A347960 -id:A347960 - cp's OEIS frontend

A327564 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)^(k_j - 1)).

1, 1, 1, 3, 1, 1, 1, 9, 4, 1, 1, 3, 1, 1, 1, 27, 1, 4, 1, 3, 1, 1, 1, 9, 6, 1, 16, 3, 1, 1, 1, 81, 1, 1, 1, 12, 1, 1, 1, 9, 1, 1, 1, 3, 4, 1, 1, 27, 8, 6, 1, 3, 1, 16, 1, 9, 1, 1, 1, 3, 1, 1, 4, 243, 1, 1, 1, 3, 1, 1, 1, 36, 1, 1, 6, 3, 1, 1, 1, 27, 64, 1, 1, 3, 1

Offset: 1

Ilya Gutkovskiy, Mar 03 2020

			a(12) = a(2^2 * 3) = (2 + 1)^(2 - 1) * (3 + 1)^(1 - 1) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences computed from indices in prime factorization.

Cf. A001221, A003557, A003958, A003959, A003968, A005117 (positions of 1's), A007947, A048250, A064478, A064549, A173557, A323363, A325126, A326297, A340368, A348038, A348039, A347960.

a[n_] := Times @@ ((#[[1]] + 1)^(#[[2]] - 1) & /@ FactorInteger[n]); Table[a[n], {n, 1, 85}]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1]++; f[k,2]--); factorback(f); \\ Michel Marcus, Mar 03 2020

a(1) = 1; a(n) = -Sum_{d|n, dA001221(n/d) * A003557(n/d) * a(d).
a(n) = A003959(n) / A048250(n) = A003968(n) / A007947(n).
a(n) = A348038(n) * A348039(n) = A340368(n) / A173557(n). - Antti Karttunen, Oct 29 2021
Dirichlet g.f.: 1/(zeta(s-1) * Product_{p prime} (1 - 1/p^(s-1) - 1/p^s)). - Amiram Eldar, Dec 07 2023

A348036 a(n) = gcd(n, A003968(n)), where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 36, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 2, 65, 66, 67, 34, 69, 70, 71, 72, 73, 74, 15, 38, 77, 78, 79, 10, 3, 82, 83, 42

Offset: 1

Antti Karttunen, Oct 19 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Differs from A007947 at the positions given by A347960.

Cf. A003959, A003968, A348030, A348037, A348038, A348039.

f[p_, e_] := p*(p + 1)^(e - 1); a[n_] := GCD[n, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Oct 20 2021 *)

A003968(n) = { my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
A348036(n) = gcd(n, A003968(n));

a(n) = gcd(n, A003968(n)).
a(n) = gcd(n, A348030(n)) = gcd(A003968(n), A348030(n)).
a(n) = n / A348037(n) = A003968(n) / A348038(n).
a(n) = A007947(n) * A348039(n).

A348037 a(n) = n / gcd(n, A003968(n)), where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1, 8, 1, 3, 1, 2, 1, 1, 1, 4, 5, 1, 9, 2, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 8, 7, 5, 1, 2, 1, 9, 1, 4, 1, 1, 1, 2, 1, 1, 3, 32, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 5, 2, 1, 1, 1, 8, 27, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 16, 1, 7, 3, 5, 1, 1, 1, 4, 1

Offset: 1

Antti Karttunen, Oct 19 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Differs from A003557 at the positions given by A347960.

Cf. A003959, A003968, A333634, A348038, A348039, A348499 (positions of 1's).

f[p_, e_] := p*(p + 1)^(e - 1); a[n_] := n / GCD[n, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Oct 20 2021 *)

A003968(n) = { my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
A348037(n) = (n/gcd(n, A003968(n)));

a(n) = n / A348036(n) = n / gcd(n, A003968(n)).

a(n) = A003557(n) / A348039(n).

A348038 a(n) = A003968(n) / gcd(n, A003968(n)), where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 9, 4, 1, 1, 3, 1, 1, 1, 27, 1, 4, 1, 3, 1, 1, 1, 9, 6, 1, 16, 3, 1, 1, 1, 81, 1, 1, 1, 2, 1, 1, 1, 9, 1, 1, 1, 3, 4, 1, 1, 27, 8, 6, 1, 3, 1, 16, 1, 9, 1, 1, 1, 3, 1, 1, 4, 243, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 6, 3, 1, 1, 1, 27, 64, 1, 1, 3, 1, 1, 1, 9, 1, 4, 1, 3, 1, 1, 1, 81, 1, 8, 4, 9, 1

Offset: 1

Antti Karttunen, Oct 20 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Differs from A327564 at the positions given by A347960.

Cf. A003968, A005117 (positions of 1's), A327564, A348036, A348037, A348039.

f[p_, e_] := p*(p + 1)^(e - 1); a[n_] := (m = Times @@ f @@@ FactorInteger[n]) / GCD[n, m]; Array[a, 100] (* Amiram Eldar, Oct 20 2021 *)

A003968(n) = {my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
A348038(n) = { my(u=A003968(n)); (u/gcd(n, u)); };

a(n) = A003968(n) / A348036(n) = A003968(n) / gcd(n, A003968(n)).

a(n) = A327564(n) / A348039(n).

A348039 a(n) = gcd(A003557(n), A327564(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6

Offset: 1

Antti Karttunen, Oct 19 2021

There is interesting regularity in the scatter plot.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003557, A003959, A003968, A007947, A327564, A348036, A348037, A348038.

Cf. A347960 (positions of terms > 1).

