti Karttunen, Nov 22 2024

A111490 a(n) = n + Sum_{k=1..n} (n mod k). Row sums of A372727.

Original entry on oeis.org

0, 1, 2, 4, 5, 9, 9, 15, 16, 21, 23, 33, 29, 41, 45, 51, 52, 68, 65, 83, 81, 91, 99, 121, 109, 128, 138, 152, 152, 180, 168, 198, 199, 217, 231, 253, 234, 270, 286, 308, 298, 338, 326, 368, 372, 384, 404, 450, 422, 463, 470, 500, 506, 558, 546, 584, 576, 610, 636

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Nov 21 2005

If the binary operation mod is defined n mod k = n if k = 0 and otherwise n - k*floor(n/k), as recommended in 'Concrete Mathematics' by Graham et. al. (p. 82), then a(n) = Sum_{k=0..n} (n mod k), for n >= 0. This definition is for example implemented in Sage, but not in Python. - Peter Luschny, Jul 19 2024
Previous name was "Sum of the element of the antidiagonals of the numerical array M(m, n) defined as follows. First row (M11, M12, ..., M1n): 1, 1, 1, 1, 1, 1, ... (all 1's). Second row (M21, M22, ..., M2n): 1, 2, 1, 2, 1, 2, ... (sequence 1, 2 repeated). Third row (M31, M32, ..., M3n): 1, 2, 3, 1, 2, 3, 1, 2, 3, ... (sequence 1, 2, 3 repeated). Fourth row (M41, M42, ..., M4n): 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, ... (sequence 1, 2, 3, 4 repeated). And so on."
Then the sequence is M(1,1), M(1,2) + M(2,1), M(1,3) + M(2,2) + M(3,1), etc., a(n) = Sum_{i=1..n} M(i, n-i+1).
This means: a(n) are the antidiagonal sums of the numerical array defined by M(n, k) = 1 + (k-1) mod n. - Michel Marcus, Sep 23 2013
The successive determinants of the arrays are the factorial numbers (A000142). - Robert G. Wilson v

			If the mod operation is defined according CMath, and n = 11, then the list
[n mod k : k = 0..n] = [11, 0, 1, 2, 3, 1, 5, 4, 3, 2, 1, 0], and the total of this list is a(11) = 33.

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, 1994, 34th printing 2022.

Robert Israel, Table of n, a(n) for n = 0..10000

Partial sums of A033879. - Gionata Neri, Sep 10 2015

Cf. A000142, A004125, A372727 (triangle).

A111490:=n->add(n mod i, i=1..n+1): seq(A111490(n), n=0..100); # Wesley Ivan Hurt, Dec 05 2014
seq(n + add(irem(n, k), k = 2..n-1), n = 0..58); # Peter Luschny, Jul 19 2024

t = Table[Flatten@Table[Range@n, {m, Ceiling[99/n]}], {n, 99}]; f[n_] := Sum[ t[[i, n - i + 1]], {i, n}]; Array[f, 58] (* Robert G. Wilson v, Nov 22 2005 *)
(* to view table *) Table[Flatten@Table[Range@n, {m, Ceiling[40/n]}], {n, 10}] // TableForm

vector(100, n, n + sum(k=2, n, n % k)) \\ Altug Alkan, Oct 12 2015

a(n) = sum(k=1, n, 2*k-sigma(k)); \\ Michel Marcus, Oct 11 2015

from math import isqrt
def A111490(n): return n*(n+1)+((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, Nov 01 2023

def a(n): return sum(n % k if k else n for k in range(n))
print([a(n) for n in range(59)])  # Peter Luschny, Jul 19 2024

def a(n): return sum(n.mod(k) for k in range(n))
print([a(n) for n in srange(59)])  # Peter Luschny, Jul 19 2024

a(n) = n + A004125(n). - Juri-Stepan Gerasimov, Aug 30 2009
a(n) = Sum_{i=1..n+1} (n mod i). - Wesley Ivan Hurt, Dec 05 2014
G.f.: 2*x/(1-x)^3 - (1-x)^(-1)*Sum_{k>=1} k*x^k/(1-x^k). - Robert Israel, Oct 11 2015
a(n) = (1 - Pi^2/12) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 04 2023

Edited and extended by Robert G. Wilson v, Nov 22 2005

Prepending a(0) = 0 and new name using a formula of Juri-Stepan Gerasimov by Peter Luschny, Jul 19 2024

A357854 Squarefree numbers with a divisor having the same sum of prime indices as their quotient.

