A013939 -id:A013939 - cp's OEIS frontend

A380315 Denominator of sum of reciprocals of all prime divisors of all positive integers <= n.

1, 2, 6, 3, 15, 30, 210, 105, 35, 70, 770, 1155, 15015, 30030, 30030, 15015, 255255, 170170, 3233230, 1616615, 4849845, 9699690, 223092870, 111546435, 22309287, 44618574, 14872858, 7436429, 215656441, 6469693230, 200560490130, 100280245065, 100280245065

Offset: 1

Ilya Gutkovskiy, Jan 20 2025

Prime divisors counted without multiplicity.

Differs from A379370 first at n=15.

			0, 1/2, 5/6, 4/3, 23/15, 71/30, 527/210, 316/105, 117/35, 283/70, 3183/770, 5737/1155, 75736/15015, ...

Cf. A000720, A007947, A013939, A024924, A028235, A284650, A379370, A379368, A380314 (numerators).

Table[DivisorSum[n, 1/# &, PrimeQ[#] &], {n, 1, 33}] // Accumulate // Denominator
Table[Sum[Floor[n/Prime[k]]/Prime[k], {k, 1, n}], {n, 1, 33}] // Denominator
nmax = 33; CoefficientList[Series[1/(1 - x) Sum[x^Prime[k]/(Prime[k] (1 - x^Prime[k])), {k, 1, nmax}], {x, 0, nmax}], x] // Denominator // Rest

a(n) = my(vp=primes(primepi(n))); denominator(sum(k=1, #vp, (n\vp[k])/vp[k])); \\ Michel Marcus, Jan 26 2025

G.f. for fractions: (1/(1 - x)) * Sum_{k>=1} x^prime(k) / (prime(k)*(1 - x^prime(k))).

a(n) is the denominator of Sum_{k=1..pi(n)} floor(n/prime(k)) / prime(k).

A383752 Product of nonzero remainders n mod p, over all primes p < n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 6, 8, 3, 8, 10, 36, 24, 8, 30, 288, 420, 1920, 2268, 640, 270, 2880, 9240, 13824, 7560, 19200, 17820, 120960, 64064, 362880, 5054400, 1881600, 475200, 165888, 464100, 6386688, 4082400, 1228800, 2120580, 34836480, 23474880, 217728000

Offset: 1

Darío Clavijo, May 28 2025

nonn

Paolo Xausa, Table of n, a(n) for n = 1..1000

Cf. A000040, A024934, A013939, A102647, A309912.

A383752[n_] := Times @@ DeleteCases[Mod[n, Prime[Range[PrimePi[n - 2]]]], 0];
Array[A383752, 50] (* Paolo Xausa, Jun 05 2025 *)

a(n) = vecprod(select(x->(x!=0), apply(lift, apply(x->Mod(n, x), primes([2,n-1]))))); \\ Michel Marcus, May 28 2025

from sympy import primerange
def a(n):
    s = 1
    for p in primerange(0, n):
        if p > (n >> 1): s *= (n-p)
        elif (x:= n % p) > 0: s*= x
    return s
print([a(n) for n in range(1,41)])

a(p) = A102647(p) if p prime.

A145191 Numbers m such that Sum_{i=1..m} omega(i)^2 is divisible by m, where omega is A001221.

Original entry on oeis.org

1, 20, 68, 903, 3876, 3890, 19096, 19122, 19127, 110990, 111004, 111007, 111010, 111013, 774276, 774277, 774278, 774279, 774303, 774313, 774314, 774315, 6615593, 70607550, 70607559, 959878582, 959878737, 959878753, 959878836, 959878846, 959878888, 959878902, 959878914

Offset: 1

Ctibor O. Zizka, Oct 03 2008

nonn

If for some m is c square, then we have RootMeanSquare(omega(1),...,omega(n)) = c.

Cf. A001221, A063971, A013939, A140480.

