A069359 -id:A069359 - cp's OEIS frontend

A329353 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329352(i) = A329352(j) for all i, j.

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 12, 18, 19, 2, 20, 2, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 29, 2, 30, 31, 32, 2, 33, 7, 34, 35, 25, 2, 36, 27, 37, 38, 39, 2, 40, 2, 41, 42, 43, 44, 45, 2, 46, 47, 48, 2, 49, 2, 50, 51, 52, 53, 54, 2, 55, 56, 57, 2, 58, 59, 60, 61, 62, 2, 63, 64, 65, 66, 67, 68, 69, 2, 70, 71, 72, 2, 73, 2, 49, 74

Offset: 1

Antti Karttunen, Nov 12 2019

nonn

Restricted growth sequence transform of A329352.
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. A010051, A019565, A069359, A305800, A329352.
Cf. also A329351.
Differs from A319682 for the first time at n=254, where a(254)=123, while A319682(254)=184.

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; };
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); };
v329353 = rgs_transform(vector(up_to, n, A329352(n)));
A329353(n) = v329353[n];

A342921 a(n) = A003415(A019565(n)).

Original entry on oeis.org

0, 1, 1, 5, 1, 7, 8, 31, 1, 9, 10, 41, 12, 59, 71, 247, 1, 13, 14, 61, 16, 87, 103, 371, 18, 113, 131, 493, 167, 719, 886, 2927, 1, 15, 16, 71, 18, 101, 119, 433, 20, 131, 151, 575, 191, 837, 1028, 3421, 24, 191, 215, 859, 263, 1241, 1504, 5153, 311, 1623, 1934, 6871, 2556, 10117, 12673, 40361, 1, 19, 20, 91, 22, 129

Offset: 0

Antti Karttunen, Apr 06 2021

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A003415, A019565, A069359, A276156, A327860, A329029, A342002.

Array[If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &[Times @@ Prime@ Flatten@ Position[Reverse@ IntegerDigits[#, 2], 1]] &, 70, 0] (* Michael De Vlieger, Apr 08 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A342921(n) = A003415(A019565(n));

a(n) = A003415(A019565(n)) = A069359(A019565(n)).

a(n) = A327860(A276156(n)) = A329029(A276156(n)) = A342002(A276156(n)).

A346469 a(n) = A340070(A276086(n)).

Original entry on oeis.org

0, 1, 1, 5, 3, 3, 1, 7, 8, 31, 3, 3, 5, 5, 5, 5, 120, 15, 25, 25, 50, 25, 75, 75, 125, 125, 125, 125, 750, 375, 1, 9, 10, 41, 3, 3, 12, 59, 71, 247, 3, 3, 5, 5, 5, 5, 15, 15, 50, 25, 25, 25, 75, 75, 375, 125, 125, 125, 375, 375, 7, 7, 7, 7, 210, 21, 7, 7, 7, 7, 21, 21, 420, 35, 35, 35, 7455, 105, 175, 175, 175, 3325

Offset: 0

Antti Karttunen, Jul 21 2021

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A003415, A069359, A276086, A327860, A329029, A340070, A345930.

Cf. also A346471.

A346469(n) = { my(s=0, t=0, m=1, p=2, e); while(n, e = (n%p); if(e, m *= (p^e); s += (1/p); t += (e/p)); n = n\p; p = nextprime(1+p)); (gcd(s,t)*m); };

a(n) = A340070(A276086(n)) = gcd(A327860(n), A329029(n)).

For n >= 1, a(n) = A327860(n) / A345930(n).

A348219 a(n) = tau(n) - omega(n) + n * Sum_{p|n, p prime} 1/p.

