A048050 -id:A048050 - cp's OEIS frontend

A244576 Sum of all proper divisors of all positive integers <= prime(n).

0, 0, 2, 7, 23, 38, 69, 89, 133, 227, 268, 397, 483, 536, 632, 821, 1018, 1125, 1355, 1511, 1633, 1890, 2077, 2406, 2906, 3150, 3263, 3509, 3680, 3960, 5026, 5319, 5854, 6003, 6909, 7130, 7761, 8345, 8681, 9381, 9986, 10351, 11456, 11771, 12212, 12481, 14128

Offset: 1

Omar E. Pol, Jun 30 2014

nonn

Also sum of all proper divisors of all positive integers <= prime(n)-1.

Also zero together with the numbers that are repeated in A244049.

Cf. A000040, A000203, A048050, A244049, A244578, A244583.

a(n) = sum(i=2, prime(n), sigma(i)-i-1); \\ Michel Marcus, Sep 29 2014

a(n) = A244049(A000040(n)-1) = A244049(A000040(n)).

a(n) ~ (Pi^2/12 - 1/2) * n^2 * log(n)^2. - Amiram Eldar, Mar 22 2024

More terms from Michel Marcus, Sep 29 2014

A347153 Sum of all divisors, except the largest of every number, of the first n odd numbers.

Original entry on oeis.org

0, 1, 2, 3, 7, 8, 9, 18, 19, 20, 31, 32, 38, 51, 52, 53, 68, 81, 82, 99, 100, 101, 134, 135, 143, 164, 165, 182, 205, 206, 207, 248, 267, 268, 295, 296, 297, 346, 365, 366, 406, 407, 430, 463, 464, 485, 520, 545, 546, 603, 604, 605, 692, 693, 694, 735, 736, 765, 830, 855

Offset: 1

Omar E. Pol, Aug 20 2021

Sum of all aliquot divisors (or aliquot parts) of the first n odd numbers.
Partial sums of the odd-indexed terms of A001065.
a(n) has a symmetric representation.

Partial sums of A346877.

Cf. A000203, A001065, A001477, A005408, A008438, A048050, A153485, A237593, A245092, A244049, A326123, A346869, A346878, A346879, A347154.

s[n_] := DivisorSigma[1, 2*n - 1] - 2*n + 1; Accumulate @ Array[s, 100] (* Amiram Eldar, Aug 20 2021 *)

a(n) = sum(k=1, n, k = 2*k-1; sigma(k)-k); \\ Michel Marcus, Aug 20 2021

from sympy import divisors
from itertools import accumulate
def A346877(n): return sum(divisors(2*n-1)[:-1])
def aupton(nn): return list(accumulate(A346877(n) for n in range(1, nn+1)))
print(aupton(60)) # Michael S. Branicky, Aug 20 2021

a(n) = A001477(n-1) + A346869(n).
G.f.: (1/(1 - x)) * Sum_{k>=0} (2*k + 1) * x^(3*k + 2) / (1 - x^(2*k + 1)). - Ilya Gutkovskiy, Aug 20 2021
a(n) = (Pi^2/8 - 1)*n^2 + O(n*log(n)). - Amiram Eldar, Mar 21 2024

A380580 Irregular tetrahedron T(s,r,k) read by rows in which the slice s is an irregular triangle, itself read by rows, in which row r lists the r-th row of A237593 sandwiched between two A380579(s+1,r+1), with s >= 0; 0 <= r <= s; k >= 0. Assume that row 0 of A237593 is empty.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 1, 1, 4, 4, 3, 1, 1, 3, 2, 2, 2, 2, 5, 5, 4, 1, 1, 4, 3, 2, 2, 3, 2, 2, 1, 1, 2, 2, 7, 7, 6, 1, 1, 6, 5, 2, 2, 5, 4, 2, 1, 1, 2, 4, 3, 3, 1, 1, 3, 3, 8, 8, 7, 1, 1, 7, 6, 2, 2, 6, 5, 2, 1, 1, 2, 5, 4, 3, 1, 1, 3, 4, 3, 3, 2, 2, 3, 3, 10, 10, 9, 1, 1, 9, 8, 2, 2, 8, 7, 2, 1, 1, 2, 7, 6, 3, 1, 1, 3, 6, 5, 3, 2, 2, 3, 5