With[{max = 10^5}, Position[Accumulate[PrimeNu[Range[max]]^2]/Range[max], ?IntegerQ] // Flatten] (* _Amiram Eldar, Sep 22 2024 *)

isok(m) = !frac(sum(i=1, m, omega(i)^2)/m); \\ Michel Marcus, Mar 15 2022

lista(nn) = {my(v = vector(nn, k, omega(k)^2)); print1(1, ", "); for (n=2, nn, v[n] += v[n-1]; if (! frac(v[n]/n), print1(n, ", ")););} \\ Michel Marcus, Mar 16 2022

listaa(nn) = {my(v = vector(nn, k, omega(k)^2)); print1(1, ", "); for (n=2, nn, v[n] += v[n-1]; if (! frac(v[n]/n), print1(n, ", "));); for (m=1, 100, last = v[nn]; v = vector(nn, k, omega(k+m*nn)^2); v[1] += last; for (n=2, nn, v[n] += v[n-1]; if (! frac(v[n]/(m*nn+n)), print1(n+m*nn, ", "));););} \\ Michel Marcus, Mar 16 2022

a(7)-a(9) from Michel Marcus, Mar 15 2022
a(10)-a(25) from Michel Marcus, Mar 16 2022
a(26)-a(33) from Amiram Eldar, Sep 22 2024

A304066 a(n) = Sum_{k=1..n} k*floor(n/prime(k)).

Original entry on oeis.org

0, 1, 3, 4, 7, 10, 14, 15, 17, 21, 26, 29, 35, 40, 45, 46, 53, 56, 64, 68, 74, 80, 89, 92, 95, 102, 104, 109, 119, 125, 136, 137, 144, 152, 159, 162, 174, 183, 191, 195, 208, 215, 229, 235, 240, 250, 265, 268, 272, 276, 285, 292, 308, 311, 319, 324, 334, 345, 362, 368, 386, 398, 404, 405, 414

Offset: 1

Ilya Gutkovskiy, May 05 2018

nonn

Partial sums of A066328.

Index entries for sequences computed from indices in prime factorization

Cf. A000040, A000720, A008472, A013939, A024916, A024924, A048803, A056239, A066328, A081401, A304038.

seq(add(k*floor(n/ithprime(k)),k=1..n),n=1..65); # Paolo P. Lava, May 14 2018

Table[Sum[k Floor[n/Prime[k]], {k, n}], {n, 65}]
nmax = 65; Rest[CoefficientList[Series[1/(1 - x) Sum[k x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]]
a[n_] := Plus @@ (PrimePi[#[[1]]] & /@ FactorInteger[n]); a[1] = 0; Accumulate[Table[a[n], {n, 65}]]

G.f.: (1/(1 - x))*Sum_{k>=1} k*x^prime(k)/(1 - x^prime(k)).
a(p^k) = a(p^k-1) + pi(p), where p is a prime and pi() = A000720.
a(n) = A056239(A048803(n)).

A332687 a(n) = Sum_{k=1..n} ceiling(n/prime(k)).

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 13, 15, 17, 19, 22, 24, 27, 29, 32, 35, 37, 39, 42, 44, 47, 50, 53, 55, 58, 60, 63, 65, 68, 70, 74, 76, 78, 81, 84, 87, 90, 92, 95, 98, 101, 103, 107, 109, 112, 115, 118, 120, 123, 125, 128, 131, 134, 136, 139, 142, 145, 148, 151, 153

Offset: 1

Ilya Gutkovskiy, Feb 19 2020

nonn

Cf. A000720, A001221, A006590, A013939, A082767, A083399.

