A048050 -id:A048050 - cp's OEIS frontend

A054014 Chowla function of n read modulo (the number of divisors of n).

0, 0, 0, 2, 0, 1, 0, 2, 0, 3, 0, 3, 0, 1, 0, 4, 0, 2, 0, 3, 2, 1, 0, 3, 2, 3, 0, 3, 0, 1, 0, 0, 2, 3, 0, 0, 0, 1, 0, 1, 0, 5, 0, 3, 2, 1, 0, 5, 1, 0, 0, 3, 0, 1, 0, 7, 2, 3, 0, 11, 0, 1, 4, 6, 2, 5, 0, 3, 2, 1, 0, 2, 0, 3, 0, 3, 2, 1, 0, 5, 4, 3, 0, 7, 2, 1, 0, 3, 0, 11, 0, 3, 2, 1, 0, 11, 0, 0, 2, 8, 0, 1, 0

Offset: 1

Asher Auel, Jan 17 2000

nonn

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

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000005, A048050, A054013, A054015.

with(numtheory): [seq((sigma(i)-i-1) mod tau(i),i=1..120)];

Array[Mod[DivisorSigma[1, #] - # - 1, DivisorSigma[0, #]] &, 103] (* Michael De Vlieger, Oct 30 2017 *)

A054014(n) = ((sigma(n)-n-1) % numdiv(n)); \\ Antti Karttunen, Oct 30 2017

a(n) = A048050(n) mod A000005(n).

A054015 a(n) is Chowla function of n read modulo (number of proper divisors of n), a(1) = 0 by convention.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 2, 0, 6, 0, 0, 2, 1, 0, 6, 0, 0, 1, 0, 0, 4, 0, 4, 2, 1, 0, 3, 1, 2, 2, 0, 0, 2, 1, 0, 1, 1, 0, 8, 0, 0, 0, 2, 0, 0, 0, 2, 2, 3, 0, 1, 0, 0, 3, 3, 0, 5, 0, 6, 3, 1, 0, 7, 1, 0, 2, 0, 0, 0, 2, 0, 1, 1, 0, 1, 0, 2, 1, 4, 0, 1, 0, 0, 2

Offset: 1

Asher Auel, Jan 17 2000

nonn

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

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors

Cf. A032741, A048050, A054013, A054014.

with(numtheory): [seq((sigma(i)-i-1) mod (tau(i)-1),i=2..120);#i>1

A054015(n) = if(1==n,0,((sigma(n)-n-1) % (numdiv(n)-1))); \\ Antti Karttunen, Oct 20 2017

a(1) = 0; for n > 1, a(n) = A048050(n) mod A032741(n).

Description clarified by Antti Karttunen, Oct 20 2017

A070037 Nonprime numbers k such that sigma(k) == k+1 (mod phi(k)).

Original entry on oeis.org

1, 4, 15, 900, 903, 28611063

Offset: 1

Labos Elemer, Apr 18 2002

Nonprime numbers k such that Chowla(k)/phi(k) is an integer.
a(7) > 3.7*10^12. - Giovanni Resta, Jul 14 2013
3^30*13*852977547249259 and 3^36*13*621820631944710643 are also terms. - Giovanni Resta, Nov 14 2019

			Below 30000000 only 5 composite numbers were found: C = {4,15,900,902,28611063}, Chowla(C) = {2,8,1920,504,17600976}, phi(C) = {2,8,240,504,17600976}, quotient = {1,1,8,1,1}.

Cf. A000010, A000203, A048050.

s2[x_] := DivisorSigma[1, x]-x-1 e0[x_] := EulerPhi[x] Do[s=s2[n]/e0[n]; If[IntegerQ[s]&&!PrimeQ[n], Print[{n, s2[n], e0[n], s}]], {n, 1, 1000000}]

is(n)=!isprime(n) && Mod(sigma(n),eulerphi(n))==n+1 \\ Charles R Greathouse IV, Jun 06 2013

A132794 Numbers n such that sigma(phi(n)) -phi(n) -1 = phi(sigma(n) -n -1).

