A000203 -id:A000203 - cp's OEIS frontend

A061256 Euler transform of sigma(n), cf. A000203.

1, 1, 4, 8, 21, 39, 92, 170, 360, 667, 1316, 2393, 4541, 8100, 14824, 26071, 46422, 80314, 139978, 238641, 408201, 686799, 1156062, 1920992, 3189144, 5238848, 8589850, 13963467, 22641585, 36447544, 58507590, 93334008, 148449417, 234829969, 370345918

Offset: 0

Vladeta Jovovic, Apr 21 2001

This is also the number of ordered triples of permutations f, g, h in Symm(n) which all commute, divided by n!. This was conjectured by Franklin T. Adams-Watters, Jan 16 2006, and proved by J. R. Britnell in 2012.
According to a message on a blog page by "Allan" (see Secret Blogging Seminar link) it appears that a(n) = number of conjugacy classes of commutative ordered pairs in Symm(n).
John McKay (email to N. J. A. Sloane, Apr 23 2013) observes that A061256 and A006908 coincide for a surprising number of terms, and asks for an explanation. - N. J. A. Sloane, May 19 2013

			1 + x + 4*x^2 + 8*x^3 + 21*x^4 + 39*x^5 + 92*x^6 + 170*x^7 + 360*x^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.
J. R. Britnell, A formal identity involving commuting triples of permutations, arXiv:1203.5079 [math.CO], 2012.
J. R. Britnell, A formal identity involving commuting triples of permutations, Preprint 2012. - _N. J. A. Sloane_, Jun 13 2012
J. R. Britnell, A formal identity involving commuting triples of permutations, Journal of Combinatorial Theory, Series A, Volume 120, Issue 4, May 2013.
E. Marberg, How to compute the Frobenius-Schur indicator of a unipotent character of a finite Coxeter system, arXiv preprint arXiv:1202.1311 [math.RT], 2012. - _N. J. A. Sloane_, Jun 10 2012
Secret Blogging Seminar, A peculiar numerical coincidence.
N. J. A. Sloane, Transforms
Tad White, Counting Free Abelian Actions, arXiv:1304.2830 [math.CO], 2013.

Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), this sequence (m=1), A275585 (m=2), A288391 (m=3), A301542 (m=4), A301543 (m=5), A301544 (m=6), A301545 (m=7), A301546 (m=8), A301547 (m=9).

Cf. A000203, A001001, A001970, A053529, A061255, A061257, A006908, A192065.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*sigma(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 08 2017

nn = 30; b = Table[DivisorSigma[1, n], {n, nn}]; CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Jun 18 2012 *)
nmax = 40; CoefficientList[Series[Product[1/QPochhammer[x^k]^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 29 2015 *)

N=66; x='x+O('x^N); gf=1/prod(j=1,N, eta(x^j)^j); Vec(gf) /* Joerg Arndt, May 03 2008 */

{a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,sigma(m)*x^m/(1-x^m+x*O(x^n))^2/m)),n))} /* Paul D. Hanna, Mar 28 2009 */

a(n) = A072169(n) / n!.
G.f.: Product_{k=1..infinity} (1 - x^k)^(-sigma(k)). a(n)=1/n*Sum_{k=1..n} a(n-k)*b(k), n>1, a(0)=1, b(k)=Sum_{d|k} d*sigma(d), cf. A001001.
G.f.: exp( Sum_{n>=1} sigma(n)*x^n/(1-x^n)^2 /n ). [Paul D. Hanna, Mar 28 2009]
G.f.: exp( Sum_{n>=1} sigma_2(n)*x^n/(1-x^n)/n ). [Vladeta Jovovic, Mar 28 2009]
G.f.: prod(n>=1, E(x^n)^n ) where E(x) = prod(k>=1, 1-x^k). [Joerg Arndt, Apr 12 2013]
a(n) ~ exp((3*Pi)^(2/3) * Zeta(3)^(1/3) * n^(2/3)/2 - Pi^(4/3) * n^(1/3) / (4 * 3^(2/3) * Zeta(3)^(1/3)) - 1/24 - Pi^2/(288*Zeta(3))) * A^(1/2) * Zeta(3)^(11/72) / (2^(11/24) * 3^(47/72) * Pi^(11/72) * n^(47/72)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Mar 23 2018

Entry revised by N. J. A. Sloane, Jun 13 2012

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

Original entry on oeis.org

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

A175254 a(n) = Sum_{k<=n} A000203(k)*(n-k+1), where A000203(m) is the sum of divisors of m.

