A332880 -id:A332880 - cp's OEIS frontend

A332881 If n = Product (p_j^k_j) then a(n) = denominator of Product (1 + 1/p_j).

1, 2, 3, 2, 5, 1, 7, 2, 3, 5, 11, 1, 13, 7, 5, 2, 17, 1, 19, 5, 21, 11, 23, 1, 5, 13, 3, 7, 29, 5, 31, 2, 11, 17, 35, 1, 37, 19, 39, 5, 41, 7, 43, 11, 5, 23, 47, 1, 7, 5, 17, 13, 53, 1, 55, 7, 57, 29, 59, 5, 61, 31, 21, 2, 65, 11, 67, 17, 23, 35

Offset: 1

Ilya Gutkovskiy, Feb 28 2020

Denominator of sum of reciprocals of squarefree divisors of n.

			1, 3/2, 4/3, 3/2, 6/5, 2, 8/7, 3/2, 4/3, 9/5, 12/11, 2, 14/13, 12/7, 8/5, 3/2, 18/17, ...

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

Cf. A001615, A008683, A017666, A048250, A007947, A109395, A187778 (positions of 1's), A306695, A308443, A308462, A332880 (numerators), A332883.

a:= n-> denom(mul(1+1/i[1], i=ifactors(n)[2])):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 28 2020

Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]] & /@ FactorInteger[n])], {n, 1, 70}] // Denominator
Table[Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}], {n, 1, 70}] // Denominator

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A332881(n) = denominator(A001615(n)/n);

Denominators of coefficients in expansion of Sum_{k>=1} mu(k)^2*x^k/(k*(1 - x^k)).
a(n) = denominator of Sum_{d|n} mu(d)^2/d.
a(n) = denominator of psi(n)/n.
a(p) = p, where p is prime.
a(n) = n / A306695(n) = n / gcd(n, A001615(n)). - Antti Karttunen, Nov 15 2021

A332882 If n = Product (p_j^k_j) then a(n) = numerator of Product (1 + 1/p_j^k_j).

Original entry on oeis.org

1, 3, 4, 5, 6, 2, 8, 9, 10, 9, 12, 5, 14, 12, 8, 17, 18, 5, 20, 3, 32, 18, 24, 3, 26, 21, 28, 10, 30, 12, 32, 33, 16, 27, 48, 25, 38, 30, 56, 27, 42, 16, 44, 15, 4, 36, 48, 17, 50, 39, 24, 35, 54, 14, 72, 9, 80, 45, 60, 2, 62, 48, 80, 65, 84, 24, 68, 45, 32, 72

Offset: 1

Ilya Gutkovskiy, Feb 28 2020

Numerator of sum of reciprocals of unitary divisors of n.

			1, 3/2, 4/3, 5/4, 6/5, 2, 8/7, 9/8, 10/9, 9/5, 12/11, 5/3, 14/13, 12/7, 8/5, 17/16, ...

Antti Karttunen, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Unitary Divisor.

Cf. A017665, A028235, A028236, A034448, A077610, A306633, A319676, A323166, A332880, A332883 (denominators).

Cf. also A348734, A348735.

a:= n-> numer(mul(1+i[1]^i[2], i=ifactors(n)[2])/n):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 28 2020

Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]]^#[[2]] & /@ FactorInteger[n])], {n, 1, 70}] // Numerator
Table[Sum[If[GCD[d, n/d] == 1,  1/d, 0], {d, Divisors[n]}], {n, 1, 70}] // Numerator

a(n) = numerator(sumdiv(n, d, if (gcd(d, n/d)==1, 1/d))); \\ Michel Marcus, Feb 28 2020

a(n) = numerator of Sum_{d|n, gcd(d, n/d) = 1} 1/d.
a(n) = numerator of usigma(n)/n.
a(p) = p + 1, where p is prime.
a(n) = A034448(n) / A323166(n). - Antti Karttunen, Nov 13 2021
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A332883(k) = zeta(2)/zeta(3) = 1.368432... (A306633). - Amiram Eldar, Nov 21 2022

A358975 Numbers that are coprime to their digital sum in base 3 (A053735).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 49, 51, 53, 55, 59, 61, 67, 69, 71, 73, 79, 81, 83, 85, 89, 91, 97, 101, 103, 107, 109, 113, 119, 121, 123, 125, 127, 129, 131, 137, 139, 141, 143, 147, 149, 151, 153, 155, 157, 159, 161, 163, 167, 169

Offset: 1

Amiram Eldar, Dec 07 2022

Numbers k such that gcd(k, A053735(k)) = 1.
All the terms are odd since if k is even then A053735(k) is even and so gcd(k, A053735(k)) >= 2.
Olivier (1975, 1976) proved that the asymptotic density of this sequence is 4/Pi^2 = 0.40528... (A185199).
The powers of 3 (A000244) are terms. These are also the only ternary Niven numbers (A064150) in this sequence.
Includes all the odd prime numbers (A065091).

			3 is a term since A053735(3) = 1, and gcd(3, 1) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Christian Mauduit, Carl Pomerance, and András Sárközy, On the distribution in residue classes of integers with a fixed sum of digits, The Ramanujan Journal, Vol. 9, No. 1-2 (2005), pp. 45-62; alternative link.
Michel Olivier, Sur la probabilité que n soit premier à la somme de ses chiffres, C. R. Math. Acad. Sci. Paris, Série A, Vol. 280 (1975), pp. 543-545.
Michel Olivier, Fonctions g-additives et formule asymptotique pour la propriété (n, f(n)) = q, Acta Arithmetica, Vol. 31, No. 4 (1976), pp. 361-384; alternative link.