Original entry on oeis.org

8, 16, 64, 256, 16384, 262144, 1048576, 4294967296

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Aug 31 2007

Used sigma(n)-n-1, namely the sum of proper divisors minus 1.
a(8) > 10^8. - Michel Marcus, Nov 01 2014
Every 2^(A000043+1) is a term. Proof sketch: Let ch=A048050 and n=2^k, then ch(phi(2^k))=phi(ch(2^k)), ch(2^(k-1))=phi(2^k-2), 2^(k-1)-2=phi(2^(k-1)-1), since phi(prime)=prime-1 the condition is satisfied by every k=A000043+1 or n=2^(A000043+1). See link. - Jon Maiga, Dec 14 2018
Conjecture: a(n)=2^(A000043(n)+1), if true the next terms are: 4294967296, 4611686018427387904, 1237940039285380274899124224... - Jon Maiga, Dec 14 2018
a(9) > 6.5*10^11. - Giovanni Resta, Dec 01 2019

Jon Maiga, Euler Phi, Chowla and Mersenne primes, 2018.

Cf. A000010, A000043, A000203, A001229, A018784, A033632, A048050, A132793.

Filtered([4..1000000],n->Sigma(Phi(n))-Phi(n)-1=Phi(Sigma(n)-n-1)); # Muniru A Asiru, Dec 16 2018

[n: n in [2..30000] | DivisorSigma(1,n) ne n+1 and DivisorSigma(1, EulerPhi(n)) - EulerPhi(n) - 1 eq EulerPhi(DivisorSigma(1,n) - n -1) ]; // G. C. Greubel, Dec 13 2018

with(numtheory); P:=proc(n) local a,i; for i from 1 to n do
a:=phi(sigma(i)-i-1); if a>0 then
if sigma(phi(i))-phi(i)-1=a then print(i);
fi; fi; od; end: P(10^7);

ch[n_]:=DivisorSigma[1,n]-n-1
test[n_]:=ch[n]!=0 && ch[EulerPhi[n]] == EulerPhi[ch[n]]
Flatten[Position[Range[300000], Integer_ ? test]] (* Jon Maiga, Dec 14 2018 *)

isok(n) = ((s=(sigma(n)-n-1)) != 0) && (sigma(eulerphi(n))-eulerphi(n)-1 == eulerphi(s)); \\ Michel Marcus, Nov 01 2014

a(1) corrected and a(6)-a(7) from Michel Marcus, Nov 01 2014

a(8) from Giovanni Resta, Dec 01 2019

A326074 Numbers n for which A326073(n) is equal to abs(1+A326146(n)).

Original entry on oeis.org

3, 6, 28, 221, 391, 496, 1189, 1421, 1961, 2419, 5429, 7811, 8128, 11659, 15049, 18871, 36581, 44461, 48689, 57721, 80851, 86519, 98431, 107869, 117739, 146171, 169511, 181829, 207761, 235421, 240199, 280151, 312131, 387349, 437669, 497951, 525991, 637981, 685801, 735349, 752249, 804101, 885119, 950821, 1009009

Offset: 1

Antti Karttunen, Jun 10 2019

nonn

Numbers n such that 1+(A001065(n)-A020639(n)) is not zero and divides 1+n-A020639(n).
Note that whenever n is even, then the above condition reduces to "(even) numbers n such that A048050(n) is not zero and divides n-1", which is a condition satisfied only by the even terms of A000396.
a(375) = 360866239 = 449 * 509 * 1579 is the first term with more than two distinct prime factors, the second is a(392) = 413733139 = 199 * 239 * 8699, and the third is a(485) = 718660177 = 41 * 853 * 20549.
Question: Are any of these terms present also in A326064 and A326148? None of the first 564 terms are. If such intersections are empty, then there are no odd perfect numbers.
If one selects only semiprimes from this sequence, one is left with 6, 221, 391, 1189, 1961, 2419, 5429, 7811, 11659, 15049, 18871, 36581, ... (555 terms out of the first 564 terms). Their smaller prime factors are: 2, 13, 17, 29, 37, 41, 61, 73, 89, 101, 113, 157, 173, 181, 197, 233, 241, 257, 269, 281, 313, ... while their larger prime factors are: 3, 17, 23, 41, 53, 59, 89, 107, 131, 149, 167, 233, 257, 269, 293, 347, 359, 383, 401, 419, 467, 503, 521, ..., and both sequences of primes seem to be monotonic.

