A065465 -id:A065465 - cp's OEIS frontend

A377845 Numbers that have more than one odd exponent larger than 1 in their prime factorization.

216, 864, 1000, 1080, 1512, 1944, 2376, 2744, 2808, 3000, 3375, 3456, 3672, 4000, 4104, 4320, 4968, 5400, 6048, 6264, 6696, 6750, 7000, 7560, 7776, 7992, 8232, 8856, 9000, 9261, 9288, 9504, 9720, 10152, 10584, 10648, 10976, 11000, 11232, 11448, 11880, 12000, 12744, 13000

Offset: 1

Amiram Eldar, Nov 09 2024

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/(p^2*(p+1))) * (1 + Sum_{p prime} (1/(p^3+p^2-1))) = 0.0035024748296318122535... .

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

Complement of the union of A335275 and A377844.
Subsequence of A295661.
Subsequences: A162142, A179671, A190011.
Cf. A065465.

q[n_] := Count[FactorInteger[n][[;; , 2]], _?(# > 1 && OddQ[#] &)] > 1; Select[Range[13000], q]

is(k) = #select(x -> x>1 && x%2, factor(k)[, 2]) > 1;

A072779 a(n) = sigma_2(n) + phi(n) * sigma(n).

Original entry on oeis.org

2, 8, 18, 35, 50, 74, 98, 145, 169, 202, 242, 322, 338, 394, 452, 589, 578, 689, 722, 882, 884, 970, 1058, 1330, 1271, 1354, 1540, 1722, 1682, 1876, 1922, 2373, 2180, 2314, 2452, 3003, 2738, 2890, 3044, 3650, 3362, 3652, 3698, 4242, 4238, 4234, 4418

Offset: 1

T. D. Noe, Jul 15 2002

This sequence is interesting because (1) a(n) >= 2 n^2, with equality only when n is prime (or 1) and (2) a(n) = 2 + 2*n^2 if and only if n is the product of two distinct primes. Note for twin primes: let n = m^2 - 1, then m-1 and m+1 are twin primes if and only if a(n) = 2 + 2*n^2. Note for the Goldbach conjecture: let n = m^2 - r^2, then m-r and m+r are primes that add to 2m if and only if a(n) = 2 + 2*n^2. See A072780 for a(n) - 2*n^2.

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Divisor Function.
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A000010, A000203, A001157, A065387, A072780.

Cf. A002117, A065465.

a072779 n = a001157 n + (a000203 n) * (a000010 n)
-- Reinhard Zumkeller, Jan 15 2013

Table[DivisorSigma[2, n]+EulerPhi[n]DivisorSigma[1, n], {n, 100}]

a(n)=sigma(n,2)+eulerphi(n)*sigma(n) \\ Charles R Greathouse IV, May 15 2013

a(n) = A001157(n) + A000203(n)*A000010(n). - Reinhard Zumkeller, Jan 15 2013

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(3) + Product_{p prime} (1 - 1/(p^2*(p+1))) = A002117 + A065465 = 2.083570742884... . - Amiram Eldar, Dec 03 2023

A329728 Partial sums of A092261.

Original entry on oeis.org

1, 4, 8, 9, 15, 27, 35, 36, 37, 55, 67, 71, 85, 109, 133, 134, 152, 155, 175, 181, 213, 249, 273, 277, 278, 320, 321, 329, 359, 431, 463, 464, 512, 566, 614, 615, 653, 713, 769, 775, 817, 913, 957, 969, 975, 1047, 1095, 1099, 1100, 1103, 1175, 1189, 1243

Offset: 1

Amiram Eldar, Nov 19 2019

nonn

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 50.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Mathematische Zeitschrift, Vol. 74, No. 1 (1960), pp. 66-80.
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Vaclav Kotesovec, Plot of a(n)/n^2 for n = 1..1000000

Cf. A065465, A092261.

Accumulate[Table[Plus @@ Select[Divisors@ n, Max @@ Last /@ FactorInteger@ # == 1 && GCD[#, n/#] == 1 &], {n, 1, 53}]] (* after Michael De Vlieger at A092261 *)

Lim_{n->oo} a(n)/n^2 = 1/2 * Product_{p prime}(1 - 1/(p^2*(p+1))) = 1/2 * A065465.

A336223 Numbers k such that the largest square dividing k is a unitary divisor of k and its square root has an even number of distinct prime divisors.

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, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 100, 101, 102, 103

Offset: 1

Amiram Eldar, Jul 12 2020

nonn

First differs from A333634 at n = 47.
Terms k of A335275 such that A000188(k) is a term of A030231.
Numbers whose powerful part (A057521) is a square term of A030231.
The squarefree numbers (A005117) are terms of this sequence since if k is squarefree, then the largest square dividing k is 1 which is a unitary divisor, sqrt(1) = 1 has 0 prime divisors, and 0 is even.
The asymptotic density of this sequence is (Product_{p prime} (1 - 1/(p^2*(p+1))) + Product_{p prime} (1 - (2*p+1)/(p^2*(p+1))))/2 = (0.881513... + 0.394391...)/2 = 0.637952807730728551636349961980617856650450613867264... (Cohen, 1964; the first product is A065465).