Offset: 0

Omar E. Pol, Jan 27 2025

The discussion of this sequence was too long to be included here, and can be found in the attached "Discussion" text file (see the first link). - N. J. A. Sloane, Jul 31 2025

Paolo Xausa, Table of n, a(n) for n = 0..14441 (slices 0..50 of the tetrahedron, flattened).
Omar E. Pol, Discussion of the Sequence A380580.
Omar E. Pol, Illustration of initial terms of A000203 in the pyramid.
Omar E. Pol, Illustration of initial terms of A001065 in the pyramid.
Omar E. Pol, Illustration of initial terms of A048050 in the pyramid.
Omar E. Pol, Illustration of initial terms of A067742 in the pyramid.
Omar E. Pol, Illustration of initial terms of A224613 (black spiders).
Omar E. Pol, Illustration of initial terms of A237593 (essentially a template for the Pop-Up pyramid).
Omar E. Pol, Prism of partitions of 10 and its companion tower (both have the same volume).
Omar E. Pol, The symmetric representation of sigma(n), n = 1..64 (the top view of the pyramid and of the tower).

See the "Discussion" text file for the cross-references.

A237593row[n_] := Join[#, Reverse[#]] & [Table[Ceiling[(n+1)/k - (k+1)/2] + Quotient[k*(k+3) - 2*n, 2*(k+1)], {k, Quotient[Sqrt[8*n + 1] - 1, 2]}]];
A380580slice[s_] := Table[Join[#, A237593row[r], #] & [{Quotient[3*s, 2] - r + 1}], {r, 0, s}];
Array[A380580slice, 10, 0] (* Paolo Xausa, Aug 19 2025 *)

Edited by N. J. A. Sloane, Jul 31 2025

A002954 Smallest number such that n-th iterate of Chowla function is 0.

Original entry on oeis.org

2, 4, 8, 15, 12, 27, 24, 36, 90, 96, 245, 288, 368, 676, 1088, 2300, 1596, 1458, 3344, 3888, 5360, 8895, 11852, 25971, 23360, 38895, 35540, 35595, 36032, 53823, 47840, 62055, 59360, 83391, 70784, 128079, 145668, 349299, 254540, 327495, 293744, 328335, 167664

Offset: 1

N. J. A. Sloane

nonn

Chowla's function (A048050) = sum of divisors of n except 1 and n.
a(83) > 10^10. - Donovan Johnson, Feb 15 2013
The first 35 terms were found by Lal and Forbes (1971). - Amiram Eldar, Mar 09 2024

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

Donovan Johnson, Table of n, a(n) for n = 1..82
M. Lal and A. Forbes, A note on Chowla's function, Math. Comp., 25 (1971), 923-925.

Cf. A048050.

chowla[n_] := DivisorSigma[1, n] - 1 - n; chowlaSeq[n_] := Module[{m = n, cnt = 0, seq = {}}, While[m > 0 && ! MemberQ[seq, m], AppendTo[seq, m]; m = chowla[m]; cnt++]; If[m == 0, AppendTo[seq, m]]; seq]; nn = 20; t = Table[0, {nn}]; left = nn; n = 1; While[left > 0, n++; cSeq = chowlaSeq[n]; c = Length[cSeq] - 1; If[cSeq[[-1]] == 0 && c <= nn && t[[c]] == 0, t[[c]] = n; left--]]; t (* T. D. Noe, Dec 29 2011 *)

a(31)-a(43) from T. D. Noe, Dec 29 2011

A053246 First differences of chowla(n).

