A048050 -id:A048050 - cp's OEIS frontend

A384230 Number of subparts in the central part of the symmetric representation of sigma(n).

1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 0, 2, 1, 0, 3, 0, 2, 0, 0, 0, 2, 1, 0, 0, 2, 0, 4, 0, 1, 0, 0, 2, 3, 0, 0, 0, 2, 0, 4, 0, 0, 4, 0, 0, 2, 1, 1, 0, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 0, 2, 1, 0, 4, 0, 0, 0, 2, 0, 3, 0, 0, 0, 0, 2, 0, 0, 2, 1, 0, 0, 4, 0, 0, 0

Offset: 1

Omar E. Pol, Jun 29 2025

This sequence shares infinitely many terms with A067742 from which first differs at a(18). It also shares with A067742 the positions of zeros and nonzeros.
Observation: consider the 2-dense sublists of divisors of n. At least for the first 88 terms a(n) coincides with the number of odd terms in the central 2-dense sublist of divisors of n. For more information see A384225 and A280940.
See the "Discussion" text file in the first link for more comments.

			See the "Discussion" text file in the first link for the examples.

Omar E. Pol, Discussion of the Sequence A384230.
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).

Cf. A001227 (number of subparts), A071561 (positions of zeros), A071562 (positions of nonzeros), A237270 (parts), A237271, A237593, A279387 (subparts), A280940, A384225, A335574, A338488, A377654.

See the "Discussion" text file in the first link for more cross-references.

a(n) = 0 if and only if A067742(n) = 0.
a(n) >= A067742(n).
(a(n) - A067742(n)) is an even number.

Edited by Omar E. Pol, Aug 24 2025

A062785 a(n) = Chowla's function of n * sigma(n).

Original entry on oeis.org

0, 0, 0, 14, 0, 60, 0, 90, 39, 126, 0, 420, 0, 216, 192, 434, 0, 780, 0, 882, 320, 468, 0, 2100, 155, 630, 480, 1512, 0, 2952, 0, 1890, 672, 1026, 576, 4914, 0, 1260, 896, 4410, 0, 5088, 0, 3276, 2496, 1800, 0, 9300, 399, 3906, 1440, 4410, 0, 7800, 1152, 7560, 1760, 2790, 0, 17976, 0, 3168, 4160, 7874, 1512

Offset: 1

Jason Earls, Jul 18 2001

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000203, A048050.

Cf. A002117, A086463.

a[n_] := Module[{s = DivisorSigma[1, n]}, s*(s - n - 1)]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Apr 01 2024 *)

a(n) = {my(s = sigma(n)); if(n == 1, 0, s*(s-n-1));}

a(n) = A000203(n)*A048050(n). - Michel Marcus, Jun 29 2018

Sum_{k=1..n} a(k) ~ (5*zeta(3)/6 - Pi^2/18) * n^3. - Amiram Eldar, Apr 01 2024

a(1) corrected by Amiram Eldar, Apr 01 2024

A062824 a(n) = Ch(A005117(n)) where Ch(n) is Chowla's function and A005117(n) are the squarefree numbers.

Original entry on oeis.org

0, 0, 0, 0, 5, 0, 7, 0, 0, 9, 8, 0, 0, 10, 13, 0, 15, 0, 41, 0, 14, 19, 12, 0, 21, 16, 0, 53, 0, 25, 0, 20, 0, 16, 22, 31, 0, 0, 33, 18, 77, 0, 26, 73, 0, 0, 39, 18, 89, 0, 43, 0, 22, 45, 32, 0, 20, 34, 49, 24, 0, 0, 113, 0, 86, 55, 0, 0, 105, 40, 0, 125, 28, 61, 24, 63, 44, 0, 46, 121, 0, 26, 69, 0, 149, 0, 50, 73, 24, 34, 75, 0

Offset: 1

Jason Earls, Jul 20 2001

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

Cf. A005117, A048050.

chowla[n_] := DivisorSigma[1, n] - n - 1; chowla[1] = 0; chowla /@ Select[Range[150], SquareFreeQ] (* Amiram Eldar, Mar 10 2024 *)

j=[0]; for(n = 2, 150, if(issquarefree(n), j = concat(j, sigma(n)-n-1))); j

a(n) = A048050(A005117(n)). - Amiram Eldar, Mar 10 2024

a(1) corrected by Amiram Eldar, Mar 10 2024

A074844 Largest difference between consecutive divisors of n is equal to the sum of divisors of n except 1 and n.

