A163870 -id:A163870 - cp's OEIS frontend

A062825 Ch(n-th nonprime) where Ch(n) is Chowla's function, cf. A048050.

0, 2, 5, 6, 3, 7, 15, 9, 8, 14, 20, 21, 10, 13, 35, 5, 15, 12, 27, 41, 30, 14, 19, 12, 54, 21, 16, 49, 53, 39, 32, 25, 75, 7, 42, 20, 45, 65, 16, 63, 22, 31, 107, 33, 40, 62, 18, 77, 57, 26, 73, 122, 39, 48, 63, 18, 89, 105, 39, 43, 139, 22, 45, 32, 91, 143, 20, 75, 34, 49, 24, 155, 72, 56, 116, 113, 105, 86, 55, 171, 105, 40, 135

Offset: 1

Jason Earls, Jul 20 2001

a(n) = A048050(A018252(n)).

a(n+1) = sum of nontrivial divisors of n-th composite number, or row sums in table A163870. - Juri-Stepan Gerasimov, Aug 06 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A048050, A002808.

a062825 1 = 0
a062825 n = sum $ a163870_row (n - 1) -- Reinhard Zumkeller, Mar 29 2014

with(numtheory): a_list := proc(n); {$1..n} minus select(isprime,{$1..n}); sort(convert(%, list)); map(f->add(d,d=(divisors(f) minus {1,f})),%) end: a_list(113); # Peter Luschny, Mar 29 2014

Reap[Do[If[!PrimeQ[k], Sow[If[k == 1, 0, DivisorSigma[1, k] - k - 1 ]]], {k, 1, 120}]][[2, 1]] (* Jean-François Alcover, Feb 12 2018 *)

j=[0]; for(n=2,200, if(isprime(n), n+1,j=concat(j, sigma(n)-n-1))); j

Definition revised and a(1) corrected by Reinhard Zumkeller, Mar 29 2014

A385476 Values of u in the quartets (1, u, v, w) of type 3; i.e., values of u for solutions to (1 - u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

Original entry on oeis.org

5, 7, 7, 9, 9, 10, 11, 11, 13, 13, 13, 13, 15, 15, 16, 16, 17, 17, 17, 19, 19, 19, 19, 21, 21, 21, 21, 22, 22, 23, 23, 25, 25, 25, 25, 25, 25, 26, 27, 27, 28, 28, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 33, 33, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 37, 37

Offset: 1

Clark Kimberling, Aug 16 2025

A 4-tuple (m, u, v, w) is a quartet of type 3 if m, u, v, w are distinct positive integers such that m < v and m*(m - u) = v*(v - w). Here, the values of u are arranged in nondecreasing order. When there is more than one solution for given m and u, the values of v are arranged in increasing order. Here, m = 1.

			First 20 quartets (1,u,v,w) of type 3:
   m    u    v    w
   1    5    2    4
   1    7    2    5
   1    7    3    5
   1    9    2    6
   1    9    4    6
   1   10    3    6
   1   11    2    7
   1   11    5    7
   1   13    2    8
   1   13    3    7
   1   13    4    7
   1   13    6    8
   1   15    2    9
   1   15    7    9
   1   16    3    5
   1   16    3    8
   1   17    2   10
   1   17    4    8
   1   17    8   10
   1   19    2   11
1(1-11) = 5(5-7), so (1, 11, 5, 7) is in the list.

Cf. A385182 (type 1), A386218 (type 2), A386631, A385246.

solnsM[m_Integer?Positive, u_Integer?Positive] :=
  Module[{n = m  (m - u), nn, sgn, ds, tups}, If[n == 0, Return[{}]];
   sgn = Sign[n]; nn = Abs[n];
   ds = Divisors[nn];
   If[sgn > 0, ds = Select[ds, # < nn/# &]];
   tups = ({m, u, nn/#, nn/# - sgn  #} & /@ ds);
   Select[tups, #[[3]] > 1 && #[[4]] > 0 && #[[2]] =!= #[[4]](*&&
     Length@DeleteDuplicates[#]==4*)&]];
(solns =
   Sort[Flatten[Map[solnsM[1, #] &, Range[2, 30]], 1]]) // ColumnForm
Map[#[[2]] &, solns] (*A385476*)
Map[#[[3]] &, solns] (*A163870*)
Map[#[[4]] &, solns] (*A385246*)
(* Peter J. C. Moses, Aug 22 2025 *)

A233773 Triangle read by rows in which row n lists the proper divisors of n together with -n.