Original entry on oeis.org

0, 0, 2, -2, 5, -5, 6, -3, 4, -7, 15, -15, 9, -1, 6, -14, 20, -20, 21, -11, 3, -13, 35, -30, 10, -3, 15, -27, 41, -41, 30, -16, 5, -7, 42, -54, 21, -5, 33, -49, 53, -53, 39, -7, -7, -25, 75, -68, 35, -22, 25, -45, 65, -49, 47, -41, 9, -31, 107, -107, 33, 7, 22, -44, 59, -77, 57, -31

Offset: 1

Asher Auel, Jan 10 2000

sign

Second differences give A053223, for n>1.

If the first term is changed to 1, this is also the first differences of A001065. - N. J. A. Sloane, Jan 17 2023

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A048050, A053222, A053223.

Cf. also A001065.

[0] cat [DivisorSigma(1,n+1) - DivisorSigma(1,n) - 1: n in [2..100]]; // G. C. Greubel, Sep 03 2018

with(numtheory): seq( sigma(i+1) - sigma(i) - 1, i=2..100); # for n>1

Chowlan[n_] := If[n == 1, 0, DivisorSigma[1, n] - n - 1]; Table[Chowlan[n + 1] - Chowlan[n], {n, 1, 100}] (* G. C. Greubel, Sep 03 2018 *)
Differences[Join[{0},Table[DivisorSigma[1,n]-n-1,{n,2,100}]]] (* Harvey P. Dale, Dec 19 2022 *)

concat([0], vector(100, n, n++; sigma(n+1) - sigma(n) -1)) \\ G. C. Greubel, Sep 03 2018

a(n) = A053222(n) - 1, for n>1

A054019 Square roots of A054018.

Original entry on oeis.org

3, 1, 4, 3, 5, 4, 5, 7, 4, 1, 8, 11, 6, 9, 8, 13, 8, 6, 10, 15, 8, 6, 11, 6, 13, 7, 19, 15, 1, 12, 21, 21, 12, 16, 20, 14, 8, 21, 1, 12, 19, 8, 20, 21, 27, 8, 14, 12, 27, 10, 29, 27, 5, 20, 16, 35, 10, 27, 35, 31, 30, 29, 3, 12, 28, 5, 1, 35, 26, 10, 20, 37, 12, 33, 18, 43, 43, 45, 22

Offset: 1

Asher Auel, Jan 19 2000

nonn

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

Cf. A048050, A054013, A054017, A054018.

chowla[n_] := DivisorSigma[1, n] - n - 1; aQ[n_] := (c = chowla[n]) > 0 && IntegerQ@Sqrt@Mod[c, n]; Sqrt @ Mod[chowla[#], #] & /@ Select[Range[1000], aQ] (* Amiram Eldar, Aug 28 2019 *)

A054020 Chowla's function of n is not divisible by the number of proper divisors of n.

Original entry on oeis.org

6, 9, 10, 15, 16, 20, 21, 22, 25, 28, 30, 33, 34, 36, 39, 42, 44, 45, 46, 48, 49, 50, 51, 54, 55, 57, 58, 60, 64, 68, 69, 70, 72, 75, 76, 78, 80, 81, 82, 84, 85, 87, 91, 93, 94, 96, 98, 99, 100, 102, 105, 106, 108, 111, 114, 115, 116, 117, 118, 120, 121, 123, 124, 126

Offset: 1

Asher Auel, Jan 19 2000

nonn

Chowla's function (A048050) = sum of divisors of n except 1 and n.

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

Complement is A054021.

Cf. A032741, A048050, A054015.

with(numtheory):
[seq(`if`((sigma(i)-i-1) mod (tau(i)-1) <> 0,i,print( )),i=2..500)];

Select[Range[2,150],!Divisible[DivisorSigma[1,#]-#-1,DivisorSigma[ 0,#]- 1]&] (* Harvey P. Dale, May 27 2014 *)

A054021 Numbers n such that Chowla's function of n is divisible by the number of proper divisors of n.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 11, 12, 13, 14, 17, 18, 19, 23, 24, 26, 27, 29, 31, 32, 35, 37, 38, 40, 41, 43, 47, 52, 53, 56, 59, 61, 62, 63, 65, 66, 67, 71, 73, 74, 77, 79, 83, 86, 88, 89, 90, 92, 95, 97, 101, 103, 104, 107, 109, 110, 112, 113, 119, 122, 125, 127, 128, 131, 134, 136

Offset: 1

Asher Auel, Jan 19 2000

nonn

Chowla's function (A048050) = sum of divisors of n except 1 and n.