Original entry on oeis.org

1, 30, 70, 154, 165, 210, 273, 286, 390, 442, 462, 561, 595, 646, 714, 741, 858, 874, 910, 1045, 1155, 1173, 1254, 1326, 1330, 1334, 1495, 1653, 1771, 1794, 1798, 1870, 1938, 2139, 2145, 2294, 2415, 2465, 2470, 2530, 2622, 2639, 2730, 2926, 2945, 2958, 3034

Offset: 1

Gus Wiseman, Oct 27 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
     1: {}
    30: {1,2,3}
    70: {1,3,4}
   154: {1,4,5}
   165: {2,3,5}
   210: {1,2,3,4}
   273: {2,4,6}
   286: {1,5,6}
   390: {1,2,3,6}
For example, 210 has factorization 14*15, and both factors have the same sum of prime indices 5, so 210 is in the sequence.

The partitions with these Heinz numbers are counted by A237258.
A subset of A319241, squarefree case of A300061.
Squarefree positions of nonzero terms in A357879.
This is the squarefree case of A357976, counted by A002219.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A033879, A033880, A064914, A181819, A235130, A237194, A276107, A300273, A321144, A357975, A357976.

sumprix[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]];
Select[Range[1000],SquareFreeQ[#]&&MemberQ[sumprix/@Divisors[#],sumprix[#]/2]&]

A318458 a(n) = n AND A001065(n), where AND is bitwise-and (A004198) & A001065 = sum of proper divisors.

Original entry on oeis.org

0, 0, 1, 0, 1, 6, 1, 0, 0, 8, 1, 0, 1, 10, 9, 0, 1, 16, 1, 20, 1, 6, 1, 0, 0, 16, 9, 28, 1, 10, 1, 0, 1, 0, 1, 36, 1, 6, 1, 32, 1, 34, 1, 40, 33, 10, 1, 0, 0, 34, 17, 36, 1, 2, 17, 0, 17, 32, 1, 44, 1, 34, 41, 0, 1, 66, 1, 0, 1, 66, 1, 72, 1, 8, 1, 64, 1, 74, 1, 64, 0, 0, 1, 4, 21, 6, 1, 88, 1, 16, 17, 76, 1, 18, 25, 0, 1, 64, 33, 100, 1, 98, 1, 104, 65

Offset: 1

Antti Karttunen, Aug 26 2018

The peculiar look of the scatterplot is partly an artifact of the logarithmic scale. Compare also to the scatterplot of A318468.

Cf. A000203, A001065, A004198, A033879, A318456, A318457.

Cf. also A318468, A318508, A318518, A318463.

[SumOfDivisors(n)-BitwiseOr(n, SumOfDivisors(n)-n): n in [1..100]]; // Vincenzo Librandi, Aug 29 2018

Table[BitAnd[n, DivisorSigma[1, n] - n], {n, 100}] (* Vincenzo Librandi, Aug 29 2018 *)

PARI
```
A318458(n) = bitand(n,sigma(n)-n);
```

a(n) = A004198(n, A001065(n)).

a(n) = A000203(n) - A318456(n) = (A000203(n)-A318457(n))/2.

A077374 Odd numbers m whose abundance by absolute value is at most 10, that is, -10 <= sigma(m) - 2m <= 10.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 15, 21, 315, 1155, 8925, 32445, 442365, 815634435

Offset: 1

Jason Earls, Nov 30 2002

Apart from {1, 3, 5, 7, 9, 11, 15, 21, 315}, subset of A088012. Probably finite. - Charles R Greathouse IV, Mar 28 2011
a(15) > 10^13. - Giovanni Resta, Mar 29 2013
The abundance of the given terms a(1..14) is: (-1, -2, -4, -6, -5, -10, -6, -10, -6, -6, 6, 6, 6, -6). See also A171929, A188263 and A188597 for numbers with abundancy sigma(n)/n close to 2. - M. F. Hasler, Feb 21 2017
a(15) > 10^22. - Wenjie Fang, Jul 13 2017

			sigma(32445) = 64896 and 32445*2 = 64890, which makes the odd number 32445 six away from perfection: A(32445) = 6 and hence in this sequence.

Eric Weisstein's World of Mathematics, Abundance.

Contains odd terms of: A191363, A088831, A125246, A088832, A141548, A087167, A125247, A088833, A101223, A223609.

