Franklin T. Adams-Watters - cp's OEIS frontend

A381233 Concatenate the sequences S(k) = [0, 1, -1, ..., k, -k] for k = 0, 1, ...

0, 0, 1, -1, 0, 1, -1, 2, -2, 0, 1, -1, 2, -2, 3, -3, 0, 1, -1, 2, -2, 3, -3, 4, -4, 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7, 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7, 8, -8, 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7, 8, -8, 9, -9

Offset: 0

N. J. A. Sloane, Mar 01 2025 [Suggested by Franklin T. Adams-Watters, Sep 21 2011]

Paolo Xausa, Table of n, a(n) for n = 0..10200 (k = 0..100)

Cf. A196199, A381232.

Flatten[Table[(-1)^j*Floor[j/2], {k, 0, 10}, {j, 2*k + 1}]] (* Paolo Xausa, Mar 01 2025 *)

A381232 Count down from k to -k for k = 0, 1, 2, ... .

Original entry on oeis.org

0, 1, 0, -1, 2, 1, 0, -1, -2, 3, 2, 1, 0, -1, -2, -3, 4, 3, 2, 1, 0, -1, -2, -3, -4, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, 8, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9

Offset: 0

N. J. A. Sloane, Mar 01 2025 [Suggested by Franklin T. Adams-Watters, Sep 21 2011]

Paolo Xausa, Table of n, a(n) for n = 0..10200 (k = 0..100)

Cf. A196199, A381233.

Flatten[Table[Range[k, -k, -1], {k, 0, 10}]] (* Paolo Xausa, Mar 01 2025 *)

from math import isqrt
def A381232(n): return (t:=isqrt(n))*(t+1)-n # Chai Wah Wu, Mar 01 2025

a(n) = -A196199(n) = floor(sqrt(n))*(floor(sqrt(n))+1)-n. - Chai Wah Wu, Mar 01 2025

A337717 Number of connected graphs, where vertices are labeled with positive integers summing to n, and where identically labeled vertices are indistinguishable and cannot be connected with an edge.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 12, 28, 65, 173, 496, 1527, 5092, 18669, 74601, 325206, 1572191, 8487232, 50962240, 343162568, 2627538015, 22853622935, 225118360626, 2539559186827, 33036542404276, 491977100299885, 8394837931641837

Offset: 0

Max Alekseyev following a suggestion from Franklin T. Adams-Watters, Sep 16 2020

nonn

Max Alekseyev, Table of n, a(n) for n = 0..30

Inverse Euler transform of A337716.

A337716 Number of graphs, where vertices are labeled with positive integers summing to n, and where identically labeled vertices are indistinguishable and cannot be connected with an edge.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 35, 77, 179, 440, 1160, 3264, 9950, 33206, 121943, 494011, 2235399, 11391306, 65287199, 422908306, 3130775625, 26490210964, 255257056748, 2825013955541, 36147331371446, 531237157370531, 8965348473026888

Offset: 0

Max Alekseyev following a suggestion from Franklin T. Adams-Watters, Sep 16 2020

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..30 from Max Alekseyev)

Cf. A337717.

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
cross(u, v) = {sum(i=1, #u, sum(j=1, #v, gcd(u[i], v[j])))}
R(n,m,u)={if(n==0, 1, sum(k=if(m==1, n, 0), n\m, my(s=0); forpart(p=k, s+=self()(n-m*k, m-1, concat(u,Vec(p)))*2^cross(p,u)*permcount(p)); s/k!))}
a(n)={R(n,n,[])} \\ Andrew Howroyd, Sep 18 2020

A320917 a(n) = sigma_2(n)*sigma_3(n)/sigma(n).

Original entry on oeis.org

1, 15, 70, 219, 546, 1050, 2150, 3315, 5299, 8190, 13542, 15330, 26690, 32250, 38220, 51491, 79170, 79485, 124166, 119574, 150500, 203130, 268710, 232050, 330771, 400350, 419020, 470850, 684546, 573300, 895622, 811395, 947940, 1187550, 1173900, 1160481, 1826210, 1862490, 1868300, 1809990