Original entry on oeis.org

4, 345, 6489, 88473

Offset: 1

Jason Earls, Sep 10 2002

No other term < 600000. - Emeric Deutsch, Aug 04 2005
No more terms < 10^9. - Lars Blomberg, Jun 04 2013
If p = 5^k - 2 is a prime > 3, then 3*p*(p+2)/5 is in this sequence (see A109080). - Charlie Neder, Oct 13 2018
a(5) > 10^13. - Giovanni Resta, Feb 15 2020

			The divisors of 345 are [1, 3, 5, 15, 23, 69, 115, 345] and the largest difference between consecutive divisors is 345-115 = 230; the sum of divisors except 1 and 345 are 3+5+15+23+69+115 = 230.

Cf. A048050, A060681.

with(numtheory): a:=proc(n) local div: div:=divisors(n): if max(seq(div[j]-div[j-1],j=2..tau(n)))=sigma(n)-1-n then n else fi end: seq(a(n),n=1..100000); # Emeric Deutsch, Aug 04 2005

More terms from Emeric Deutsch, Aug 04 2005

A157195 a(n) = 0 if n is 1 or a prime, otherwise a(n) = product of the proper divisors of n.

Original entry on oeis.org

0, 0, 0, 2, 0, 6, 0, 8, 3, 10, 0, 144, 0, 14, 15, 64, 0, 324, 0, 400, 21, 22, 0, 13824, 5, 26, 27, 784, 0, 27000, 0, 1024, 33, 34, 35, 279936, 0, 38, 39, 64000, 0, 74088, 0, 1936, 2025, 46, 0, 5308416, 7, 2500, 51, 2704, 0, 157464, 55, 175616, 57, 58, 0, 777600000

Offset: 1

Jaroslav Krizek, Feb 24 2009, Feb 27 2009

nonn

a(n) = 0 if and only if n is a noncomposite number (cf. A008578). - Omar E. Pol, Aug 01 2012

			For n = 15 a(15) = 15 = 3*5.

Cf. A007955, A008578, A048050.

If[#==1||PrimeQ[#],0,Times@@Most[Divisors[#]]]&/@Range[60] (* Harvey P. Dale, Jan 24 2014 *)

a(n) = {if ((n == 1) || isprime(n), return (0)); d = divisors(n); prod(i = 2, #d - 1, d[i]);} \\ Michel Marcus, Aug 05 2013

from math import isqrt
from sympy import divisor_count
def A157195(n): return 0 if (c:=divisor_count(n)) <= 2 else (isqrt(n) if (c:=divisor_count(n)) & 1 else 1)*n**(c//2-1) # Chai Wah Wu, Jun 25 2022

a(pq) = pq, p,q = distinct primes. a(p^k) = p^((1/2*k*(k-1)), p = prime, k = integer >=2. a(c) = A007955(c)/c, c = composite number.

Edited by N. J. A. Sloane, Mar 03 2009

Definition clarified by Harvey P. Dale, Jan 24 2014

A163871 The n-th composite plus the sum of its nontrivial divisors.

Original entry on oeis.org

6, 11, 14, 12, 17, 27, 23, 23, 30, 38, 41, 31, 35, 59, 30, 41, 39, 55, 71, 62, 47, 53, 47, 90, 59, 55, 89, 95, 83, 77, 71, 123, 56, 92, 71, 97, 119, 71, 119, 79, 89, 167, 95, 103, 126, 83, 143, 125, 95, 143, 194, 113, 123, 139, 95, 167, 185, 120, 125, 223, 107, 131, 119, 179

Offset: 1

Juri-Stepan Gerasimov, Aug 06 2009

Trivial divisors of a number are 1 and the number itself, see A048050.

			a(1) = 4 + 2 =  6;
a(2) = 6 + 5 = 11;
a(3) = 8 + 6 = 14.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A027750.