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

Complement is A054020.

Cf. A032741, A048050, A054015.

with(numtheory):
[seq(`if`((sigma(i)-i-1) mod (tau(i)-1)=0,i,print( )),i=2..1000)];

Select[Range[2,150],Divisible[DivisorSigma[1,#]-1-#,DivisorSigma[ 0,#]-1]&] (* Harvey P. Dale, Aug 13 2018 *)

A054023 Chowla function of n is not divisible by the number of divisors of n.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 25, 26, 28, 30, 33, 34, 38, 40, 42, 44, 45, 46, 48, 49, 52, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 90, 92, 93, 94, 96, 99, 100, 102, 104, 105, 106, 108, 110, 112, 114

Offset: 1

Asher Auel, Jan 19 2000

nonn

Chowla's function (A048050) = sum of divisors of n except 1 and n.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Complement is A054022.

Cf. A000005, A048050, A054014.

with(numtheory):
[seq(`if`((sigma(i)-i-1) mod tau(i) <> 0,i,print( )),i=1..1000)];

cfQ[n_]:=!Divisible[DivisorSigma[1,n]-1-n,DivisorSigma[0,n]]; Select[ Range[ 150],cfQ] (* Harvey P. Dale, Jul 22 2014 *)

A069896 GCD of consecutive values of Chowla's function.

Original entry on oeis.org

0, 0, 2, 2, 5, 5, 6, 3, 1, 7, 15, 15, 9, 1, 2, 14, 20, 20, 21, 1, 1, 13, 35, 5, 5, 3, 3, 27, 41, 41, 30, 2, 1, 1, 6, 54, 21, 1, 1, 49, 53, 53, 39, 1, 1, 25, 75, 1, 7, 2, 5, 45, 65, 1, 1, 1, 1, 31, 107, 107, 33, 1, 2, 2, 1, 77, 57, 1, 1, 73, 122, 122, 39, 3, 3, 9, 1, 89, 105, 3, 1, 43

Offset: 1

Labos Elemer, Apr 10 2002

			Chowla's function values from 92 to 96 are 75,34,49,24,155: successive values are relatively primes.

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

Cf. A048050, A000203, A001650.

m = 100; s = Array[If[# == 1, 0, DivisorSigma[1, #] - # - 1] &, {m}]; GCD[s[[1 ;; m - 1]], s[[2 ;; m]]] (* Amiram Eldar, Aug 28 2019 *)

a(n) = gcd(sigma(n+1)-(n+1)-1, sigma(n)-n-1), for n > 1.

a(1) corrected by Amiram Eldar, Aug 28 2019

cp's OEIS Frontend

A244576 Sum of all proper divisors of all positive integers <= prime(n).

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

Extensions

A347153 Sum of all divisors, except the largest of every number, of the first n odd numbers.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A380580 Irregular tetrahedron T(s,r,k) read by rows in which the slice s is an irregular triangle, itself read by rows, in which row r lists the r-th row of A237593 sandwiched between two A380579(s+1,r+1), with s >= 0; 0 <= r <= s; k >= 0. Assume that row 0 of A237593 is empty.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A002954 Smallest number such that n-th iterate of Chowla function is 0.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Extensions

A053246 First differences of chowla(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A054019 Square roots of A054018.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A054020 Chowla's function of n is not divisible by the number of proper divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A054021 Numbers n such that Chowla's function of n is divisible by the number of proper divisors of n.

Views

Author

Keywords

Comments

Links