Offset: 1

Franklin T. Adams-Watters, Oct 24 2018

Multiplicative, because products and quotients of multiplicative functions are always multiplicative. a(p^k) for fixed k is trivially a rational function of p. The proofs below show that a(n) is always an integer, and hence a(p^k) is a polynomial in p. (Note that a(n^k) is not equal to this polynomial when n is composite.)
Proof from Robert Gerbicz that this is always an integer:
First consider r=sigma_2(p^k)/sigma(p^k). Let x=p^k, then
r=(p^(2*k+2)-1)/(p^2-1)*(p-1)/(p^(k+1)-1)=(p^2*x^2-1)/(p^2-1)*(p-1)/(p*x-1)=(p*x+1)/(p+1)
Case a: k is even, then x==1 mod (p+1), so (p*x+1)==-1+1==0 mod (p+1). So here even r is an integer.
Case b: k is odd, then
  sigma_3(p^k)=(p^3*x^3-1)/(p^3-1), and we must multiple this by r, and here
  p^3*x^3-1==0 mod (p+1)
  p^3-1==-2 mod (p+1)
  This means that if p=2 then sigma_3(p^k) is divisible by (p+1), so we have proved the theorem.
  If p is odd, then (p+1)/2 divides sigma_3(p^k), but we have another factor of 2 in (p*x+1), because p and x is odd.
It appears that a stronger result is also true: sigma_3(p^k) is divisible by (p+1) if k is odd.
Proof from Giovanni Resta that this is always an integer:
We have sigma_2(p^k)*sigma_3(p^k)/sigma(p^k) =
((p^(2+2*k)-1)/(p^2-1) * (p^(3+3*k)-1)/(p^3-1)) / ((p^(1+k)-1)/(p-1)) =
((p^(k+1)+1)*(p^(3*k+3)-1)) / ((p+1)*(p^3-1)),
because p^(2+2*k)-1 is the difference of two squares and we can use x^2-1 = (x+1)*(x-1).
We observe that p^(3*k+3)-1 is always divisible by p^3-1 (it is indeed sigma_3(p^k)).
Now, if k is even, k+1 is odd, so p^(k+1)+1 is divisible by p+1 giving 1-p+p^2-p^3... +p^k as result, and we are done.
If k is odd, k+1 is even and in general p^(k+1)+1 is not divisible by p+1 (just as p^2+1 is not divisible by p+1).
However, if k is odd, then 3*k+3 is even, so p^(3*k+3)-1 beside being divisible by p^3-1 is also divisible by p+1. Since p+1 and p^3-1 have no common factors, then the ratio (p^(3*k+3)-1) / ((p+1)*(p^3-1)) is an integer and we are done.

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

Cf. A000203, A001157, A001158, A013663, A320416.

a[n_] := DivisorSigma[2, n] * DivisorSigma[3, n] / DivisorSigma[1, n]; Array[a, 40] (* Amiram Eldar, Aug 01 2019 *)
(#[[2]]#[[3]])/#[[1]]&/@Table[DivisorSigma[k,n],{n,40},{k,3}] (* Harvey P. Dale, Aug 14 2024 *)

PARI
```
a(n) = sigma(n,2)*sigma(n,3)/sigma(n)
```

a(n) = sigma_2(n)*sigma_3(n)/sigma(n).

Sum_{k=1..n} a(k) ~ c * n^5, where c = (Pi^6*zeta(5)/2700) * Product_{p prime} (1 - 2/p^2 + 2/p^3 - 2/p^4 + 1/p^6) = 0.1662831668... . - Amiram Eldar, Dec 01 2022

A320059 Sum of divisors of n^2 that do not divide n.

Original entry on oeis.org

0, 4, 9, 24, 25, 79, 49, 112, 108, 199, 121, 375, 169, 375, 379, 480, 289, 808, 361, 919, 709, 895, 529, 1591, 750, 1239, 1053, 1711, 841, 2749, 961, 1984, 1681, 2095, 1719, 3660, 1369, 2607, 2323, 3847, 1681, 5091, 1849, 4039, 3673, 3799, 2209, 6519, 2744, 5374

Offset: 1

Franklin T. Adams-Watters, Oct 04 2018

sigma(n^2) is always odd, so this sequence has the opposite parity from sigma(n): even if n is a square or twice a square, odd otherwise.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000203, A065764, A072861, A054785.

[DivisorSigma(1, n^2) - DivisorSigma(1, n): n in [1..70]]; // Vincenzo Librandi, Oct 05 2018

map(n -> numtheory:-sigma(n^2)-numtheory:-sigma(n), [$1..100]); # Robert Israel, Oct 04 2018

Table[DivisorSigma[1, n^2] - DivisorSigma[1, n], {n, 70}] (* Vincenzo Librandi, Oct 05 2018 *)

PARI
```
a(n) = sigma(n^2)-sigma(n)
```

from _future_ import division
from sympy import factorint
def A320059(n):
    c1, c2 = 1, 1
    for p, a in factorint(n).items():
        c1 *= (p**(2*a+1)-1)//(p-1)
        c2 *= (p**(a+1)-1)//(p-1)
    return c1-c2 # Chai Wah Wu, Oct 05 2018

a(n) = sigma(n^2) - sigma(n).
a(n) = A065764(n) - A000203(n).
a(n) = n^2 iff n is prime. - Altug Alkan, Oct 04 2018

A298734 a(n) = n-th term in periodic sequence repeating the divisors of n in decreasing order.

