A069359 -id:A069359 - cp's OEIS frontend

A328260 a(n) = n * omega(n).

0, 2, 3, 4, 5, 12, 7, 8, 9, 20, 11, 24, 13, 28, 30, 16, 17, 36, 19, 40, 42, 44, 23, 48, 25, 52, 27, 56, 29, 90, 31, 32, 66, 68, 70, 72, 37, 76, 78, 80, 41, 126, 43, 88, 90, 92, 47, 96, 49, 100, 102, 104, 53, 108, 110, 112, 114, 116, 59, 180, 61, 124, 126, 64, 130, 198, 67, 136, 138, 210

Offset: 1

Ilya Gutkovskiy, Oct 09 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Mikhail R. Gabdullin and Vitalii V. Iudelevich, Numbers of the form kf(k), arXiv:2201.09287 [math.NT] (2022).
Eric Weisstein's World of Mathematics, Distinct Prime Factors

Cf. A001221, A001222, A007947, A038040, A061397, A066959, A069359, A246655 (fixed points), A293548, A298473.

[0] cat [n*(#PrimeDivisors(n)):n in [2..70]]; // Marius A. Burtea, Oct 10 2019

Table[n PrimeNu[n], {n, 1, 70}]
nmax = 70; CoefficientList[Series[Sum[Prime[k] x^Prime[k]/(1 - x^Prime[k])^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n)=n*omega(n) \\ Charles R Greathouse IV, Mar 16 2022

G.f.: Sum_{k>=1} prime(k) * x^prime(k) / (1 - x^prime(k))^2.
a(n) = bigomega(rad(n)^n).
a(n) = Sum_{d|n} A061397(n/d) * d.
Define f(x) = #{n <= x: a(n) <= x}. Gabdullin & Iudelevich show that f(x) ~ x/log log x. - Charles R Greathouse IV, Mar 16 2022

A329031 a(n) = A327860(A328841(n)).

Original entry on oeis.org

0, 1, 1, 5, 1, 5, 1, 7, 8, 31, 8, 31, 1, 7, 8, 31, 8, 31, 1, 7, 8, 31, 8, 31, 1, 7, 8, 31, 8, 31, 1, 9, 10, 41, 10, 41, 12, 59, 71, 247, 71, 247, 12, 59, 71, 247, 71, 247, 12, 59, 71, 247, 71, 247, 12, 59, 71, 247, 71, 247, 1, 9, 10, 41, 10, 41, 12, 59, 71, 247, 71, 247, 12, 59, 71, 247, 71, 247, 12, 59, 71, 247, 71, 247, 12, 59

Offset: 0

Antti Karttunen, Nov 07 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..32768
Index entries for sequences related to primorial base

Cf. A003415, A069359, A276086, A327860, A328571, A328841, A329032.

Cf. A060735 (the positions of ones).

A329031(n) = { my(s=0, m=1, p=2); while(n, if(n%p, m *= p; s += (1/p)); n = n\p; p = nextprime(1+p)); (s*m); };

a(n) = A327860(A328841(n)) = A003415(A276086(A328841(n))).

a(n) = A003415(A328571(n)) = A069359(A328571(n)).

A329039 If n = Product p_i^e_i, a(n) = n * Sum ((e_i - 1)/p_i).

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 8, 3, 0, 0, 6, 0, 0, 0, 24, 0, 6, 0, 10, 0, 0, 0, 24, 5, 0, 18, 14, 0, 0, 0, 64, 0, 0, 0, 30, 0, 0, 0, 40, 0, 0, 0, 22, 15, 0, 0, 72, 7, 10, 0, 26, 0, 36, 0, 56, 0, 0, 0, 30, 0, 0, 21, 160, 0, 0, 0, 34, 0, 0, 0, 96, 0, 0, 15, 38, 0, 0, 0, 120, 81, 0, 0, 42, 0, 0, 0, 88, 0, 30, 0, 46, 0, 0, 0, 192, 0, 14, 33, 70, 0, 0, 0, 104, 0

Offset: 1

Antti Karttunen, Nov 07 2019

nonn

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

Cf. A003415, A069359.

Cf. A005117 (positions of zeros).