Cf. A000203, A001065, A020639, A048050, A061228, A325960, A325961, A326064, A326073, A326146, A326148.

Cf. A000396 (a subsequence, the even terms of this sequence if there are no odd perfect numbers).

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A326073(n) = gcd(1+n-A020639(n), 1+sigma(n)-A020639(n)-n);
A326146(n) = (sigma(n)-A020639(n)-n);
isA326074(n) = (A326073(n)==abs(1+A326146(n)));

A062972 Numbers k such that the Chowla function of k is divisible by phi(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269

Offset: 1

Jason Earls, Jul 24 2001

nonn

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

Sequence contains all primes; see A070037 for nonprime terms. - Charles R Greathouse IV, Apr 14 2010

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

Cf. A000010, A048050, A070037.

chowla[1] = 0; chowla[n_] := DivisorSigma[1, n] - n - 1; Select[Range[270], Divisible[chowla[#], EulerPhi[#]] &] (* Amiram Eldar, Dec 01 2019 *)

j=[]; for(n=1,600, if(Mod(sigma(n)-n-1,eulerphi(n)) == 0,j=concat(j,n))); j

A202279 Numbers k such that the sum of digits^3 of k equals Sum_{d|k, 1

Original entry on oeis.org

142, 160, 1375, 6127, 12643, 51703, 86833, 103039, 104647, 112093, 137317, 218269, 261883, 266923, 449881, 505891, 617569, 907873

Offset: 1

Michel Lagneau, Dec 15 2011

The sequence is finite because the restricted sum of divisors of n, for n composite, is at least sqrt(n), while the sum of the cubes of the digits of n is at most 9^3*log_10(n+1). - Giovanni Resta, Oct 05 2018

			160 is in the sequence because 1^3 + 6^3 + 0^3 = 217, and the sum of the divisors 1< d<160 is 2 + 4 + 5 + 8 + 10 + 16 + 20 + 32 + 40 + 80 = 217.

Cf. A070308, A202279, A202147, A202285, A202240.

A055012 := proc(n)
        add(d^3,d=convert(n,base,10)) ;
end proc:
A048050 := proc(n)
        if n > 1 then
        numtheory[sigma](n)-1-n ;
        else
                0;
        end if;
end proc:
isA202279 := proc(n)
        A055012(n) = A048050(n) ;
end proc:
for n from 1 do
        if isA202279(n) then
                printf("%d,\n",n);
        end if;
end do; # R. J. Mathar, Dec 15 2011

Q[n_]:=Module[{a=Total[Rest[Most[Divisors[n]]]]}, a == Total[IntegerDigits[n]^3]]; Select[Range[2, 5*10^7], Q]
Select[Range[1000000],DivisorSigma[1,#]-#-1==Total[IntegerDigits[#]^3]&] (* Harvey P. Dale, Jul 19 2014 *)

{n: A055012(n) = A048050(n)}. - R. J. Mathar, Dec 15 2011

A346869 Sum of all divisors, except the smallest and the largest of every number, of the first n odd numbers.

Original entry on oeis.org

0, 0, 0, 0, 3, 3, 3, 11, 11, 11, 21, 21, 26, 38, 38, 38, 52, 64, 64, 80, 80, 80, 112, 112, 119, 139, 139, 155, 177, 177, 177, 217, 235, 235, 261, 261, 261, 309, 327, 327, 366, 366, 388, 420, 420, 440, 474, 498, 498, 554, 554, 554, 640, 640, 640, 680, 680, 708, 772, 796

Offset: 1

Omar E. Pol, Aug 18 2021

Partial sums of the odd-indexed terms of Chowla's function A048050.

a(n) has a symmetric representation.

Partial sums of A346879.

Cf. A000203, A005408, A008438, A048050, A237593, A245092, A244049, A326123, A346870.

a:= proc(n) option remember; `if`(n=1, 0,
      a(n-1)+numtheory[sigma](2*n-1)-2*n)
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Aug 19 2021

s[1] = 0; s[n_] := DivisorSigma[1, 2*n - 1] - 2*n; Accumulate @ Array[s, 50] (* Amiram Eldar, Aug 19 2021 *)
Accumulate[Join[{0},Table[DivisorSigma[1,n]-n-1,{n,3,151,2}]]] (* Harvey P. Dale, Jul 29 2023 *)

from sympy import divisors
from itertools import accumulate
def A346879(n): return sum(divisors(2*n-1)[1:-1])
def aupton(nn): return list(accumulate(A346879(n) for n in range(1, nn+1)))
print(aupton(60)) # Michael S. Branicky, Aug 19 2021

A346879 Sum of the divisors, except the smallest and the largest, of the n-th odd number.