Cf. A064150, A053735, A185199, A332880.
Subsequences: A000244, A065091.
Similar sequences: A094387, A339076, A358976, A358977, A358978.

q[n_] := CoprimeQ[n, Plus @@ IntegerDigits[n, 3]]; Select[Range[200], q]

PARI
```
is(n) = gcd(n, sumdigits(n, 3)) == 1;
```

A372606 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} phi(k*j).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 2, 4, 5, 6, 4, 6, 10, 9, 10, 2, 8, 10, 14, 13, 12, 6, 6, 16, 18, 22, 17, 18, 4, 12, 12, 24, 26, 28, 23, 22, 6, 12, 24, 20, 44, 34, 40, 31, 28, 4, 12, 20, 36, 28, 52, 46, 48, 37, 32, 10, 12, 30, 36, 60, 40, 76, 62, 66, 45, 42, 4, 20, 20, 42, 52, 72, 52, 92, 74, 74, 55, 46

Offset: 1

Seiichi Manyama, May 07 2024

			Square array T(n,k) begins:
   1,  1,  2,  2,  4,  2,   6, ...
   2,  3,  4,  6,  8,  6,  12, ...
   4,  5, 10, 10, 16, 12,  24, ...
   6,  9, 14, 18, 24, 20,  36, ...
  10, 13, 22, 26, 44, 28,  60, ...
  12, 17, 28, 34, 52, 40,  72, ...
  18, 23, 40, 46, 76, 52, 114, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=1..2 give: A002088, A049690.
Main diagonal gives A372608.
Cf. A000010, A372619.
Cf. A007947, A048250, A332880, A332881.

T[n_, k_] := Sum[EulerPhi[k*j], {j, 1, n}]; Table[T[k, n-k+1], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, May 10 2024 *)

PARI
```
T(n, k) = sum(j=1, n, eulerphi(k*j));
```

T(n,k) ~ (3/Pi^2) * c(k) * n^2, where c(k) = k * A007947(k)/A048250(k) = k * A332881(k) / A332880(k) is the multiplicative function defined by c(p^e) = p^(e+1)/(p+1). - Amiram Eldar, May 10 2024

A373439 Numerator of sum of reciprocals of square divisors of n.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 5, 1, 1, 1, 21, 1, 10, 1, 5, 1, 1, 1, 5, 26, 1, 10, 5, 1, 1, 1, 21, 1, 1, 1, 25, 1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 21, 50, 26, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 85, 1, 1, 1, 5, 1, 1, 1, 25, 1, 1, 26, 5, 1, 1, 1, 21, 91, 1, 1, 5, 1

Offset: 1

Ilya Gutkovskiy, Jun 05 2024

			1, 1, 1, 5/4, 1, 1, 1, 5/4, 10/9, 1, 1, 5/4, 1, 1, 1, 21/16, 1, 10/9, 1, 5/4, 1, 1, 1, 5/4, 26/25, ...

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

Cf. A007406, A017665, A017667, A028235, A035316, A332880, A373440 (denominators).

nmax = 85; CoefficientList[Series[Sum[x^(k^2)/(k^2 (1 - x^(k^2))), {k, 1, nmax}], {x, 0, nmax}], x] // Rest // Numerator
f[p_, e_] := (p^2 - p^(-2*Floor[e/2]))/(p^2-1); a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Jun 26 2024 *)

a(n) = numerator(sumdiv(n, d, if (issquare(d), 1/d))); \\ Michel Marcus, Jun 05 2024

Numerators of coefficients in expansion of Sum_{k>=1} x^(k^2)/(k^2*(1 - x^(k^2))).
a(n) is the numerator of Sum_{d^2|n} 1/d^2.
From Amiram Eldar, Jun 26 2024: (Start)
Let f(n) = a(n)/A373440(n). Then:
f(n) is multiplicative with f(p^e) = (p^2 - p^(-2*floor(e/2)))/(p^2-1).
Dirichlet g.f. of f(n): zeta(s) * zeta(2*s+2).
Sum_{k=1..n} f(k) ~ zeta(4) * n. (End)

A370783 a(n) is the numerator of the sum of the reciprocals of the squarefree divisors of the powerful part of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 6, 1, 4, 3, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 8, 6, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 4, 3, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1

Offset: 1

Amiram Eldar, Mar 02 2024

			Fractions begin with: 1, 1, 1, 3/2, 1, 1, 1, 3/2, 4/3, 1, 1, 3/2, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Arithmetical Functions over the Powerful Part of an Integer, ResearchGate, 2024.

Cf. A005117, A057521, A157289, A295295, A332880, A370784 (denominators).

a[n_] := Numerator[Times @@ (1 + 1/Select[FactorInteger[n], Last[#] > 1 &][[;; , 1]])]; Array[a, 100]

a(n) = {my(f = factor(n)); numerator(prod(i = 1, #f~, if(f[i,2] == 1, 1, 1 + 1/f[i,1])));}

a(n) = A332880(A057521(n)).
Let f(n) = a(n)/A370784(n):
f(n) is multiplicative with f(p) = 1 and f(p^e) = 1 + 1/p for e >= 2.
f(n) = 1 if and only if n is squarefree (A005117).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} f(k) = zeta(3)/zeta(6) = 1.181564... (A157289) (Jakimczuk, 2024).

cp's OEIS Frontend

A332881 If n = Product (p_j^k_j) then a(n) = denominator of Product (1 + 1/p_j).

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A332882 If n = Product (p_j^k_j) then a(n) = numerator of Product (1 + 1/p_j^k_j).

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A358975 Numbers that are coprime to their digital sum in base 3 (A053735).

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A372606 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} phi(k*j).

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A373439 Numerator of sum of reciprocals of square divisors of n.

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A370783 a(n) is the numerator of the sum of the reciprocals of the squarefree divisors of the powerful part of n.

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