A329039(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, ((f[i, 2]-1)/f[i, 1])));

a(n) = A003415(n) - A069359(n).

A329351 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329350(i) = A329350(j) for all i, j.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 4, 12, 9, 17, 2, 18, 2, 19, 20, 21, 22, 23, 2, 24, 25, 26, 2, 27, 2, 28, 29, 30, 2, 31, 32, 33, 34, 35, 2, 36, 37, 38, 39, 18, 2, 40, 2, 41, 42, 43, 44, 45, 2, 46, 47, 48, 2, 49, 2, 50, 51, 52, 44, 53, 2, 54, 55, 56, 2, 57, 58, 59, 60, 61, 2, 62, 63, 64, 65, 66, 29, 67, 2, 68, 69

Offset: 1

Antti Karttunen, Nov 12 2019

nonn

Restricted growth sequence transform of A329350.
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A069359(i) = A069359(j).

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

Cf. A069359, A305800, A329350.

Cf. also A329353, A329381.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A329350(n) = { my(m=1); fordiv(n,d,if(isprime(n/d), m *= A276086(d))); (m); };
v329351 = rgs_transform(vector(up_to, n, A329350(n)));
A329351(n) = v329351[n];

A329352 a(n) = Product_{d|n} A019565(d)^A010051(n/d).

Original entry on oeis.org

1, 2, 2, 3, 2, 18, 2, 5, 6, 30, 2, 75, 2, 90, 60, 7, 2, 210, 2, 105, 180, 126, 2, 245, 10, 210, 14, 525, 2, 66150, 2, 11, 252, 66, 300, 1155, 2, 198, 420, 385, 2, 173250, 2, 825, 2940, 990, 2, 847, 30, 3234, 132, 1155, 2, 15246, 420, 2695, 396, 2310, 2, 2223375, 2, 6930, 1540, 13, 700, 64350, 2, 195, 1980, 171990, 2, 5005, 2, 390, 32340, 975, 1260

Offset: 1

Antti Karttunen, Nov 12 2019

nonn

			The divisors of 30 are [1, 2, 3, 5, 6, 10, 15, 30], of which only d = 6, 10 and 15 are such that 30/d is a prime, thus a(n) = A019565(6) * A019565(10) * A019565(15) = 15 * 21 * 210 = 66150.

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A010051, A019565, A048675, A069359, A329353 (rgs-transform).
Cf. also A329350.
Differs from A300832 for the first time at n=30, where a(30) = 66150, while A300832(30) = 132300.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A329352(n) = { my(m=1); fordiv(n,d,if(isprime(n/d), m *= A019565(d))); (m); };

a(n) = Product_{d|n} A019565(d)^A010051(n/d).

For all n, A048675(a(n)) = A069359(n).

A345930 a(n) = A344756(A276086(n)).

Original entry on oeis.org

1, 1, 1, 2, 7, 1, 1, 1, 1, 13, 41, 2, 9, 11, 37, 2, 47, 3, 11, 7, 43, 19, 53, 4, 13, 17, 49, 11, 59, 1, 1, 1, 1, 17, 55, 1, 1, 1, 1, 106, 317, 19, 73, 92, 289, 127, 359, 13, 87, 113, 331, 148, 401, 11, 101, 134, 373, 169, 443, 2, 11, 13, 47, 2, 61, 17, 69, 86, 277, 121, 347, 2, 83, 107, 319, 2, 389, 31, 97, 128, 19

Offset: 1

Antti Karttunen, Jul 19 2021

Antti Karttunen, Table of n, a(n) for n = 1..13860
Index entries for sequences related to primorial base

Cf. A003415, A069359, A276086, A327860, A329029, A344756, A346469.

Cf. also A342002, A346474, A346475 (compare the scatter plots).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A344756(n) = { my(u=A003415(n)); (u/gcd(u,A069359(n))); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A345930(n) = A344756(A276086(n));

a(n) = A344756(A276086(n)).

a(n) = A327860(n) / A346469(n) = A327860(n) / gcd(A327860(n), A329029(n)).

A349338 Dirichlet convolution of A000010 (Euler totient phi) with A080339 (characteristic function of noncomposite numbers).

