A087943 -id:A087943 - cp's OEIS frontend

A379473 a(n) is the highest power of 3 dividing the sum of divisors of n.

1, 3, 1, 1, 3, 3, 1, 3, 1, 9, 3, 1, 1, 3, 3, 1, 9, 3, 1, 3, 1, 9, 3, 3, 1, 3, 1, 1, 3, 9, 1, 9, 3, 27, 3, 1, 1, 3, 1, 9, 3, 3, 1, 3, 3, 9, 3, 1, 3, 3, 9, 1, 27, 3, 9, 3, 1, 9, 3, 3, 1, 3, 1, 1, 3, 9, 1, 9, 3, 9, 9, 3, 1, 3, 1, 1, 3, 3, 1, 3, 1, 9, 3, 1, 27, 3, 3, 9, 9, 9, 1, 3, 1, 9, 3, 9, 1, 9, 3, 1, 3, 27, 1, 3, 3

Offset: 1

Antti Karttunen, Dec 27 2024

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n).

Cf. A000203, A000244, A038500, A354100, A379484 [= a(A379481(n))].
Cf. A329963 (positions of 1's), A087943 (of terms > 1).
Cf. also A082903.

a[n_] := 3^IntegerExponent[DivisorSigma[1, n], 3]; Array[a, 100] (* Amiram Eldar, Dec 27 2024 *)

PARI
```
A379473(n) = (3^valuation(sigma(n),3));
```

Multiplicative with a(p^e) = A038500((p^(e+1)-1)/(p-1)).
a(n) = A038500(A000203(n)).
a(n) = A000244(A354100(n)).

A382123 a(n) = sigma(n)sigma(2n)/3 for n >= 1.

Original entry on oeis.org

1, 7, 16, 35, 36, 112, 64, 155, 169, 252, 144, 560, 196, 448, 576, 651, 324, 1183, 400, 1260, 1024, 1008, 576, 2480, 961, 1372, 1600, 2240, 900, 4032, 1024, 2667, 2304, 2268, 2304, 5915, 1444, 2800, 3136, 5580, 1764, 7168, 1936, 5040, 6084, 4032, 2304, 10416, 3249, 6727, 5184, 6860

Offset: 1

Paul D. Hanna, Apr 06 2025

nonn

For n >= 1, 2*A329963(n) = A087943(k) for some k; this is a consequence of the prime factorization properties of the numbers listed in A329963 and A087943 (see the comments in both entries). That is, two times any term found in A329963 (numbers k such that sigma(k) is not divisible by 3) equals a term found in A087943 (numbers k such that 3 divides sigma(k)). Therefore sigma(n)*sigma(2*n) is divisible by 3 for n >= 1.

Equals the logarithmic derivative of A382124.

Cf. A382124, A382125, A087943, A329963, A347108.

Cf. A000203, A062731.

{a(n) = sigma(n)*sigma(2*n)/3}
for(n=1,52, print1(a(n),", "))

a(n) = A000203(n) * A062731(n) / 3.

Sum_{k=1..n} a(k) ~ 2*zeta(3)*n^3/3. - Vaclav Kotesovec, Apr 06 2025

A382124 G.f. A(x) = exp( Sum_{n>=1} sigma(n)sigma(2n)/3 * x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

Original entry on oeis.org

1, 1, 4, 9, 22, 44, 105, 200, 425, 825, 1634, 3072, 5917, 10846, 20153, 36436, 65882, 116831, 207293, 361502, 629539, 1083068, 1856251, 3150554, 5328137, 8933266, 14920357, 24745481, 40869317, 67089425, 109697089, 178379353, 288953043, 465805681, 748079686, 1196148976, 1905801579, 3024212984

Offset: 0

Paul D. Hanna, Apr 06 2025

nonn

Compare with g.f. for partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.
Equals the self-convolution cube root of A382125.
Conjecture: a(n) == A382125(3*n) (mod 3) for n >= 0.

			G.f.: A(x) = 1 + x + 4*x^2 + 9*x^3 + 22*x^4 + 44*x^5 + 105*x^6 + 200*x^7 + 425*x^8 + 825*x^9 + 1634*x^10 + 3072*x^11 + 5917*x^12 + ...
RELATED SERIES.
A(x)^3 = 1 + 3*x + 15*x^2 + 52*x^3 + 180*x^4 + 555*x^5 + 1696*x^6 + 4809*x^7 + 13410*x^8 + ... + A382125(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A382125, A382123, A156302, A347108, A000203 (sigma), A000041 (partitions).