A002808 := proc(n) local resul,i ; i := 1 ; resul := 4 ; while i < n do resul := resul+1 ; while isprime(resul) do resul := resul+1 ; od ; i := i+1 ; od; RETURN(resul) ; end:
A048050 := proc(n) if n <= 3 then 0; else numtheory[sigma](n)-n-1 ; fi; end:
A163871 := proc(n) A002808(n)+A048050(A002808(n)) ; end: seq(A163871(n),n=1..80) ; # R. J. Mathar, Aug 11 2009

#+Total[Most[Rest[Divisors[#]]]]&/@Select[Range[4,200],!PrimeQ[#]&] (* Harvey P. Dale, Oct 28 2013 *)

a(n) = A002808(n) + A062825(n+1).

a(4) corrected by R. J. Mathar, Aug 11 2009

A164114 Numbers k such that Chowla(k) + phi(k) is prime.

Original entry on oeis.org

3, 6, 10, 12, 20, 22, 24, 44, 46, 54, 58, 66, 68, 70, 78, 80, 82, 84, 88, 90, 106, 116, 120, 136, 138, 154, 156, 160, 166, 168, 174, 178, 184, 186, 188, 190, 192, 212, 226, 234, 246, 250, 252, 258, 262, 270, 284, 286, 300, 306, 318, 320, 328, 330, 332, 336, 346, 352, 356

Offset: 1

Juri-Stepan Gerasimov, Aug 10 2009

nonn

Indices such that the sum of the nontrivial divisors and of the Euler totient function at that index is prime.

			n=3 is in the sequence because Chowla(3) + phi(3) = 0 + 2 = 2 (a prime);
n=6 is in the sequence because Chowla(6) + phi(5) = 5 + 2 = 7 (a prime).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000010, A000040, A000203, A048050.

[n: n in [1..400] | IsPrime((SumOfDivisors(n)+EulerPhi(n))-n-1)]; // Vincenzo Librandi, Sep 12 2017

A048050 := proc(n) if n = 1 then 0; else numtheory[sigma](n)-n-1 ; fi; end:
A000010 := proc(n) numtheory[phi](n) ; end: isA164114 := proc(n) isprime( A000010(n)+A048050(n)) ; end:
for n from 1 to 400 do if isA164114(n) then printf("%d,",n): fi; od: # R. J. Mathar, Aug 27 2009

f[n_] := Plus @@ Divisors[n] - n - 1; Select[Range[100], PrimeQ[f[#] + EulerPhi[#]] &] (* G. C. Greubel, Sep 11 2017 *)

isok(n) = isprime(sigma(n)+eulerphi(n)-n-1); \\ Michel Marcus, Sep 12 2017

{k: A048050(k)+A000010(k) in A000040}.

34 and 60 removed, 54 inserted by R. J. Mathar, Aug 27 2009

A213675 a(n) = Chowla's function(n) + anti-Chowla's function(n).

Original entry on oeis.org

0, 0, 0, 2, 2, 5, 5, 9, 5, 14, 5, 20, 10, 16, 16, 17, 17, 36, 5, 32, 18, 34, 19, 42, 27, 22, 36, 54, 5, 57, 21, 67, 40, 26, 41, 62, 25, 66, 42, 77, 14, 91, 27, 50, 88, 52, 29, 99, 46, 89, 28, 104, 53, 81, 53, 82, 58, 88, 51, 174, 16, 70, 110, 65, 59, 119, 87, 124, 34, 128

Offset: 1

Juri-Stepan Gerasimov, Mar 04 2013

See A216982 for definition of anti-Chowla's function.