Original entry on oeis.org

1, 2, 3, 3, 5, 5, 7, 6, 8, 9, 11, 8, 13, 13, 14, 12, 17, 14, 19, 14, 20, 21, 23, 16, 24, 25, 24, 20, 29, 22, 31, 24, 32, 33, 34, 22, 37, 37, 38, 28, 41, 32, 43, 32, 38, 45, 47, 32, 48, 44, 50, 38, 53, 42, 54, 40, 56, 57, 59, 36, 61, 61, 54, 48, 64, 52, 67, 50, 68, 58, 71, 44, 73, 73, 68, 56, 76, 62, 79, 56, 72, 81

Offset: 1

Antti Karttunen, Nov 17 2021

nonn

Möbius transform of A230593.

The number of integers k from 1 to n such that gcd(n, k) is a noncomposite number. - Amiram Eldar, Jun 21 2025

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

Cf. A000010, A005117, A008683, A069359, A080339, A085548, A117494, A230593, A348976, A349435.

a[n_] := DivisorSum[n, Boole[!CompositeQ[#]] * EulerPhi[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 17 2021 *)

A349338(n) = sumdiv(n, d, eulerphi(n/d)*((1==d)||isprime(d)));

a(n) = {my(f = factor(n), p = f[,1], e = f[,2]); n * vecprod(apply(x -> 1-1/x, p)) * (1 + vecsum(apply(x -> 1/x, p - vector(#e, i, e[i] == 1)~)));} \\ Amiram Eldar, Jun 21 2025

a(n) = Sum_{d|n} A000010(n/d) * A080339(d).
a(n) = Sum_{d|n} A008683(n/d) * A230593(d).
a(n) = Sum_{d|n} A349435(n/d) * A348976(d).
a(n) = A000010(n) + A117494(n). [Because A117494 is the Möbius transform of A069359]
For all n >= 1, a(A005117(n)) = A348976(A005117(n)).
Sum_{k=1..n} a(k) ~ 3 * (1 + A085548) * n^2 / Pi^2. - Vaclav Kotesovec, Nov 20 2021

A180253 Call two divisors of n adjacent if the larger is a prime times the smaller. a(n) is the sum of elements of all pairs of adjacent divisors of n.

Original entry on oeis.org

0, 3, 4, 9, 6, 24, 8, 21, 16, 36, 12, 64, 14, 48, 48, 45, 18, 87, 20, 96, 64, 72, 24, 144, 36, 84, 52, 128, 30, 216, 32, 93, 96, 108, 96, 229, 38, 120, 112, 216, 42, 288, 44, 192, 174, 144, 48, 304, 64, 201, 144, 224, 54, 276, 144, 288, 160, 180, 60, 552, 62, 192, 232, 189

Offset: 1

Vladimir Shevelev, Aug 20 2010

The pairs of adjacent divisors of n are counted in A062799(n).

For each divisor d of n we can check in how many pairs it occurs. For each prime divisor p of n, see the exponent of p in the factorization of d. If it's positive (p|d) then it occurs once more. If d*p doesn't divide n, add one to the frequency as well. - David A. Corneth, Dec 17 2018

			a(4) = (1 + 2) + (2 + 4) = 9.
a(120) = a(3*5*2^3) = 4*6*(3*8 + 4*4 + 4*2 + 3) = 1224.

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

Cf. A001221, A062799, A069359, A179926, A180026, A323599, A328260, A329354, A329375.

divisorSumPrime[n_] := DivisorSum[n, 1+1/# &, PrimeQ[#] &]; a[n_] := DivisorSum[n, #*divisorSumPrime[#]& ]; Array[a, 70] (* Amiram Eldar, Dec 17 2018 *)

a(n) = sumdiv(n, d, d*sumdiv(d, p, isprime(p)*(1+1/p))); \\ Michel Marcus, Dec 17 2018

a(n) = my(f = factor(n), res = 0); fordiv(n, d, for(i = 1, #f~, v = valuation(d, f[i, 1]); res+=(d * ((v > 0) + (v < f[i, 2]))))); res \\ David A. Corneth, Dec 17 2018