Original entry on oeis.org

1, 5, 13, 28, 49, 82, 123, 179, 248, 335, 434, 561, 702, 867, 1056, 1276, 1514, 1791, 2088, 2427, 2798, 3205, 3636, 4127, 4649, 5213, 5817, 6477, 7167, 7929, 8723, 9580, 10485, 11444, 12451, 13549, 14685, 15881, 17133, 18475, 19859, 21339, 22863, 24471, 26157

Offset: 1

Jaroslav Krizek, Mar 14 2010

Partial sums of A024916. - Omar E. Pol, Jul 03 2014
a(n) is also the volume of the stepped pyramid with n levels described in A245092. - Omar E. Pol, Aug 12 2015
Also the alternating row sums of A262612. - Omar E. Pol, Nov 23 2015
From Omar E. Pol, Jan 20 2021: (Start)
Convolution of A000203 and A000027.
Convolution of A340793 and the nonzero terms of A000217.
Antidiagonal sums of A319073.
Row sums of A274824. (End)
Row sums of A345272. - Omar E. Pol, Jun 14 2021
Also the alternating row sums of A353690. - Omar E. Pol, Jun 05 2022

			For n = 4: a(4) = sigma(1)*4 + sigma(2)*3 + sigma(3)*2 + sigma(4)*1 = 1*4 + 3*3 + 4*2 + 7*1 = 28.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 2209 terms from Indranil Ghosh)

Cf. A000203, A000217, A006218, A024916, A072481, A237593, A244050, A245092, A262612, A274824, A319073, A340793, A345272, A353690.

Cf. A143128, A353908.

b:= proc(n) option remember; `if`(n<1, [0$2],
      (p-> p+[numtheory[sigma](n), p[1]])(b(n-1)))
    end:
a:= n-> b(n+1)[2]:
seq(a(n), n=1..45);  # Alois P. Heinz, Oct 07 2021

Table[Sum[DivisorSigma[1, k] (n - k + 1), {k, n}], {n, 45}] (* Michael De Vlieger, Nov 24 2015 *)

a(n) = sum(x=1, n, sigma(x)*(n-x+1)) \\ Michel Marcus, Mar 18 2013

from math import isqrt
def A175254(n): return (((s:=isqrt(n))**2*(s+1)*((s+1)*(2*s+1)-6*(n+1))>>1) + sum((q:=n//k)*(-k*(q+1)*(3*k+2*q+1)+3*(n+1)*(2*k+q+1)) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 21 2023

Conjecture: a(n) = Sum_{k=0..n} A006218(n-k). - R. J. Mathar, Oct 17 2012
a(n) = A000330(n) - A072481(n). - Omar E. Pol, Aug 12 2015
a(n) ~ Pi^2*n^3/36. - Vaclav Kotesovec, Sep 25 2016
G.f.: (1/(1 - x)^2)*Sum_{k>=1} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 03 2017
a(n) = Sum_{k=1..n} Sum_{i=1..k} k - (k mod i). - Wesley Ivan Hurt, Sep 13 2017
a(n) = A244050(n)/4. - Omar E. Pol, Jan 22 2021
a(n) = (n+1)*A024916(n) - A143128(n). - Vaclav Kotesovec, May 11 2022

Corrected by Jaroslav Krizek, Mar 17 2010

More terms from Michel Marcus, Mar 18 2013

A062068 a(n) = d(sigma(n)), where d(k) is the number of divisors function (A000005) and sigma(k) is the sum of divisor function (A000203).

Original entry on oeis.org

1, 2, 3, 2, 4, 6, 4, 4, 2, 6, 6, 6, 4, 8, 8, 2, 6, 4, 6, 8, 6, 9, 8, 12, 2, 8, 8, 8, 8, 12, 6, 6, 10, 8, 10, 4, 4, 12, 8, 12, 8, 12, 6, 12, 8, 12, 10, 6, 4, 4, 12, 6, 8, 16, 12, 16, 10, 12, 12, 16, 4, 12, 8, 2, 12, 15, 6, 12, 12, 15, 12, 8, 4, 8, 6, 12, 12, 16, 10, 8, 3, 12, 12, 12, 12, 12

Offset: 1

Amarnath Murthy, Jun 13 2001

			a(5) = d(sigma(5)) = d(6) = 4.

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

Cf. A000005, A000203, A062068, A062069.