Original entry on oeis.org

1, 1, 3, 4, 5, 3, 7, 1, 1, 5, 11, 1, 13, 7, 3, 16, 17, 1, 19, 10, 21, 11, 23, 1, 25, 13, 3, 4, 29, 3, 31, 16, 33, 17, 5, 1, 37, 19, 3, 1, 41, 21, 43, 22, 9, 23, 47, 3, 49, 25, 3, 4, 53, 3, 5, 1, 57, 29, 59, 1, 61, 31, 9, 64, 65, 33, 67, 34, 69, 5, 71, 1, 73, 37, 15, 4, 77, 3, 79, 1, 81, 41, 83, 1, 85, 43, 3, 1, 89, 10, 7, 46, 93, 47

Offset: 1

Franklin T. Adams-Watters, Jan 25 2018

nonn

			The divisors of 6 are 1, 2, 3, 6, which reversed is 6,3,2,1; repeating that produces the sequence 6, 3, 2, 1, 6, 3, 2, 1, 6, 3, 2, 1, ...; the 6th term in that sequence is 3, so a(6) = 3.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A122377 (n/a(n)), A033950 (indices of 1's).

with(numtheory):
a:= n-> n/(l-> l[1+irem(n-1, nops(l))])(sort([divisors(n)[]])):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 29 2018

Table[PadRight[{},n,Reverse[Divisors[n]]][[-1]],{n,100}] (* Harvey P. Dale, Jul 21 2024 *)

a(n) = my(d=Vecrev(divisors(n))); if (n % #d, d[n % #d], 1); \\ Michel Marcus, Jan 26 2018

A295218 Number of partitions of 2*n-1 into four squares.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 3, 2, 2, 3, 2, 3, 2, 3, 3, 4, 2, 4, 3, 3, 3, 4, 3, 4, 4, 4, 4, 4, 2, 5, 5, 4, 3, 6, 4, 5, 4, 5, 5, 5, 3, 6, 6, 5, 5, 6, 4, 5, 5, 5, 6, 8, 4, 6, 6, 7, 5, 7, 5, 7, 7, 6, 6, 6, 5, 8, 8, 6, 5, 10, 6, 8, 6, 7, 7, 8, 5, 8, 10, 7, 8, 8, 6, 8, 7, 9, 9, 11, 5, 8, 10, 7, 7

Offset: 1

Franklin T. Adams-Watters, Nov 17 2017

nonn

This is a bisection of A002635.
While A002635 contains each positive integer infinitely often, here a number can appear only finitely many times.
By the Jacobi theorem, a(n) >= A000203(n)/48 >= (1+n)/48, which implies the previous comment. - Robert Israel, Nov 21 2017

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Jacobi four square theorem

Cf. A002635, A295159.

N:= 100: # to get a(1)...a(N)
V:= Array(0..2*N-1):
for a from 0 while 4*a^2 <= 2*N-1 do
  for b from a while a^2 + 3*b^2 <= 2*N-1 do
     for c from b while a^2 + b^2 + 2*c^2 <= 2*N-1 do
       for d from c while a^2 + b^2 + c^2 + d^2 <= 2*N-1 do
         t:= a^2 + b^2 + c^2 + d^2;
         V[t]:= V[t]+1
od od od od:
seq(V[2*i-1],i=1..N); # Robert Israel, Nov 21 2017

A289276 Numbers k such that phi(k) (the totient function A000010) is a power of the number of divisors of k (A000005).

Original entry on oeis.org

1, 2, 3, 5, 8, 10, 17, 18, 24, 30, 34, 63, 76, 85, 128, 136, 170, 257, 315, 333, 364, 380, 436, 444, 514, 640, 680, 972, 1285, 1542, 1820, 1824, 1836, 1875, 2142, 2220, 2907, 3285, 3488, 3796, 4369, 4788, 4860

Offset: 1

Franklin T. Adams-Watters, Jun 30 2017

A019434 is a subsequence. - David A. Corneth, Jun 30 2017

Is the frequency of e such that A000005(a(n))^e = A000010(a(n)) finite? - David A. Corneth, Jul 01 2017

Charles R Greathouse IV, Table of n, a(n) for n = 1..5319 (first 526 terms from Antti Karttunen)

Cf. A000005, A000010, A019434, A020488, A032447, A036913, A051281, A068559, A068560, A114063, A286627.