a(n) = Sum_{d|n} d*Sum_{p|d} (1 + 1/p) where p is restricted to primes.
a(n) = Sum_{d|n} A069359(d) + Sum_{d|n} d*A001221(d).
a(n) = A323599(n) + A329354(n) = A323599(n) + A328260(n) + A329375(n). - Antti Karttunen, Nov 15 2019
a(p^k) = (p^k - 1)*(p + 1)/(p - 1).
a(p_1*p_2*...*p_m) = m*(p_1 + 1)*(p_2 + 1)*...*(p_m + 1).
a(p*q^k) = (p + 1)*(2*q^k + 3*q^(k - 1) + 3*q^(k - 2) + ... + 3*q + 2).
a(p*q*r^k) = (p + 1)*(q + 1)*(3*r^k + 4*r^(k - 1) + 4*r^(k - 2) + ... + 4*r + 3) and similar for a larger number of distinct prime factors of n.

Definition rephrased, entries checked, one example added. - R. J. Mathar, Oct 25 2010

A292786 a(n) = psi(n) - phi(n).

Original entry on oeis.org

0, 2, 2, 4, 2, 10, 2, 8, 6, 14, 2, 20, 2, 18, 16, 16, 2, 30, 2, 28, 20, 26, 2, 40, 10, 30, 18, 36, 2, 64, 2, 32, 28, 38, 24, 60, 2, 42, 32, 56, 2, 84, 2, 52, 48, 50, 2, 80, 14, 70, 40, 60, 2, 90, 32, 72, 44, 62, 2, 128, 2, 66, 60, 64, 36, 124, 2, 76, 52, 120, 2, 120, 2, 78, 80, 84, 36, 144

Offset: 1

Altug Alkan, Sep 23 2017

Even numbers that are not the terms of this sequence are 12, 102, 114, 130, ...

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

Cf. A000010, A001615, A051612, A069359, A070915, A088245, A291784.

psi[n_] := If[n < 1, 0, n Sum[ MoebiusMu[d]^2/d, {d, Divisors@ n}]]; Array[psi@# - EulerPhi@# &, 87] (* Robert G. Wilson v, Sep 23 2017 *)

a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
a(n) = a001615(n) - eulerphi(n); \\ after Charles R Greathouse IV at A001615

a(n) = A001615(n) - A000010(n).
a(n) = 2 iff n is prime.
a(n) = 2*A069359(n) iff n is in A070915.
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c =  9/(2*Pi^2) = 0.455945... (A088245). - Amiram Eldar, Dec 05 2023

A344182 a(n) = A344026(n) XOR A344028(n).

Original entry on oeis.org

0, 0, 0, 6, 0, 0, 5, 8, 0, 0, 0, 26, 15, 26, 18, 40, 0, 0, 0, 22, 0, 0, 63, 56, 9, 14, 31, 34, 82, 124, 119, 64, 0, 0, 0, 50, 0, 0, 45, 88, 0, 0, 0, 98, 99, 38, 234, 88, 29, 114, 29, 202, 35, 34, 136, 160, 162, 444, 406, 130, 393, 430, 452, 224, 0, 0, 0, 42, 0, 0, 97, 120, 0, 0, 0, 46, 215, 222, 130, 136, 0, 0, 0, 202

Offset: 0

Antti Karttunen, May 16 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to binary expansion of n

Cf. A003415, A003714 (positions of zeros), A005940, A069359, A344026, A344028.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A344182(n) = { my(u=A005940(1+n)); bitxor(A003415(u),A069359(u)); };

a(n) = A344026(n) XOR A344028(n) = A003415(A005940(1+n)) XOR A069359(A005940(1+n)).

cp's OEIS Frontend

A328260 a(n) = n * omega(n).

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A329031 a(n) = A327860(A328841(n)).

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A329039 If n = Product p_i^e_i, a(n) = n * Sum ((e_i - 1)/p_i).

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A329351 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329350(i) = A329350(j) for all i, j.

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A329352 a(n) = Product_{d|n} A019565(d)^A010051(n/d).

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A345930 a(n) = A344756(A276086(n)).

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A349338 Dirichlet convolution of A000010 (Euler totient phi) with A080339 (characteristic function of noncomposite numbers).

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A180253 Call two divisors of n adjacent if the larger is a prime times the smaller. a(n) is the sum of elements of all pairs of adjacent divisors of n.

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