Original entry on oeis.org

0, 0, 0, 0, 3, 0, 0, 8, 0, 0, 10, 0, 5, 12, 0, 0, 14, 12, 0, 16, 0, 0, 32, 0, 7, 20, 0, 16, 22, 0, 0, 40, 18, 0, 26, 0, 0, 48, 18, 0, 39, 0, 22, 32, 0, 20, 34, 24, 0, 56, 0, 0, 86, 0, 0, 40, 0, 28, 64, 24, 11, 44, 30, 0, 46, 0, 26, 104, 0, 0, 50, 24, 34, 80, 0, 0, 80, 36

Offset: 1

Omar E. Pol, Aug 18 2021

a(n) has a symmetric representation.

			For n = 5 the 5th odd number is 9 and the divisors of 9 are [1, 3, 9] and the sum of the divisors of 9 except the smaller and the largest is 3, so a(5) = 3.
For n = 6 the 6th odd number is 11 and the divisors of 11 are [1, 11] and the sum of the divisors of 11 except the smaller and the largest is 0, so a(6) = 0.

Bisection of A048050.
Partial sums give A346869.
Cf. A000203, A005408, A008438, A237593, A245092, A244049, A326123, A346880.

a[1] = 0; a[n_] := DivisorSigma[1, 2*n - 1] - 2*n; Array[a, 100] (* Amiram Eldar, Aug 19 2021 *)

from sympy import divisors
def a(n): return sum(divisors(2*n-1)[1:-1])
print([a(n) for n in range(1, 79)]) # Michael S. Branicky, Aug 19 2021

a(n) = A048050(2*n-1).

A347154 Sum of all divisors, except the largest of every number, of the first n positive even numbers.

Original entry on oeis.org

1, 4, 10, 17, 25, 41, 51, 66, 87, 109, 123, 159, 175, 203, 245, 276, 296, 351, 373, 423, 477, 517, 543, 619, 662, 708, 774, 838, 870, 978, 1012, 1075, 1153, 1211, 1285, 1408, 1448, 1512, 1602, 1708, 1752, 1892, 1938, 2030, 2174, 2250, 2300, 2456, 2529, 2646, 2760

Offset: 1

Omar E. Pol, Aug 20 2021

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

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

Partial sums of A346878.

Cf. A000203, A005843, A048050, A062731, A237593, A245092, A244049, A326124, A346870, A346877, A346880, A347153.

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

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

from sympy import divisors
from itertools import accumulate
def A346878(n): return sum(divisors(2*n)[:-1])
def aupton(nn): return list(accumulate(A346878(n) for n in range(1, nn+1)))
print(aupton(51)) # Michael S. Branicky, Aug 20 2021

from math import isqrt
def A347154(n): return (t:=isqrt(m:=n>>1))**2*(t+1) - sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))-3*((s:=isqrt(n))**2*(s+1) - sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1)-n*(n+1) # Chai Wah Wu, Nov 02 2023

a(n) = n + A346870(n).

a(n) = (5*Pi^2/24 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, May 15 2023

cp's OEIS Frontend

A054014 Chowla function of n read modulo (the number of divisors of n).

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A054015 a(n) is Chowla function of n read modulo (number of proper divisors of n), a(1) = 0 by convention.

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A070037 Nonprime numbers k such that sigma(k) == k+1 (mod phi(k)).

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A132794 Numbers n such that sigma(phi(n)) -phi(n) -1 = phi(sigma(n) -n -1).

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A326074 Numbers n for which A326073(n) is equal to abs(1+A326146(n)).

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A062972 Numbers k such that the Chowla function of k is divisible by phi(k).

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A202279 Numbers k such that the sum of digits^3 of k equals Sum_{d|k, 1

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A346869 Sum of all divisors, except the smallest and the largest of every number, of the first n odd numbers.