			36 is a term since the largest square dividing 36 is 36, which is a unitary divisor, sqrt(36) = 6, 6 = 2 * 3 has 2 distinct prime divisors, and 2 is even.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Some asymptotic formulas in the theory of numbers, Trans. Amer. Math. Soc., Vol. 112 (1964), pp. 214-227.

Intersection of A333634 and A335275.

Cf. A000188, A001221, A005117, A030231, A057521, A065465.

seqQ[n_] := EvenQ @ Length[(e = Select[FactorInteger[n][[;; , 2]], # > 1 &])] && AllTrue[e, EvenQ[#] &]; Select[Range[100], seqQ]

A377844 Numbers that have a single odd exponent larger than 1 in their prime factorization.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 108, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 224, 232, 243, 248, 250, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 432, 440, 456, 459, 472, 480, 486, 488, 500

Offset: 1

Amiram Eldar, Nov 09 2024

First differs from A295661, A325990 and A376142 at n = 24: A295661(24) = A325990(24) = A376142(24) = 216 = 2^3 * 3^3 is not a term of this sequence.
Differs from A060476 by having the terms 432, 648, 1728, ..., and not having the terms 1, 216, 256, 768, 864, ... .
The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p^2*(p+1))) * Sum_{p prime} (1/(p^3+p^2-1)) = 0.11498368544519741081... .

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

Subsequence of A295661.
Subsequences: A065036, A143610, A163569.
Cf. A060476, A065465, A325990, A376142, A377845.

q[n_] := Count[FactorInteger[n][[;; , 2]], _?(# > 1 && OddQ[#] &)] == 1; Select[Range[500], q]

is(k) = #select(x -> x>1 && x%2, factor(k)[, 2]) == 1;

A386401 a(n) = numerator(sigma(n)*phi(n)/n^2).

Original entry on oeis.org

1, 3, 8, 7, 24, 2, 48, 15, 26, 18, 120, 7, 168, 36, 64, 31, 288, 13, 360, 21, 128, 90, 528, 5, 124, 126, 80, 6, 840, 16, 960, 63, 320, 216, 1152, 91, 1368, 270, 448, 9, 1680, 32, 1848, 105, 208, 396, 2208, 31, 342, 93, 256, 147, 2808, 20, 576, 45, 320, 630, 3480, 56

Offset: 1

Stefano Spezia, Jul 20 2025

a(n)/A386402(n) = sigma(n)*phi(n)*(1/n^2) is a multiplicative function since it is the product of three multiplicative functions.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Exercise 5.3.21 on page 169.

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

Cf. A000010, A000203, A000290, A062354, A065465.

Cf. A386402 (denominators).

a[n_]:=Numerator[DivisorSigma[1,n]EulerPhi[n]/n^2]; Array[a,60]

a(n) = {my(f = factor(n)); numerator(sigma(f) * eulerphi(f) / n^2);} \\ Amiram Eldar, Jul 21 2025

From Amiram Eldar, Jul 21 2025: (Start)
Let f(n) = a(n)/A386402(n) = sigma(n)*phi(n)/n^2. Then:
f(n) = A062354(n)/n^2.
f(n) is multiplicative with f(p^e) = 1 - 1/p^(e+1).
Dirichlet g.f. of f(n): zeta(s) * zeta(s+1) * Product_{p prime} (1 - 1/p^(s+1) - 1/p^(s+2)+ 1/p^(2*s+2)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} f(k) = Product_{p prime} (1 - 1/(p^2*(p+1))) = 0.881513... (A065465). (End)

A007517 a(n) = phi(n) * (sigma(n) - n).

Original entry on oeis.org

0, 1, 2, 6, 4, 12, 6, 28, 24, 32, 10, 64, 12, 60, 72, 120, 16, 126, 18, 176, 132, 140, 22, 288, 120, 192, 234, 336, 28, 336, 30, 496, 300, 320, 312, 660, 36, 396, 408, 800, 40, 648, 42, 800, 792, 572, 46, 1216, 336, 860, 672, 1104, 52, 1188, 680

Offset: 1

Walter Nissen

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A001065, A059956, A065465.

List([1..60],n->Phi(n)*(Sigma(n)-n)); # Muniru A Asiru, Mar 22 2018

with(numtheory): seq(phi(n)*(sigma(n)-n),n=1..60); # Muniru A Asiru, Mar 22 2018

Table[EulerPhi[n](DivisorSigma[1,n]-n),{n,60}] (* Harvey P. Dale, Mar 16 2013 *)

for(n=1,50, print1(eulerphi(n)*(sigma(n) - n), ", ")) \\ G. C. Greubel, Mar 22 2018

a(n) = A000010(n)*A001065(n). - Michel Marcus, Mar 22 2018

Sum_{k=1..n} a(k) ~ c * n^2 / 3, where c = A065465 - A059956 = 0.273586... . - Amiram Eldar, Dec 04 2023

A078087 Continued fraction expansion of Product_{p prime} (1 - 1/(p^2*(p+1))).