Table[DivisorSigma[0, DivisorSigma[1, n]], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)

PARI
```
for(n=1,120,print(numdiv(sigma(n))))
```

{ for (n=1, 1000, write("b062068.txt", n, " ", numdiv(sigma(n))) ) } \\ Harry J. Smith, Jul 31 2009

a(n) = A000005(A000203(n)) = A062069(n) + A076360(n). - Amiram Eldar, Mar 16 2025

Corrected and extended by Jason Earls, Jun 16 2001

A294898 Deficiency minus binary weight: a(n) = A033879(n) - A000120(n) = A005187(n) - A000203(n).

Original entry on oeis.org

0, 0, 0, 0, 2, -2, 3, 0, 3, 0, 7, -6, 9, 1, 2, 0, 14, -5, 15, -4, 7, 5, 18, -14, 16, 7, 10, -3, 24, -16, 25, 0, 16, 12, 19, -21, 33, 13, 18, -12, 37, -15, 38, 1, 8, 16, 41, -30, 38, 4, 26, 3, 48, -16, 33, -11, 30, 22, 53, -52, 55, 23, 16, 0, 44, -14, 63, 8, 39, -7, 66, -53, 69, 31, 22, 9, 54, -16, 73, -28, 38, 35, 78, -59, 58

Offset: 1

Antti Karttunen, Nov 25 2017

sign

"Least deficient numbers" or "almost perfect numbers" are those k for which A033879(k) = 1, or equally, for which a(k) = -A048881(k-1). The only known solutions are powers of 2 (A000079), all present also in A295296. See also A235796 and A378988. - Antti Karttunen, Dec 16 2024

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

Cf. A000120, A000203, A001065, A005187, A011371, A013661, A033879, A048881, A235796, A294896, A294899, A297114 (Möbius transform), A317844 (difference from a(n)), A326133, A326138, A324348 (a(n) applied to Doudna sequence), A379008 (a(n) applied to prime shift array), A378988.

Cf. A295296 (positions of zeros), A295297 (parity of a(n)).

Array[IntegerExponent[(2 #)!, 2] - DivisorSigma[1, #] &, 85] (* Michael De Vlieger, Nov 26 2017 *)

A294898(n) = ((2*n-sigma(n))-hammingweight(n)); \\ Antti Karttunen, Dec 16 2024

(define (A294898 n) (- (A005187 n) (A000203 n)))

a(n) = A005187(n) - A000203(n).
a(n) = A011371(n) - A001065(n).
a(n) = A033879(n) - A000120(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1 - zeta(2)/2 = 0.177532... . - Amiram Eldar, Feb 22 2024

Name edited by Antti Karttunen, Dec 16 2024

A046692 Dirichlet inverse of sigma function (A000203).

Original entry on oeis.org

1, -3, -4, 2, -6, 12, -8, 0, 3, 18, -12, -8, -14, 24, 24, 0, -18, -9, -20, -12, 32, 36, -24, 0, 5, 42, 0, -16, -30, -72, -32, 0, 48, 54, 48, 6, -38, 60, 56, 0, -42, -96, -44, -24, -18, 72, -48, 0, 7, -15, 72, -28, -54, 0, 72, 0, 80, 90, -60, 48, -62, 96, -24, 0, 84, -144, -68, -36, 96, -144, -72, 0, -74, 114, -20, -40, 96, -168

Offset: 1

Andrew R. Feist (arf22540(AT)cmsu2.cmsu.edu)

			a(36) = a(2^2*3^2) = 2*3 = 6.

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 39.
Andrew R. Feist, Fun With the Sigma-Function, unpub.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)
G. P. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408.

Cf. A000203, A053822, A053825, A053826, A178448.

t := 1; a := proc(n,t) local t1,d; t1 := 0; for d from 1 to n do if n mod d = 0 then t1 := t1+d^t*mobius(d)*mobius(n/d); fi; od; t1; end;

a[n_] := (k = 0; Do[If[Mod[n, d] == 0, k = k + d*MoebiusMu[d]*MoebiusMu[n/d]], {d, 1, n}]; k); Table[a[n], {n, 1, 78}](* Jean-François Alcover, Oct 13 2011, after Maple *)
f[p_, e_] := Which[e == 1, -p-1, e == 2, p, e >= 3, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

a(n)=if(n<1,0,direuler(p=2,n,(1-X)*(1-p*X))[n]) /* Ralf Stephan */

seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sigma(n)))} \\ Andrew Howroyd, Aug 05 2018