Original entry on oeis.org

1, 2, 2, 4, 2, 7, 2, 7, 5, 9, 2, 14, 2, 11, 10, 12, 2, 19, 2, 18, 12, 15, 2, 26, 7, 17, 12, 22, 2, 36, 2, 21, 16, 21, 14, 37, 2, 23, 18, 34, 2, 46, 2, 30, 28, 27, 2, 48, 9, 39, 22, 34, 2, 51, 18, 42, 24, 33, 2, 71, 2, 35, 34, 38, 20, 66, 2, 42, 28, 64, 2, 70, 2, 41, 44, 46

Offset: 1

Wesley Ivan Hurt, Oct 07 2021

nonn

For each divisor d of n, add n/d if d is prime, otherwise add 1. For example, a(9) = 5 can be found using its divisors 1,3,9 to get 1 + 9/3 + 1 = 5.

If p is prime, then a(p) = 2 since we have a(p) = tau(p) - omega(p) + p/p = 2 - 1 + 1 = 2.

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

Cf. A000005 (tau), A000010 (phi), A001221 (omega), A005171, A007503, A010051, A069359, A348203, A386438.

A348219(n) = sumdiv(n,d,(n/d)^isprime(d)); \\ Antti Karttunen, Nov 11 2021

A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A348219(n) = (A069359(n)+numdiv(n)-omega(n)); \\ Antti Karttunen, Nov 11 2021

a(n) = Sum_{d|n} (n/d)^c(d), where c is the prime characteristic (A010051).
a(n) = A000005(n) - A001221(n) + A069359(n).
a(prime(n)) = 2.
From Wesley Ivan Hurt, Jul 21 2025: (Start)
a(n) = Sum_{d|n} (c(d) + phi(d)*omega(n/d)), where c = A005171.
a(n) = A007503(n) - A386438(n). (End)

A322068 a(n) = (1/2)Sum_{p prime <= n} floor(n/p) floor(1 + n/p).

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 10, 11, 15, 18, 25, 26, 36, 37, 46, 54, 62, 63, 78, 79, 93, 103, 116, 117, 137, 142, 157, 166, 184, 185, 216, 217, 233, 247, 266, 278, 308, 309, 330, 346, 374, 375, 416, 417, 443, 467, 492, 493, 533, 540, 575, 595, 625, 626, 671, 687, 723, 745

Offset: 0

Daniel Suteu, Nov 25 2018

Partial sums of A069359.

Wikipedia, Prime Zeta Function

Cf. A000217, A000720, A013939, A013940, A069359.

with(numtheory): seq(add(i*pi(floor(n/i)), i=1..n), n=0..60); # Ridouane Oudra, Oct 16 2019

a[n_] := Module[{s=0, p=2}, While[p<=n, s += (Floor[n/p] * Floor[1 + n/p]); p=NextPrime[p]]; s]/2; Array[a, 100, 0] (* Amiram Eldar, Nov 25 2018 *)

a(n) = my(s=0); forprime(p=2, n, s+=(n\p)*(1+n\p)); s/2;

a(n) = sum(k=1, sqrtint(n), k*(k+1) * (primepi(n\k) - primepi(n\(k+1))))/2 + sum(k=1, n\(sqrtint(n)+1), if(isprime(k), (n\k)*(1+n\k), 0))/2;

a(n) ~ A085548 * n*(n+1)/2.
a(n) = Sum_{p prime <= n} A000217(floor(n/p)).
a(n) = (Sum_{k=1..floor(sqrt(n))} k*(k+1) * (pi(floor(n/k)) - pi(floor(n/(k+1)))) + Sum_{p prime <= floor(n/(1+floor(sqrt(n))))} floor(n/p)*floor(1+n/p))/2, where pi(x) is the prime-counting function (A000720).
a(n) = Sum_{i=1..n} i*pi(floor(n/i)), where pi(n) = A000720(n). - Ridouane Oudra, Oct 16 2019

A349337 Dirichlet inverse of A230593.

Original entry on oeis.org

1, -3, -4, 3, -6, 13, -8, -3, 4, 19, -12, -16, -14, 25, 25, 3, -18, -17, -20, -22, 33, 37, -24, 19, 6, 43, -4, -28, -30, -87, -32, -3, 49, 55, 49, 33, -38, 61, 57, 25, -42, -113, -44, -40, -29, 73, -48, -22, 8, -25, 73, -46, -54, 21, 73, 31, 81, 91, -60, 125, -62, 97, -37, 3, 85, -165, -68, -58, 97, -163, -72, -52

Offset: 1

Antti Karttunen, Nov 15 2021

sign

Coincides with A347084 on all squarefree numbers (A005117), but also on n=81, where a(81) = A347084(81) = 4. Question: Are there any other such numbers?