A213675 := proc(n)
    A048050(n)+A216982(n) ;
end proc; # R. J. Mathar, Mar 15 2013

a(n)=if(n<4,0,my(k=valuation(n,2));sigma(n)+sigma(2*n+1)+sigma(2*n-1)+sigma(n>>k)<<(k+1)-3-23*n\/3) \\ Charles R Greathouse IV, Mar 05 2013

a(n) = A048050(n) + A216982(n).

a(16) corrected by Charles R Greathouse IV, Mar 05 2013

A222563 Primes p such that the sum of divisors (excluding 1 and p - 1) of p - 1 and the sum of divisors (excluding 1 and p + 1) of p + 1 are both prime.

Original entry on oeis.org

5, 59, 83, 239, 281, 359, 443, 479, 521, 599, 761, 839, 1163, 1319, 1361, 1583, 1619, 1721, 1787, 1871, 1877, 2003, 2063, 2339, 2927, 2969, 3251, 3371, 3407, 3671, 3767, 3917, 4001, 4013, 4229, 4283, 4397, 4451, 4463, 4649, 4679, 5147, 5261, 6287, 6329, 6659, 6689

Offset: 1

Gerasimov Sergey, Feb 25 2013

nonn

			83 is in the sequence because: it is prime, the sum of divisors (excluding 1 and 82) of 82 is 2 + 41 = 43, which is prime, and the sum of divisors (excluding 1 and 84) of 84 is 2 + 3 + 4 + 6 + 7 + 12 + 14 + 21 + 28 + 42 = 139, which is also prime.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A048050, A085842.

Select[Prime[Range[2,900]],AllTrue[{Total[Most[Rest[Divisors[#-1]]]], Total[ Most[Rest[Divisors[#+1]]]]},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 29 2016 *)

is(n)=isprime(n)&&isprime(sigma(n-1)-n)&&isprime(sigma(n+1)-n-2) \\ Charles R Greathouse IV, Feb 25 2013

Extended and a(4) and a(6) inserted by Charles R Greathouse IV, Feb 25 2013

A247131 Numbers n > 0 such that a record number of composite numbers k have n as the sum of the nontrivial divisors of k.

Original entry on oeis.org

1, 2, 5, 20, 30, 48, 72, 90, 114, 120, 168, 210, 300, 330, 360, 390, 420, 510, 630, 720, 780, 840, 1050, 1260, 1470, 1560, 1680, 1890, 2100, 2310, 2520, 2730, 3150, 3360, 3570, 3990, 4200, 4410, 4620, 5250, 5460, 6090, 6510, 6720, 6930, 7770, 7980, 8190, 9030, 9240, 10710, 10920, 11550, 13020, 13650, 13860, 15540

Offset: 1

Daniel Lignon, Nov 22 2014

nonn

A prime number has no nontrivial divisors so their sum is = 0. That's why we take only composite numbers.

			For 1, there are no numbers.
For 2, there is 1 number: 4.
For 5, there are 2 numbers: 6 and 25.
For 20, there are 3 numbers: 18, 51, 91.

Amiram Eldar, Table of n, a(n) for n = 1..139

Cf. A145899 (similar but with all divisors), A238895 (similar but with proper divisors), A048050 (Chowla's function: sum of nontrivial divisors).

ch[1] = 0; ch[n_] := DivisorSigma[1, n] - n - 1; m = 300; v = Table[0, {m}]; Do[c = ch[k]; If[1 <= c <= m, v[[c]]++], {k, 1, m^2}]; s = {}; vm = -1; Do[If[v[[k]] > vm, vm = v[[k]]; AppendTo[s, k]], {k, 1, m}]; s (* Amiram Eldar, Nov 05 2019 *)

Obviously a(n) = A238895(n)-1.

cp's OEIS Frontend

A384230 Number of subparts in the central part of the symmetric representation of sigma(n).

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A062785 a(n) = Chowla's function of n * sigma(n).

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A062824 a(n) = Ch(A005117(n)) where Ch(n) is Chowla's function and A005117(n) are the squarefree numbers.

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A074844 Largest difference between consecutive divisors of n is equal to the sum of divisors of n except 1 and n.

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A157195 a(n) = 0 if n is 1 or a prime, otherwise a(n) = product of the proper divisors of n.

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A163871 The n-th composite plus the sum of its nontrivial divisors.

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A164114 Numbers k such that Chowla(k) + phi(k) is prime.

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