Join[{1},Select[Range[2,5000],IntegerQ[Log[DivisorSigma[0,#],EulerPhi[#]]]&]] (* Harvey P. Dale, Aug 06 2017 *)

ispowerof(n, k)= if(k==1, return(n==1)); while(n>=k, if(n%k!=0, return(0)); n\=k); n==1
isa(n) = ispowerof(eulerphi(n),numdiv(n)) \\ Quick program, fast enough for early values.

is(n) = if(n==1, return(1)); my(f = factor(n); phi = eulerphi(f), ndiv = numdiv(f), e = logint(phi, ndiv)); ndiv^e == phi \\ David A. Corneth, Jun 30 2017, changed per suggestion of Charles R Greathouse IV

isA289276(n)= if(n==1, return(1)); my(phi = eulerphi(n), ndiv = numdiv(n), v = valuation(phi, ndiv)); ndiv^v == phi; \\ (A variant of above program). - Antti Karttunen, Jun 30 2017

list(lim)=my(v=List([1])); forfactored(n=2,lim\1, my(phi = eulerphi(n), ndiv = numdiv(n)); if(ndiv^valuation(phi,ndiv) == phi, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Jul 01 2017

A288574 Total number of distinct primes in all representations of 2*n+1 as a sum of 3 odd primes.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 4, 4, 6, 7, 9, 10, 12, 15, 17, 16, 19, 19, 23, 25, 26, 26, 28, 33, 32, 35, 43, 39, 41, 45, 45, 48, 54, 55, 52, 60, 59, 56, 75, 67, 67, 81, 74, 76, 92, 83, 85, 100, 96, 81, 106, 103, 91, 121, 108, 98, 131, 120, 116, 143, 133, 129, 151, 144, 124, 163

Offset: 0

Franklin T. Adams-Watters and R. J. Mathar, Jun 28 2017

nonn

That is, a representation 2n+1 = p+p+p counts as 1, as p+p+q counts as 2, and p+q+r counts as 3. If each representation is counted once, we simply get A007963.

Indranil Ghosh (first 200 terms), Hugo Pfoertner, Table of n, a(n) for n = 0..10000

A288573 appears to be an erroneous version of this sequence.

Cf. A007963, A054860, A087916.

A288574 := proc(n)
    local a, i, j, k, p, q, r,pqr ;
    a := 0 ;
    for i from 2 do
        p := ithprime(i) ;
        for j from i do
            q := ithprime(j) ;
            for k from j do
                r := ithprime(k) ;
                if p+q+r = 2*n+1 then
                    pqr := {p,q,r} ;
                    a := a+nops(pqr) ;
                elif p+q+r > 2*n+1 then
                    break;
                end if;
            end do:
            if p+2*q > 2*n+1 then
                break;
            end if;
        end do:
        if 3*p > 2*n+1 then
            break;
        end if;
    end do:
    return a;
end proc:
seq(A288574(n),n=0..80) ; # R. J. Mathar, Jun 29 2017

Table[x = 2 n + 1; max = PrimePi[x]; Total[Length /@ Tally /@ DeleteDuplicates[Sort /@ Select[Tuples[Range[2, max], 3], Prime[#[[1]]] + Prime[#[[2]]] + Prime[#[[3]]] == x &]]], {n, 0, 100}] (* Robert Price, Apr 22 2025 *)

a(n)={my(p,q,r,cnt);n=2*n+1;
forprime(p=3,n\3,forprime(q=p,(n-p)\2,
if(isprime(r=n-p-q), cnt+=if(p===q&&p==r,1,if(p==q||q==r,2,3)))));cnt}
\\ Franklin T. Adams-Watters, Jun 28 2017

from sympy import primerange, isprime
def a(n):
    n=2*n + 1
    c=0
    for p in primerange(3, n//3 + 1):
        for q in primerange(p, (n - p)//2 + 1):
            r=n - p - q
            if isprime(r): c+=1 if p==q and p==r else 2 if p==q or q==r else 3
    return c
print([a(n) for n in range(66)]) # Indranil Ghosh, Jun 29 2017

cp's OEIS Frontend

User: Franklin T. Adams-Watters

A381233 Concatenate the sequences S(k) = [0, 1, -1, ..., k, -k] for k = 0, 1, ...

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Mathematica

A381232 Count down from k to -k for k = 0, 1, 2, ... .

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A337717 Number of connected graphs, where vertices are labeled with positive integers summing to n, and where identically labeled vertices are indistinguishable and cannot be connected with an edge.

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A337716 Number of graphs, where vertices are labeled with positive integers summing to n, and where identically labeled vertices are indistinguishable and cannot be connected with an edge.

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Author

Keywords

Links

Crossrefs

Programs

PARI

A320917 a(n) = sigma_2(n)*sigma_3(n)/sigma(n).

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Mathematica

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A320059 Sum of divisors of n^2 that do not divide n.

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A298734 a(n) = n-th term in periodic sequence repeating the divisors of n in decreasing order.

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Maple

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A295218 Number of partitions of 2*n-1 into four squares.

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Maple

A289276 Numbers k such that phi(k) (the totient function A000010) is a power of the number of divisors of k (A000005).