a(p) = -p-1, a(p^2) = p, a(p^k) = 0 for k > 2.
Dirichlet g.f.: 1/(zeta(s)*zeta(s-1)). - Benedict W. J. Irwin, Jul 10 2018
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} sigma(k)*A(x^k). - Ilya Gutkovskiy, May 11 2019
From Peter Bala, Jan 17 2024: (Start)
a(n) = Sum_{d divides n} d*mu(d)*mu(n/d). See Brown, p. 408.
a(n) = - Sum_{d divides n, d < n} a(d)*sigma_1(n/d).
a(n) = Sum_{d divides n} d*a(d)*J_2(n/d), where the Jordan totient function J_2(n) = A007434(n).
a(n) = Sum_{d divides n} d*A007427(d)*phi(n/d), where A007427 is the Dirichlet inverse of the tau function.
More generally, a(n) = Sum_{d divides n} d*sigma_[r]^(-1)(d)*J_(r+1)(n/d), where sigma_[r]^(-1) denotes the Dirichlet inverse of the function sigma_[r] = Sum_{d divides n} d^r.
a(n) = Sum_{k = 1..n} gcd(k, n)*A007427(gcd(k, n)).
a(n) = Sum_{1 <= j, k <= n} gcd(j, k, n)*a(gcd(j, k, n)). (End)
Sum_{k=1..n} abs(a(k)) ~ 45*n^2/Pi^4. - Vaclav Kotesovec, May 30 2024

Corrected by T. D. Noe, Nov 13 2006

A000385 Convolution of A000203 with itself.

Original entry on oeis.org

1, 6, 17, 38, 70, 116, 185, 258, 384, 490, 686, 826, 1124, 1292, 1705, 1896, 2491, 2670, 3416, 3680, 4602, 4796, 6110, 6178, 7700, 7980, 9684, 9730, 12156, 11920, 14601, 14752, 17514, 17224, 21395, 20406, 24590, 24556, 28920, 27860, 34112, 32186, 38674, 37994, 43980, 42136, 51646, 47772, 56749, 55500, 64316, 60606, 73420, 67956, 80500, 77760, 88860, 83810, 102284, 92690, 108752, 105236, 120777, 112672, 135120, 123046, 145194, 138656, 157512, 146580, 177515, 159396, 185744, 179122

Offset: 1

N. J. A. Sloane

Convolution of A340793 and A024916. - Omar E. Pol, Feb 17 2021

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jacques Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Nicolae Ciprian Bonciocat, Congruences for the Convolution of Divisor sum function, Bull. Greek Math. Soc., Vol 47 (2003), pp. 19-29.
MathOverflow, Searching for a proof for a series identity.
Srinivasa Ramanujan, On certain arithmetical functions, Transactions of the Cambridge Philosophical Society, 22, No.9 (1916), 169- 184 (see Table IV, line 1).
Jacques Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]

Cf. A000203, A024916, A001158, A340793.

Column k=2 of A319083 (shifted).

a000385 n = sum $ zipWith (*) sigmas $ reverse sigmas where
   sigmas = take n a000203_list
-- Reinhard Zumkeller, Sep 20 2011

f:= n -> 5/12*numtheory:-sigma[3](n+1)-(5+6*n)/12*numtheory:-sigma(n+1):
map(f, [$1..100]); # Robert Israel, Sep 17 2018

a[n_] := Sum[DivisorSigma[1, k] DivisorSigma[1, n-k+1], {k, 1, n}];
Array[a, 100] (* Jean-François Alcover, Aug 01 2018 *)

a(n) = sum(k=1, n, sigma(k)*sigma(n-k+1)); \\ Michel Marcus, Nov 10 2016

a(n) = my(f = factor(n+1)); (5 * sigma(f, 3) - (6*n + 5) * sigma(f)) / 12; \\ Amiram Eldar, Jan 04 2025

from sympy import factorint
def A000385(n):
    f = factorint(n+1).items()
    return(5*prod((p**(3*(e+1))-1)//(p**3-1) for p,e in f)-(5+6*n)*prod((p**(e+1)-1)//(p-1) for p, e in f))//12 # Chai Wah Wu, Jul 25 2024

a(n) = Sum_{k=1..n} A000203(k)*A000203(n-k+1).
G.f.: (1/x)*(Sum_{k>=1} k*x^k/(1 - x^k))^2. - Ilya Gutkovskiy, Nov 10 2016
a(5*n+1)==0 (mod 5) and a(7*n+6)==0 (mod 7). See Bonciocat link. - Michel Marcus, Nov 10 2016
a(n) = (5/12)*A001158(n+1) - ((5+6*n)/12)*A000203(n+1). - Robert Israel, Sep 17 2018
Sum_{k=1..n} a(k) ~ Pi^4 * n^4 / 864. - Vaclav Kotesovec, Apr 02 2019