Cf. A033879, A033880, A000203, A088012.

Select[Range[1, 10^6, 2], -10 <= DivisorSigma[1, #] - 2 # <= 10 &] (* Michael De Vlieger, Feb 22 2017 *)

forstep(n=1,442365,2,if(abs(sigma(n)-2*n)<=10,print1(n,",")))

a(14) from Farideh Firoozbakht, Jan 12 2004

A103977 Zumkeller deficiency of n: Let d_1 ... d_k be the divisors of n. Then a(n) = min_{ e_1 = +-1, ... e_k = +-1 } | Sum_i e_i d_i |.

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 6, 1, 5, 2, 10, 0, 12, 4, 6, 1, 16, 1, 18, 0, 10, 8, 22, 0, 19, 10, 14, 0, 28, 0, 30, 1, 18, 14, 22, 1, 36, 16, 22, 0, 40, 0, 42, 4, 12, 20, 46, 0, 41, 7, 30, 6, 52, 0, 38, 0, 34, 26, 58, 0, 60, 28, 22, 1, 46, 0, 66, 10, 42, 0, 70, 1, 72, 34, 26, 12, 58, 0, 78, 0

Offset: 1

Yasutoshi Kohmoto, Jan 01 2007

nonn

Like the ordinary deficiency (A033879) obtains 0's only at perfect numbers (A000396), the Zumkeller deficiency obtains 0's only at integer-perfect numbers, A083207. See the formula section. Unlike the ordinary deficiency, this obtains only nonnegative values. See A378600 for another version. - Antti Karttunen, Dec 03 2024

			a(6) = 1 + 2 + 3 - 6 = 0.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 10000 terms from Amiram Eldar)

Cf. A125732, A125733, A005835, A023196, A033879, A083206, A083207 (positions of 0's), A263837, A378643 (Dirichlet inverse), A378644 (Möbius transform), A378645, A378646, A378647 (an analog of A000027), A378648 (an analog of sigma), A378649 (an analog of Euler phi), A379503 (positions of 1's), A379504, A379505.
Cf. A378600 (signed variant).
Cf. also A058377, A119347.

A103977 := proc(n) local divs,a,acandid,filt,i,p,sigs ; divs := convert(numtheory[divisors](n),list) ; a := add(i,i=divs) ; for sigs from 0 to 2^nops(divs)-1 do filt := convert(sigs,base,2) ; while nops(filt) < nops(divs) do filt := [op(filt), 0] ; od ; acandid := 0 ; for p from 0 to nops(divs)-1 do if op(p+1,filt) = 0 then acandid := acandid-op(p+1,divs) ; else acandid := acandid+op(p+1,divs) ; fi ; od: acandid := abs(acandid) ; if acandid < a then a := acandid ; fi ; od: RETURN(a) ; end: seq(A103977(n),n=1..80) ; # R. J. Mathar, Nov 27 2007
# second Maple program:
a:= proc(n) option remember; local l, b; l, b:= [numtheory[divisors](n)[]],
      proc(s, i) option remember; `if`(i<1, s,
        min(b(s+l[i], i-1), b(abs(s-l[i]), i-1)))
      end: b(0, nops(l))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Dec 05 2024

a[n_] := Module[{d = Divisors[n], c, p, m}, c = CoefficientList[Product[1 + x^i, {i, d}], x]; p = -1 + Position[c, ?(# > 0 &)] // Flatten; m = Length[p]; If[OddQ[m], If[(d = p[[(m + 1)/2]] - p[[(m - 1)/2]]) == 1, 0, d], p[[m/2 + 1]] - p[[m/2]]]]; Array[a, 100] (* _Amiram Eldar, Dec 11 2019 *)

nonzerocoefpositions(p) = { my(v=Vec(p), lista=List([])); for(i=1,#v,if(v[i], listput(lista,i))); Vec(lista); }; \\ Doesn't need to be 0-based, as we use their differences only.
A103977(n) = { my(p=1); fordiv(n, d, p *= (1 + 'x^d)); my(plist=nonzerocoefpositions(p), m = #plist, d); if(!(m%2), plist[1+(m/2)]-plist[m/2], d = plist[(m+1)/2]-plist[(m-1)/2]; if(1==d,0,d)); }; \\ Antti Karttunen, Dec 03 2024, after Mathematica-program by Amiram Eldar