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

Cf. A005117, A069359, A230593.

Cf. also A347084, A349434, A349435.

up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA230593(n) = sumdiv(n, d, ((1==d)||isprime(d))*(n/d));
v349337 = DirInverseCorrect(vector(up_to,n,A230593(n)));
A349337(n) = v349337[n];

For n > 1, a(n) = -Sum_{d|n, 1A230593(d) * A349337(n/d).

A386438 a(n) = sigma(n) + omega(n) - n * Sum_{p|n, p prime} 1 / p.

Original entry on oeis.org

1, 3, 4, 6, 6, 9, 8, 12, 11, 13, 12, 20, 14, 17, 18, 24, 18, 26, 20, 30, 24, 25, 24, 42, 27, 29, 32, 40, 30, 44, 32, 48, 36, 37, 38, 63, 38, 41, 42, 64, 42, 58, 44, 60, 56, 49, 48, 86, 51, 60, 54, 70, 54, 77, 58, 86, 60, 61, 60, 109, 62, 65, 76, 96, 68, 86, 68, 90, 72, 88, 72, 137, 74, 77, 86, 100, 80, 100, 80, 132, 95, 85, 84, 145, 88, 89, 90, 130, 90, 144

Offset: 1

Wesley Ivan Hurt, Jul 21 2025

nonn

For each divisor d of n, add 1 if n/d is prime, else add d.

Cf. A000010 (phi), A000203 (sigma), A001221 (omega), A005171, A007503, A010051, A069359, A348219.

Table[Sum[d^(1 - PrimePi[n/d] + PrimePi[n/d - 1]), {d, Divisors[n]}], {n, 100}]

a(n) = Sum_{d|n} d^c(n/d), where c = A005171.
a(n) = Sum_{d|n} (d + c(d) - phi(d)*omega(n/d)), where c = A010051.
a(n) = A000203(n) + A001221(n) - A069359(n).
a(n) = A007503(n) - A348219(n).

A304404 If n = Product (p_j^k_j) then a(n) = Product (n/p_j^k_j).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 12, 1, 14, 15, 1, 1, 18, 1, 20, 21, 22, 1, 24, 1, 26, 1, 28, 1, 900, 1, 1, 33, 34, 35, 36, 1, 38, 39, 40, 1, 1764, 1, 44, 45, 46, 1, 48, 1, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 3600, 1, 62, 63, 1, 65, 4356, 1, 68, 69, 4900, 1, 72, 1, 74, 75

Offset: 1

Ilya Gutkovskiy, May 12 2018

nonn

			a(60) = a(2^2*3*5) = (60/2^2) * (60/3) * (60/5) = 15 * 20 * 12 = 3600.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Ilya Gutkovskiy, Logarithmic scatter plot of a(n) up to n=30000
Index entries for sequences computed from indices in prime factorization
Index entries for sequences computed from exponents in factorization of n

Cf. A000961 (positions of ones), A001221, A003557, A007774 (fixed points), A028234, A028236, A051119, A062509, A066504, A069359, A072195, A141809, A205959, A284600, A290480.

a[n_] := Times @@ (n/#[[1]]^#[[2]] & /@ FactorInteger[n]); Table[a[n], {n, 75}]
Table[n^(PrimeNu[n] - 1), {n, 75}]

A304404(n) = (n^(omega(n)-1)); \\ Antti Karttunen, Aug 06 2018

from sympy.ntheory.factor_ import primenu
def A304404(n): return int(n**(primenu(n)-1)) # Chai Wah Wu, Jul 12 2023

a(n) = n^(omega(n)-1), where omega() = A001221.

a(n) = A062509(n)/n.