Table[Sum[Ceiling[n/Prime[k]], {k, 1, n}], {n, 1, 60}]
Table[n + Sum[PrimeNu[k], {k, 1, n - 1}], {n, 1, 60}]
nmax = 60; CoefficientList[Series[x/(1 - x)^2 + (x/(1 - x)) Sum[x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
With[{nmax = 100}, Range[nmax] + Join[{0}, Accumulate[Table[PrimeNu[k], {k, 1, nmax - 1}]]]] (* Amiram Eldar, Sep 21 2024 *)

a(n) = sum(k=1, n, ceil(n/prime(k))); \\ Michel Marcus, Feb 21 2020

lista(nmax) = my(s = 1); for(n = 2, nmax, print1(s, ", "); s += omega(n-1) + 1); \\ Amiram Eldar, Sep 21 2024

G.f.: x/(1 - x)^2 + (x/(1 - x)) * Sum_{k>=1} x^prime(k) / (1 - x^prime(k)).
a(n) = n + Sum_{k=1..n-1} omega(k), where omega = A001221.
a(n) = n - omega(n) + Sum_{k=1..n} pi(floor(n/k)), where pi = A000720.
a(n) = n + A013939(n-1) for n >= 2. - Amiram Eldar, Sep 21 2024

A346423 Numbers m such that Sum_{k=1..m} omega(k) = sigma(m).

Original entry on oeis.org

11, 230, 52830, 160908, 6134334960

Offset: 1

Metin Sariyar, Jul 16 2021

Numbers k such that A013939(k) = sigma(k).

			The sum of number of distinct primes dividing numbers up to 11 is 1+1+1+1+2+1+1+1+2+1 = sigma(11), so 11 is a term.

Cf. A000203, A013939.

s=0;Do[s=s+PrimeNu[n];If[DivisorSigma[1,n]==s,Print[n]],{n,2,160908}]

a(5) from Martin Ehrenstein, Aug 22 2021

A380408 a(n) = Sum_{k>=0} floor(n/(2k)!).

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24, 25, 27, 28, 30, 31, 33, 34, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 74, 75, 77, 78, 80, 81, 83, 84, 86, 87, 89, 90, 92, 93, 95, 96, 98, 99, 101, 102, 104, 105, 107

Offset: 0

Akiva Weinberger, Jan 23 2025

nonn

Partial sum of A060832 except for the first term in the sum.
Congruent to A034968(n) mod 2. Therefore, the parity of a(n) is the parity of the n-th permutation of k elements (k>=n) in lexicographic order.
For even n, a(n) equals A059563(n/2) whenever cosh(1)*n - a(n) < 1. The first time this fails is n=70, as a(70)=107 but A059563(35)=108. For small n, such failures appear to be very rare; however, the asymptotic density of these failures approaches 1.

Cf. A060832, A034968, A013936, A013939, A038663.

a(n) = round(sumpos(k=0, n\(2*k)!)); \\ Michel Marcus, Jan 24 2025

a(n) = cosh(1)*n - f(n) where f(n) = Sum_{k>=0} fract(n/(2k)!). Here, fract() is the fractional part. The error term f(n) is unbounded above, and the greatest lower bound is 0 (even excluding n=0). The first values for which f(n) > s for s=1,2,3 are f(13)=1.06005, f(407) = 2.03382, and f(22319) = 3.01669. The error is almost periodic: for large m, f(n) is approximately f(n+(2m)!). If n is odd, f(n) > 1/2. f(n) alternately rises and descends, that is, f(2*n)f(2*n+2) for all n.

cp's OEIS Frontend

A380315 Denominator of sum of reciprocals of all prime divisors of all positive integers <= n.

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A383752 Product of nonzero remainders n mod p, over all primes p < n.

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A145191 Numbers m such that Sum_{i=1..m} omega(i)^2 is divisible by m, where omega is A001221.

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A304066 a(n) = Sum_{k=1..n} k*floor(n/prime(k)).

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A332687 a(n) = Sum_{k=1..n} ceiling(n/prime(k)).

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A346423 Numbers m such that Sum_{k=1..m} omega(k) = sigma(m).

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A380408 a(n) = Sum_{k>=0} floor(n/(2k)!).

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