{1}~Join~Array[GCD @@ Map[Times @@ # &, Transpose@ Map[{#1^(#2 - 1), (#1 + 1)^(#2 - 1)} & @@ # &, FactorInteger[#]]] &, 105, 2] (* Michael De Vlieger, Oct 20 2021 *)

A003968(n) = {my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
A007947(n) = factorback(factorint(n)[, 1]);
A348036(n) = gcd(n, A003968(n));
A348039(n) = (A348036(n)/A007947(n));

A003557(n) = (n/factorback(factorint(n)[, 1]));
A327564(n) = { my(f=factor(n)); for(k=1, #f~, f[k, 1]++; f[k, 2]--); factorback(f); }; \\ From A327564
A348039(n) = gcd(A003557(n), A327564(n));

a(n) = gcd(A003557(n), A327564(n)).
a(n) = A348036(n) / A007947(n).
a(n) = A003557(n) / A348037(n).
a(n) = A327564(n) / A348038(n).

A348499 Numbers k such that A003968(k) is a multiple of k, where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 72, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119

Offset: 1

Antti Karttunen, Oct 29 2021

nonn

This is a subsequence of A333634. See comments in A347892.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Union of A005117 and A347892 (terms that are not squarefree).
Subsequence of A333634.
Positions of ones in A348037.
Cf. A003968, A348030, A348036, A347960.

f[p_, e_] := p*(p + 1)^(e - 1); s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[120], Divisible[s[#], #] &] (* Amiram Eldar, Oct 29 2021 *)

A003968(n) = {my(f=factor(n)); for (i=1, #f~, p= f[i, 1]; f[i, 1] = p*(p+1)^(f[i, 2]-1); f[i, 2] = 1); factorback(f); }
isA348499(n) = !(A003968(n)%n);

A348169 Positive integers which can be represented as A(x^2 + y^2 + z^2) = B(xy + xz + y*z) with positive integers x, y, z, A, B and gcd(A,B)=1.

Original entry on oeis.org

3, 12, 18, 27, 30, 42, 48, 72, 75, 77, 98, 108, 120, 147, 154, 162, 168, 192, 243, 255, 260, 264, 270, 272, 273, 285, 288, 297, 300, 308, 338, 363, 378, 392, 432, 450, 462, 480, 490, 494, 507, 510, 513, 588, 616, 630, 648, 672, 675, 693, 702, 714, 722, 750, 754, 768, 798

Offset: 1

Alexander Kritov, Oct 04 2021

nonn

The sequence represents a generalization of cases A033428 (k=1), A347960 (k=2), A347969 (k=5) with all possible k given by A331605. Instead of integer k, it utilizes the ratio B/A.

			a(6)=42: the quintuple (x,y,z) A,B is 1,2,4 (2,3) because 42 = 2*(1^2 + 2^2 + 4^2) = 3*(1*4 + 1*2 + 2*4).
  a(n)    (x,y,z)     A,  B
    3     (1,1,1)     1,  1
   12     (2,2,2)     1,  1
   18     (1,1,4)     1,  2
   27     (3,3,3)     1,  1
   30     (1,1,2)     5,  6
   42     (1,2,4)     2,  3
   48     (4,4,4)     1,  1
   72     (1,2,2)     8,  9  [also (2,2,8) 1, 2]
   75     (5,5,5)     1,  1
   77     (1,1,3)     7, 11
   98     (1,4,9)     1,  2
  108     (6,6,6)     1,  1
  120     (2,2,4)     5,  6
  147     (7,7,7)     1,  1
  154     (1,2,3)    11, 14
  162     (3,3,12)    1,  2
  168     (2,4,8)     2,  3
  192     (8,8,8)     1,  1
  243     (9,9,9)     1,  1
  255     (1,1,7)     5, 17
  260     (2,5,6)     4,  5
  264     (1,4,4)     8, 11
  270     (2,5,5)     5,  6
  272     (2,2,3)    16, 17
  288     (4,4,2)     8,  9  [also (4,4,16) 1, 2]

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Alexander Kritov, Source code

Cf. A331605, A347969, A347960.

The sequence contains A033428 (A=B=1), A347969 (B=2*A), A347960 (B=5*A).

C
```
/* See links */
```

from itertools import islice, count
from math import gcd
from sympy import divisors, integer_nthroot
def A348169(): # generator of terms
    for n in count(1):
        for d in divisors(n,generator=False):
            x, x2 = 1, 1
            while 3*x2 <= d:
                y, y2 = x, x2
                z2 = d-x2-y2
                while z2 >= y2:
                    z, w = integer_nthroot(z2,2)
                    if w:
                        A = n//d
                        B, u = divmod(n,x*(y+z)+y*z)
                        if u == 0 and gcd(A,B) == 1:
                            yield n
                            break
                    y += 1
                    y2 += 2*y-1
                    z2 -= 2*y-1
                else:
                    x += 1
                    x2 += 2*x-1
                    continue
                break
            else:
                continue
            break
A348169_list = list(islice(A348169(),57)) # Chai Wah Wu, Nov 26 2021

cp's OEIS Frontend

A327564 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)^(k_j - 1)).

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A348036 a(n) = gcd(n, A003968(n)), where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

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A348037 a(n) = n / gcd(n, A003968(n)), where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

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A348038 a(n) = A003968(n) / gcd(n, A003968(n)), where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

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A348039 a(n) = gcd(A003557(n), A327564(n)).

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A348499 Numbers k such that A003968(k) is a multiple of k, where A003968 is multiplicative with a(p^e) = p*(p+1)^(e-1).

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A348169 Positive integers which can be represented as A*(x^2 + y^2 + z^2) = B*(x*y + x*z + y*z) with positive integers x, y, z, A, B and gcd(A,B)=1.

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A348169 Positive integers which can be represented as A(x^2 + y^2 + z^2) = B(xy + xz + y*z) with positive integers x, y, z, A, B and gcd(A,B)=1.