A348203 a(n) = n - omega(n) + n * Sum_{p|n} 1/p.

Original entry on oeis.org

1, 2, 3, 5, 5, 9, 7, 11, 11, 15, 11, 20, 13, 21, 21, 23, 17, 31, 19, 32, 29, 33, 23, 42, 29, 39, 35, 44, 29, 58, 31, 47, 45, 51, 45, 64, 37, 57, 53, 66, 41, 80, 43, 68, 67, 69, 47, 86, 55, 83, 69, 80, 53, 97, 69, 90, 77, 87, 59, 119, 61, 93, 91, 95, 81, 124, 67, 104, 93, 126

Offset: 1

Wesley Ivan Hurt, Oct 06 2021

nonn

For 1 <= k <= n, if k is a prime divisor of n then add n/k, otherwise add 1. For example, a(6) = 9 since the values of k from 1 to 6 would be: 1 + 6/2 + 6/3 + 1 + 1 + 1 = 9.

If p is prime, then a(p) = p since we have a(p) = p - omega(p) + phi(1)*omega(p/1) + phi(p)*omega(p/p) = p - 1 + 1*1 + (p-1)*0 = p.

Cf. A000010 (phi), A001221 (omega), A010051, A069359.

Table[n - PrimeNu[n] + Sum[EulerPhi[k]*PrimeNu[n/k] (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 80}]

a(n) = Sum_{k=1..n} (n/k)^(c(k) * (1 - ceiling(n/k) + floor(n/k))), where c is the prime characteristic (A010051).
a(n) = n - A001221(n) + A069359(n).
a(prime(n)) = prime(n).

A370513 Complement of A323599.

Original entry on oeis.org

2, 5, 29, 39, 53, 59, 73, 95, 119, 123, 125, 129, 137, 145, 147, 149, 157, 159, 163, 173, 179, 191, 199, 207, 209, 213, 219, 221, 235, 251, 257, 263, 265, 269, 271, 279, 291, 293, 299, 303, 305, 325, 327, 329, 343, 345, 347, 359, 365, 367, 369, 375, 385, 395, 397

Offset: 1

Torlach Rush, Feb 20 2024

nonn

Terms of this sequence are not solutions of Sum_{d|k} A069359(d), k >= 1.
Proof that 2 is not a solution of Sum_{d|k} A069359(d), k >= 1: (Start)
If 2 is a solution then the only summands of the above are either (0,2) or (0,1,1).
(0,2) cannot be the only summands. If 2 is a summand then it is also a divisor of a(n) and A069359(2) = 1. If 2 is a summand then so must 1 be a summand.
(0,1,1) cannot be the only summands. There must exist an additional summand A069359(p_1*p_2) where p_1 and p_2 (primes) contribute to each 1 in (0,1,1).
(End)
To prove that 5 is not a solution of Sum_{d|k} A069359(d), k >= 1 we need to show that each of the following summands cannot exist: (0,5), (0,1,4), (0,1,2,2), (0,1,1,3), (0,1,1,1,2). (0,1,1,1,1,1). Following from the above proof this is elementary.

			2 is a term because it is not a solution of Sum_{d|k} A069359(d), k >= 1. See proof in Comments.

Cf. A069359, A323599.

cp's OEIS Frontend

A329353 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329352(i) = A329352(j) for all i, j.

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PARI

A342921 a(n) = A003415(A019565(n)).

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A346469 a(n) = A340070(A276086(n)).

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A348219 a(n) = tau(n) - omega(n) + n * Sum_{p|n, p prime} 1/p.

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A322068 a(n) = (1/2)*Sum_{p prime <= n} floor(n/p) * floor(1 + n/p).

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A349337 Dirichlet inverse of A230593.

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A386438 a(n) = sigma(n) + omega(n) - n * Sum_{p|n, p prime} 1 / p.

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A304404 If n = Product (p_j^k_j) then a(n) = Product (n/p_j^k_j).

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A322068 a(n) = (1/2)Sum_{p prime <= n} floor(n/p) floor(1 + n/p).