If n=p (prime), then a(n)=p-1. If n=2^m, then a(n)=1. [Corrected by R. J. Mathar, Nov 27 2007]
a(n) = 0 iff n is a Zumkeller number (A083207). - Amiram Eldar, Jan 05 2020
From Antti Karttunen, Dec 03 2024: (Start)
a(n) = A033879(n) iff n is a non-abundant number (A263837).
a(n) = abs(A378600(n)).
a(n) = 2*A378647(n) - A378648(n). [Analogously to A033879(n) = 2*n - sigma(n)]
a(n) = 0 <=> A083206(n) > 0.
(End)
a(p^e) = p^e - (1+p+...+p^(e-1)) = (p^e*(p-2) + 1)/(p-1) for prime p. - Jianing Song, Dec 05 2024
a(n) = 1 <=> A379504(n) > 0. - Antti Karttunen, Jan 07 2025

More terms from R. J. Mathar, Nov 27 2007

Name "Zumkeller deficiency" coined by Antti Karttunen, Dec 03 2024

A326057 a(n) = gcd(A003961(n)-2n, A003961(n)-sigma(n)), where A003961(n) is fully multiplicative function with a(prime(k)) = prime(k+1).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 1, 1, 1, 43, 1, 3, 5, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 3, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 19, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 3, 3, 5, 7, 1, 1, 3, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 1, 1

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

Terms a(n) larger than 1 and equal to A252748(n) occur at n = 6, 28, 69, 91, 496, ..., see A326134. See also A349753.

Records 1, 3, 43, 45, 2005, 79243, ... occur at n = 1, 6, 28, 360, 496, 8128, ...

Cf. A000203, A000360, A003961, A033879, A033880, A252748, A286385, A326134.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326060, A326062, A349753.

Array[GCD[#3 - #1, #3 - #2] & @@ {2 #, DivisorSigma[1, #], Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1]} &, 78] (* Michael De Vlieger, Feb 22 2021 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A252748(n) = (A003961(n) - (2*n));
A286385(n) = (A003961(n) - sigma(n));
A326057(n) = gcd(A252748(n), A286385(n));

a(n) = gcd(A252748(n), A286385(n)) = gcd(A003961(n) - 2n, A003961(n) - A000203(n)).

a(n) = gcd(A252748(n), A033879(n)) = gcd(A286385(n), A033879(n)). [Also A033880 can be used] - Antti Karttunen, May 06 2024

A325314 a(n) = n - A162296(n), where A162296(n) is the sum of divisors of n that have a square factor.

Original entry on oeis.org

1, 2, 3, 0, 5, 6, 7, -4, 0, 10, 11, -4, 13, 14, 15, -12, 17, -9, 19, -4, 21, 22, 23, -24, 0, 26, -9, -4, 29, 30, 31, -28, 33, 34, 35, -43, 37, 38, 39, -32, 41, 42, 43, -4, -9, 46, 47, -64, 0, -25, 51, -4, 53, -54, 55, -40, 57, 58, 59, -36, 61, 62, -9, -60, 65, 66, 67, -4, 69, 70, 71, -111, 73, 74, -25, -4, 77, 78, 79, -88, -36, 82, 83

Offset: 1

Antti Karttunen, Apr 21 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A013661, A033879, A162296, A228058, A325313, A325315, A325320.

Array[# - DivisorSigma[1, #] + Times @@ (1 + FactorInteger[#][[;; , 1]]) - Boole[# == 1] &, 83] (* Michael De Vlieger, Jun 06 2019, after Amiram Eldar at A162296 *)

A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325314(n) = (n - A162296(n));

a(n) = n - A162296(n).
a(n) = A033879(n) + A325313(n).
a(A228058(n)) = -A325320(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1 - zeta(2)/2 = 0.1775329665... . - Amiram Eldar, Feb 22 2024

cp's OEIS Frontend

A286385 a(n) = A003961(n) - A000203(n).

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A357976 Numbers with a divisor having the same sum of prime indices as their quotient.

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A153881 1 followed by -1, -1, -1, ... .

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A111490 a(n) = n + Sum_{k=1..n} (n mod k). Row sums of A372727.

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A357854 Squarefree numbers with a divisor having the same sum of prime indices as their quotient.

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A318458 a(n) = n AND A001065(n), where AND is bitwise-and (A004198) & A001065 = sum of proper divisors.

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A077374 Odd numbers m whose abundance by absolute value is at most 10, that is, -10 <= sigma(m) - 2m <= 10.