Cf. A087943, A329963.

nmax=37; CoefficientList[Series[Exp[Sum[DivisorSigma[1,n]DivisorSigma[1,2*n] * x^n/(3n) ,{n,nmax}]],{x,0,nmax}],x] (* Stefano Spezia, Apr 06 2025 *)

{a(n) = my(A = exp( sum(m=1,n, sigma(m)*sigma(2*m)/3*x^m/m ) +x*O(x^n) ));
polcoef(A,n)}
for(n=0,30, print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = exp( (1/3) * Sum_{n>=1} sigma(n)*sigma(2*n) * x^n/n ).
(2) A(x) = exp( (1/3) * Sum_{n>=1} (1/n) * Sum_{k>=1} sigma(2*n*k) * x^(n*k) ).
(3) a(n) = (1/n) * Sum_{k=1..n} sigma(k)*sigma(2*k)/3 * a(n-k) for n > 0, with a(0) = 1.

A088535 n representable as x^2+xy+y^2 and such that 3 divides sigma(n).

Original entry on oeis.org

49, 147, 169, 196, 361, 441, 507, 588, 637, 676, 784, 931, 961, 1083, 1183, 1225, 1323, 1369, 1444, 1519, 1521, 1764, 1813, 1849, 1911, 2028, 2107, 2352, 2527, 2548, 2704, 2793, 2883, 2989, 3136, 3211, 3249, 3283, 3549, 3577, 3675, 3721, 3724, 3844, 3871

Offset: 1

Benoit Cloitre, Nov 16 2003

nonn

n in A003136 and in A087943

More terms from David Wasserman, Aug 09 2005

A274685 Odd numbers n such that sigma(n) is divisible by 5.

Original entry on oeis.org

19, 27, 29, 57, 59, 79, 87, 89, 95, 109, 133, 135, 139, 145, 149, 171, 177, 179, 189, 199, 203, 209, 229, 237, 239, 247, 261, 267, 269, 285, 295, 297, 319, 323, 327, 343, 349, 351, 359, 377, 379, 389, 395, 399, 409, 413, 417, 419, 435, 437, 439, 445, 447, 449, 459, 475, 479, 493, 499

Offset: 1

M. F. Hasler, Jul 02 2016

nonn

The subsequence of odd terms in A274397.
If n is in the sequence and gcd(n,m)=1 for some odd m, then n*m is also in the sequence. One might call "primitive" those terms which are not of this form, i.e., not a "coprime" multiple of an earlier term. The list of these primitive terms is (19, 27, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 343, 349, 359, 379, 389, 409, 419, 439, 449, 479, 499, ...). The primitive terms are the primes and powers of primes within the sequence. If a prime power p^k (k >= 1) is in the sequence, then p^(m(k+1)-1) is in the sequence for any m >= 1, since 1+p+...+p^(m(k+1)-1) = (1+p+...+p^k)(1+p^(k+1)+...+p^((m-1)*(k+1))). For example, with the prime p=19 we also have all odd powers 19^3, 19^5, ..., and with 27 = 3^3, we also have 27^5, 27^9, ... in the sequence.
On the other hand, for any prime p <> 5 there is an exponent k in {1, 3, 4} such that p^k is in this sequence (and therewith all higher powers of the form given above).
One may notice that there are many pairs of the form (30k-3, 30k-1), e.g., 27,29; 57,59; 87,89; 177,179; 237,239; 295,299; ... Indeed, it is likely that 30k-1 is prime and in this case, if 10k-1 is also prime, then sigma(30k-3) = 40k is divisible by 5 and sigma(30k-1) = 30k is also divisible by 5.

			Some values of a(2^k): a(2) = 27, a(4) = 57, a(8) = 89, a(16) = 171, a(32) = 297, a(64) = 545, a(128) = 1029, a(256) = 1937, a(512) = 3625, a(1024) = 6939, a(2048) = 13257, a(4096) = 25483, a(8192) = 49319, a(16384) = 95695, a(32768) = 185991, a(65536) = 362725, a(131072) = 708887, a(262144) = 1388367, a(524288) = 2722639, a(1048576) = 5346681, a(2097152) = 10514679, a(4194304) = 20698531, a(8388608) = 40790203.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000203 (sigma), A028983 (sigma even), A087943 (sigma = 3k), A248150 (sigma = 4k); A028982 (sigma is odd), A248151 (sigma is not divisible by 4); A272930(sigma(sigma(k)) = nk).

Select[Range[1, 500, 2], Divisible[DivisorSigma[1, #], 5] &] (* Michael De Vlieger, Jul 16 2016 *)

is_A274685(n)=sigma(n)%5==0&&bittest(n,0)

forstep(n=1,999,2,sigma(n)%5||print1(n","))

a(n) ~ 2n. - Charles R Greathouse IV, Jul 16 2016

A374223 Numbers k such that sigma(k) and sopfr(k) are both multiples of 3, where sigma is the sum of divisors, and sopfr is the sum of prime factors with repetition.