More terms from Sean A. Irvine, Nov 14 2010

A054640 a(n) is the sum of the divisors of the n-th primorial: a(n) = A000203(A002110(n)).

Original entry on oeis.org

1, 3, 12, 72, 576, 6912, 96768, 1741824, 34836480, 836075520, 25082265600, 802632499200, 30500034969600, 1281001468723200, 56364064623820800, 2705475101943398400, 146095655504943513600, 8765739330296610816000, 543475838478389870592000, 36956357016530511200256000

Offset: 0

Labos Elemer, May 15 2000

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Rafael Jakimczuk, Two Topics in Number Theory: Sum of Divisors of the Primorial and Sum of Squarefree Parts, International Mathematical Forum, Vol. 12, No. 7 (2017), pp. 331-338.

Cf. A000203, A002110, A005867, A008864, A028815, A051674, A054641, A070826, A073004, A074107.

[1/2*&*[(1+NthPrime(k)): k in [0..n-1]]: n in [1..19]]; // Vincenzo Librandi, May 08 2017

a:= n-> mul(1+ithprime(j), j=1..n): seq(a(n), n=0..20); # Zerinvary Lajos, Aug 24 2008

Table[Product[1 + Prime[i], {i,n-1}], {n,100}] (* Geoffrey Critzer, Dec 01 2014 *)

a(n)=prod(i=1,n,prime(i)+1) \\ Charles R Greathouse IV, Feb 13 2013

def A054640(n): return product(nth_prime(j)+1 for j in range(1,n+1))
[A054640(n) for n in range(41)] # G. C. Greubel, Aug 05 2024

a(n+1) = a(n)*(prime(n) + 1) = a(n)*A028815(n) (quotient=n-th prime+1 starting with 2).
a(n) ~ (6/Pi^2) * exp(gamma) * A002110(n) * log(prime(n)) + O(A002110(n)) (Jakimczuk, 2017). - Amiram Eldar, Feb 17 2021
a(n) = a(n-1) * A008864(n). - Flávio V. Fernandes, Mar 20 2021
a(n) = A002110(n) + A074107(n), a(n) <= A070826(1+n) [= A002110(1+n)/2] < A051674(n). - Antti Karttunen, Nov 19 2024

a(0)=1 prepended by Alois P. Heinz, Apr 01 2021

A324054 a(n) = A000203(A005940(1+n)).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 13, 15, 8, 18, 24, 28, 31, 39, 40, 31, 12, 24, 32, 42, 48, 72, 78, 60, 57, 93, 124, 91, 156, 120, 121, 63, 14, 36, 48, 56, 72, 96, 104, 90, 96, 144, 192, 168, 248, 234, 240, 124, 133, 171, 228, 217, 342, 372, 403, 195, 400, 468, 624, 280, 781, 363, 364, 127, 18, 42, 56, 84, 84, 144, 156, 120, 112, 216, 288, 224

Offset: 0

Antti Karttunen, Feb 14 2019

nonn

As noted by David A. Corneth, the function f(n) = a(n-1) [that is, the offset-1 version of this sequence] seems to be "almost multiplicative". Sequence A324109 gives the positions n where f(n) satisfies the multiplicativity in a sense that f(n) = f(p(1)^e(1)) * ... * f(p(k)^e(k)), when n = p(1)^e(1) * ... * p(k)^e(k), and A324110 the positions where this does not hold.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..70327
Index entries for sequences related to binary expansion of n
Michael De Vlieger, Annotated fan-style binary tree diagram showing 16 levels, where blue represents A336834(n) = 1 and yellow A336834(n) = 0. Labels are a(n) for the 8 smallest levels.

Cf. A000203, A005940, A038712, A324055, A324056, A324057, A324058, A324108, A324109, A324110, A324111, A324112, A356321.