Original entry on oeis.org

0, 1, 7, 2, 3, 1, 1, 1, 7, 1, 1, 6, 1, 5, 1, 1, 3, 2, 1, 1, 3, 2, 1, 1, 1, 1, 13, 1, 16, 1, 1, 16, 1, 80, 1, 1, 1, 1, 7, 5, 1, 4, 1, 33, 3, 8, 1, 8, 1, 16, 11, 1, 2, 6, 1, 19, 1, 12, 5, 11, 1, 7, 5, 1, 1, 1, 2, 5, 1, 4, 1, 3, 4, 4, 4, 1, 11, 1, 2, 5, 4, 12, 3, 1, 4, 1, 3, 1, 168, 1, 4, 1, 1

Offset: 0

Benoit Cloitre, Dec 02 2002

Cf. A065465 (decimal expansion).

digits = 93;
$MaxExtraPrecision = 4 digits;
terms = 4 digits;
LR = Join[{0, 0, 0}, LinearRecurrence[{-2, -1, 1, 1}, {-3, 4, -5, 3}, terms + 10]];
r[n_Integer] := LR[[n]];
c = Exp[NSum[r[n] PrimeZetaP[n - 1]/(n - 1), {n, 4, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10]];
ContinuedFraction[c][[;; digits]] (* Jean-François Alcover, Aug 01 2019 *)

contfrac(prodeulerrat(1 - 1/(p^2*(p+1)))) \\ Amiram Eldar, Mar 14 2021

Offset changed by Andrew Howroyd, Jul 05 2024

A336224 Numbers k such that the largest square dividing k is a unitary divisor of k and its square root has an even number of prime divisors (counted with multiplicity).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 48, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 100

Offset: 1

Amiram Eldar, Jul 12 2020

Terms k of A335275 such that A000188(k) is a term of A028260.
Numbers whose powerful part (A057521) is the square of a term of A028260.
The squarefree numbers (A005117) are terms of this sequence since if k is squarefree, then the largest square dividing k is 1 which is a unitary divisor, sqrt(1) has 0 prime divisors, and 0 is even.
The asymptotic density of this sequence is (5 * Product_{p prime} (1 - 1/(p^2*(p+1))) + 2 * Product_{p prime} (1 + 1/(p^2*(p+1))))/10 = (5 * 0.881513... + 2 * 1.125606...)/10 = 0.665878294481337275662425136416469977597382409701642... (Cohen, 1964; the first product is A065465).

			16 is a term since the largest square dividing 16 is 16, which is a unitary divisor, sqrt(16) = 4, 4 = 2 * 2 has 2 prime divisors, and 2 is even.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Some asymptotic formulas in the theory of numbers, Trans. Amer. Math. Soc., Vol. 112 (1964), pp. 214-227.

Intersection of A335275 and A336222.

Cf. A000188, A001222, A005117, A028260, A057521, A065465.

seqQ[n_] := AllTrue[(e = FactorInteger[n][[;; , 2]]), # == 1 || EvenQ[#] &] && EvenQ @ Total[Select[e, # > 1 &]/2]; Select[Range[100], seqQ]

A357684 The squarefree part (A007913) of numbers whose squarefree part is a unitary divisor (A335275).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 09 2022

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
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. A000290, A005117, A007913, A008833, A065465, A069891, A335275.

s[n_] := If[AllTrue[(f = FactorInteger[n])[[;; , 2]], # == 1 || EvenQ[#] &], i = Position[f[[;; , 2]], 1] // Flatten; Times @@ f[[i, 1]], Nothing]; Array[s, 100]

s(n) = {my(f = factor(n), ans = 1); for(k = 1, #f~, if(f[k,2] > 1 && f[k,2]%2, ans = 0)); if(ans, ans = prod(k = 1, #f~, if(f[k,2] == 1, f[k,1], 1))) };
for(n = 1, 100, if(s(n) > 0, print1(s(n), ", ")))

a(n) = A007913(A335275(n)).
a(n) = 1 iff A335275(n) is a square (A000290).
a(n) = A335275(n) iff A335275(n) is squarefree (A005117).
Sum_{k, a(k) <= x} ~ c*x^2 + o(x^2), where c = (3/Pi^2) * Sum_{k>=1} f(k)/k^4 = 0.32103327852028541131..., and f(k) = Product_{p prime | k} (p/(p+1)) (Jakimczuk, 2017).
Sum_{k=1..n} a(k) ~ c'*x^2 + o(x^2), where c' = c / (A065465)^2 = 0.41313480468422995583... .

cp's OEIS Frontend

A377845 Numbers that have more than one odd exponent larger than 1 in their prime factorization.

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A072779 a(n) = sigma_2(n) + phi(n) * sigma(n).

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A329728 Partial sums of A092261.

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A336223 Numbers k such that the largest square dividing k is a unitary divisor of k and its square root has an even number of distinct prime divisors.

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A377844 Numbers that have a single odd exponent larger than 1 in their prime factorization.

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A386401 a(n) = numerator(sigma(n)*phi(n)/n^2).

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A007517 a(n) = phi(n) * (sigma(n) - n).

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A078087 Continued fraction expansion of Product_{p prime} (1 - 1/(p^2*(p+1))).

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