Original entry on oeis.org

8, 14, 20, 24, 26, 35, 38, 42, 44, 50, 60, 62, 65, 68, 72, 74, 77, 78, 86, 92, 95, 105, 110, 114, 116, 119, 122, 125, 126, 132, 134, 143, 146, 150, 155, 158, 160, 161, 164, 170, 180, 185, 186, 188, 194, 195, 196, 203, 204, 206, 209, 212, 215, 216, 218, 221, 222, 230, 231, 234, 236, 242, 254, 258, 275, 276, 278, 280

Offset: 1

Antti Karttunen, Jul 08 2024

nonn

Numbers such that the multiplicities of prime factors of the forms 3m+1 (A002476) and 3m-1 (A003627) are equal modulo 3, and some of the former has exponent of the form 3k+2, or some of the latter has odd exponent.
If m and n are in the sequence and gcd(m,n)=1, then m*n is also in sequence.
Term k is included <=> term 3*k is included.

Antti Karttunen, Table of n, a(n) for n = 1..12000

Cf. A000203, A001414, A002476, A003627, A374222 (characteristic function).
Indices of multiples of 3 in A374126.
Intersection of A087943 and A289142.

PARI
```
isA374223 = A374222;
```

A274062 Even numbers such that the sum of the odd divisors is a Fibonacci number F and the sum of the even divisors is 2F.

Original entry on oeis.org

2, 14, 18, 230, 238, 4958, 53430, 57930, 64506, 65586, 68226, 70730, 77270, 78638, 81926, 84986, 88826, 90446, 91306, 1006350, 1248054, 1341950, 18177726, 19033854, 19603430, 21044030, 22356798, 22395522, 22876730, 23954170, 24241966, 24840710, 24883910, 25285666, 25306246

Offset: 1

Michel Lagneau, Jun 09 2016

nonn

a(n) == 2,6,10 (mod 12) i.e. a(n) == 2 (mod 4) so this sequence is a subsequence of A016825 (of which 3|sigma(A016825(n))).
The corresponding Fibonacci numbers F are 1, 8, 13, 144, 144, 2584, 46368, 46368, 46368, 46368,... with index 1 (or 2), 6, 7, 12, 12, 18, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 30, 30, 30.
The sequence is generalizable with the following definition: even numbers such that the sum of the odd divisors is a Fibonacci number F and the sum of the even divisors is (2^k -2)*F = A000918(k)*F with k>1. The corresponding sequences b(n,k) are of the form b(n,k) = a(n)*2^(k-2) where a(n) is the primitive sequence.

			18 is in the sequence because: its divisors are {1, 2, 3, 6, 9, 18}; the sum of its odd divisors is 1 + 3 + 9 = 13, a Fibonacci number, and the sum of its even divisors is 2 + 6 + 18 = 26 = 2*13.

Cf. A000045, A000918, A016825, A087943.

with(numtheory):
for n from 2 by 2  to 10^7 do:
   y:=divisors(n):n1:=nops(y):s0:=0:s1:=0:
     for k from 1 to n1 do:
       if irem(y[k], 2)=0
        then
        s0:=s0+ y[k]:
        else
        s1:=s1+ y[k]:
      fi:
     od:
     if s0=2*s1
      then
      ii:=0:
        x:=sqrt(5*s1^2+4):y:=sqrt(5*s1^2-4):
         if x=floor(x) or y=floor(y)
          then
          printf ( "%d %d \n",n,s1):
           else
          fi:
        fi:
     od:

t = Fibonacci@ Range@ 40; Select[Range[2, 2*10^6, 4], Function[d, And[Total@ Select[d, EvenQ] == 2 #, MemberQ[t, #]] &@ Total@ Select[d, OddQ]]@ Divisors@ # &] (* Michael De Vlieger, Jun 09 2016 *)

isok(n) = sod = sumdiv(n, d, d*(d % 2)); (2*sod == sumdiv(n, d, d*(1-(d % 2)))) && (issquare(5*sod^2-4) || issquare(5*sod^2+4)); \\ Michel Marcus, Jun 09 2016

a(23)-a(35) from Michel Marcus, Jun 14 2016

cp's OEIS Frontend

A379473 a(n) is the highest power of 3 dividing the sum of divisors of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A382123 a(n) = sigma(n)*sigma(2*n)/3 for n >= 1.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

A382124 G.f. A(x) = exp( Sum_{n>=1} sigma(n)*sigma(2*n)/3 * x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A088535 n representable as x^2+xy+y^2 and such that 3 divides sigma(n).

Views

Author

Keywords

Formula

Extensions

A274685 Odd numbers n such that sigma(n) is divisible by 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A374223 Numbers k such that sigma(k) and sopfr(k) are both multiples of 3, where sigma is the sum of divisors, and sopfr is the sum of prime factors with repetition.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A274062 Even numbers such that the sum of the odd divisors is a Fibonacci number F and the sum of the even divisors is 2F.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A382123 a(n) = sigma(n)sigma(2n)/3 for n >= 1.

A382124 G.f. A(x) = exp( Sum_{n>=1} sigma(n)sigma(2n)/3 * x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.