Cf. also A106737, A290077 (tau and phi similarly permuted).

nn = 76; a[0] = 1; Do[Set[a[n], Prime[1 + DigitCount[n, 2, 0]]*a[n - 2^Floor@ Log2@ n]], {n, nn}]; Array[DivisorSigma[1, a[#]] &, nn, 0] (* Michael De Vlieger, Aug 03 2022 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A324054(n) = sigma(A005940(1+n));

A324054(n) = { my(p=2,mp=p*p,m=1); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, if(3==(n%4),mp *= p,m *= (mp-1)/(p-1))); n>>=1); (m); };

from math import prod
from itertools import accumulate
from collections import Counter
from sympy import prime
def A324054(n): return prod(((p:=prime(len(a)+1))**(b+1)-1)//(p-1) for a, b in Counter(accumulate(bin(n)[2:].split('1')[:0:-1])).items()) # Chai Wah Wu, Mar 10 2023

a(n) = A000203(A005940(1+n)).

a(n) = A324056(n) * A038712(1+n).

A062069 a(n) = sigma(d(n)), where d(k) is the number of divisors function (A000005) and sigma(k) is the sum of divisors function (A000203).

Original entry on oeis.org

1, 3, 3, 4, 3, 7, 3, 7, 4, 7, 3, 12, 3, 7, 7, 6, 3, 12, 3, 12, 7, 7, 3, 15, 4, 7, 7, 12, 3, 15, 3, 12, 7, 7, 7, 13, 3, 7, 7, 15, 3, 15, 3, 12, 12, 7, 3, 18, 4, 12, 7, 12, 3, 15, 7, 15, 7, 7, 3, 28, 3, 7, 12, 8, 7, 15, 3, 12, 7, 15, 3, 28, 3, 7, 12, 12, 7, 15, 3, 18, 6, 7, 3, 28, 7, 7, 7, 15, 3

Offset: 1

Amarnath Murthy, Jun 13 2001

nonn

a(1) = 1, a(p) = 3 for p = primes (A000040), a(pq) = 7 for pq = product of two distinct primes (A006881), a(pq...z) = 2^(k+1)-1 = A000225(k+1) for pq...z = product of k (k > 2) distinct primes p,q,...,z (A120944), a(p^k) = sigma(k+1) = A000203(k+1) for p^k = prime powers (A000961(n) for n > 1). Sequence {1,3,4,12} is finite sequence of numbers n such that sigma(tau(n)) = n. [Jaroslav Krizek, Jul 16 2009]
For semiprime n, a(n) is either 4 or 7. Also a(n) = d(n) + omega(n) + mu(n), the sum of three core sequences A000005, A001221 and A008683. When n is semiprime, a(n) is completely defined by the Mobius function as: a(n) = 4 + 3*mu(n). a(n) also has the fractal-like identities a(d(n)) = d(n) and a(n) = sigma(a(d(n))). - Wesley Ivan Hurt, Sep 02 2013
If n is a triprime (A014612), d(n) is 4, 6, or 8 and a(n) = sigma(d(n)) is 7, 12, or 15 respectively. Then a(n) = -d(n)^2/4 + 5*d(n) - 9. - Wesley Ivan Hurt, Sep 08 2013

			sigma(d(12)) = sigma(6) = 12.

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

Cf. A000005, A000203, A001221, A008683, A062068.

A062069:= (n-> numtheory[sigma](numtheory[tau](n))):
seq (A062069(n), n=1..40); # Jani Melik, Jan 25 2011

Table[DivisorSigma[1, DivisorSigma[0, n]], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)

v=[]; for(n=1,150,v=concat(v, sigma(numdiv(n)))); v

{ for (n=1, 1000, write("b062069.txt", n, " ", sigma(numdiv(n))) ) } \\ Harry J. Smith, Jul 31 2009

a(n) = A000203(A000005(n)). - Wesley Ivan Hurt, Sep 09 2013

More terms from Jason Earls, Jun 19 2001

cp's OEIS Frontend

A061256 Euler transform of sigma(n), cf. A000203.

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

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A175254 a(n) = Sum_{k<=n} A000203(k)*(n-k+1), where A000203(m) is the sum of divisors of m.

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A062068 a(n) = d(sigma(n)), where d(k) is the number of divisors function (A000005) and sigma(k) is the sum of divisor function (A000203).

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A294898 Deficiency minus binary weight: a(n) = A033879(n) - A000120(n) = A005187(n) - A000203(n).

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A046692 Dirichlet inverse of sigma function (A000203).

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