Original entry on oeis.org

-1, 1, -2, 1, -3, 1, 2, -4, 1, -5, 1, 2, 3, -6, 1, -7, 1, 2, 4, -8, 1, 3, -9, 1, 2, 5, -10, 1, -11, 1, 2, 3, 4, 6, -12, 1, -13, 1, 2, 7, -14, 1, 3, 5, -15, 1, 2, 4, 8, -16, 1, -17, 1, 2, 3, 6, 9, -18, 1, -19, 1, 2, 4, 5, 10, -20, 1, 3, 7, -21, 1, 2, 11, -22, 1, -23

Offset: 1

Omar E. Pol, Dec 31 2013

The same as A027750 but with the last term of every row multiplied by -1.

The sum of row n gives the abundance of n.

			Written as an irregular triangle in which row n has length A000005(n) the sequence begins:
-1;
1, -2;
1, -3;
1, 2, -4;
1, -5;
1, 2, 3, -6;
1, -7;
1, 2, 4, -8;
1, 3, -9;
1, 2, 5, -10;
1, -11;
1, 2, 3, 4, 6, -12;
...

Eric Weisstein's World of Mathematics, Abundance
Eric Weisstein's World of Mathematics, Quasiperfect Number

Row sums give A033880.
Cf. A000005, A000203, A000396, A001065, A005100, A005101, A027750, A027751, A033879.
Cf. A137510, A163870.

Flatten[Table[Join[{Most[Divisors[n]]},{-n}],{n,30}]] (* Harvey P. Dale, Feb 20 2016 *)

A328337 The number whose binary indices are the nontrivial divisors of n (greater than 1 and less than n).

Original entry on oeis.org

0, 0, 0, 2, 0, 6, 0, 10, 4, 18, 0, 46, 0, 66, 20, 138, 0, 294, 0, 538, 68, 1026, 0, 2222, 16, 4098, 260, 8266, 0, 16950, 0, 32906, 1028, 65538, 80, 133422, 0, 262146, 4100, 524954, 0, 1056870, 0, 2098186, 16660, 4194306, 0, 8423598, 64, 16777746, 65540

Offset: 1

Gus Wiseman, Oct 15 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The nontrivial divisors of 18 are {2, 3, 6, 9}, so a(18) = 2^1 + 2^2 + 2^5 + 2^8 = 294.

Removing zeros gives binary indices of rows of A163870.
The version for all divisors is A034729.
The version for proper divisors is A247146.
Cf. A000005, A000120, A027750, A029931, A033676, A048793, A060775, A070824, A070939, A275700, A326031.

Table[Total[(2^DeleteCases[Divisors[n],1|n])/2],{n,100}]

from sympy import divisors
def A328337(n): return sum(1<<(d-1) for d in divisors(n,generator=True) if 1Chai Wah Wu, Jul 15 2022

A000120(a(n)) = A070824(n).
A070939(a(n)) = A032742(n).
A001511(a(n)) = A107286(n).

A328459 Sorted positions of first appearances in A328458 (maximum run-length of nontrivial divisors) of each positive integer in the image.

Original entry on oeis.org

1, 2, 6, 12, 60, 420, 504, 840, 2520, 27720, 360360, 720720, 4084080

Offset: 0

Gus Wiseman, Oct 17 2019

			The sequence of terms > 1 together with their nontrivial divisors begins:
    2: {}
    6: {2,3}
   12: {2,3,4,6}
   60: {2,3,4,5,6,10,12,15,20,30}
  420: {2,3,4,5,6,7,10,12,14,15,20,21,28,30,35,42,60,70,84,105,140,210}
  504: {2,3,4,6,7,8,9,12,14,18,21,24,28,36,42,56,63,72,84,126,168,252}

Positions of first appearances in A328458.
The version for all divisors is A051451.
Cf. A000005, A033676, A060775, A070824, A119313, A129308, A181063, A199970, A163870, A328448, A328449.

dav=Table[Switch[n,1,1,_,Max@@Length/@Split[DeleteCases[Divisors[n],1|n],#2==#1+1&]],{n,1000}];
Table[Position[dav,i][[1,1]],{i,Union[dav]}]//Sort

a(12) from Robert Israel, Mar 31 2023

cp's OEIS Frontend

A062825 Ch(n-th nonprime) where Ch(n) is Chowla's function, cf. A048050.

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A385476 Values of u in the quartets (1, u, v, w) of type 3; i.e., values of u for solutions to (1 - u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

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A233773 Triangle read by rows in which row n lists the proper divisors of n together with -n.

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A328337 The number whose binary indices are the nontrivial divisors of n (greater than 1 and less than n).

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A328459 Sorted positions of first appearances in A328458 (maximum run-length of nontrivial divisors) of each positive integer in the image.

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