A013939 -id:A013939 - cp's OEIS frontend

A115228 Nonsquarefree numbers n such that 2n+1 is also nonsquarefree (A013929).

4, 12, 24, 40, 49, 60, 76, 84, 112, 121, 144, 148, 162, 171, 175, 180, 184, 212, 220, 256, 264, 292, 312, 328, 364, 387, 400, 412, 416, 420, 423, 436, 472, 480, 490, 508, 512, 544, 580, 612, 616, 625, 637, 652, 684, 688, 712, 722, 724, 760, 796, 808, 812

Offset: 1

Don Reble, Mar 05 2006

For any distinct primes p, q with q odd, contains all n such that n == 0 (mod p^2) and n == -1/2 (mod q^2). - Robert Israel, Oct 21 2016

			24 is in the sequence because 2^2 divides 24 and 7^2 divides 24*2 + 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A013939, A065474, A115170.

select(n -> not numtheory:-issqrfree(n) and not numtheory:-issqrfree(2*n+1), [$1..2000]); # Robert Israel, Oct 21 2016

fQ[n_] := ! SquareFreeQ[n] && ! SquareFreeQ[2 n + 1]; Select[Range[1000], fQ] (* Robert G. Wilson v, Oct 21 2016 *)

isok(n) = !issquarefree(n) && ! issquarefree(2*n+1); \\ Michel Marcus, Oct 22 2016

a(n) ~ n/(1 - 14/Pi^2 + 3*k/2 ) as n -> infinity, where k is the Feller-Tornier constant (A065474). - Robert Israel, Oct 21 2016

Corrected by Zak Seidov, Oct 21 2016

A181966 Sum of the sizes of normalizers of all prime order cyclic subgroups of the symmetric group S_n.

Original entry on oeis.org

0, 2, 12, 72, 480, 4320, 35280, 322560, 3265920, 39916800, 479001600, 6706022400, 93405312000, 1482030950400, 24845812992000, 418455797760000, 7469435990016000, 147254595231744000, 2919482409811968000, 63255452212592640000, 1430546380807864320000

Offset: 1

Olivier Gérard, Apr 04 2012

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
Math.stackexchange.com, Normalizer of the cyclic group in S_n

Cf. A181954 for the number of such subgroups.

Cf. A013939, A181967.

List([1..7], n->Sum(Filtered( ConjugacyClassesSubgroups( SymmetricGroup(n)), x->IsPrime( Size( Representative(x))) ), x->Size(x)*Size( Normalizer( SymmetricGroup(n), Representative(x))) )); # Andrew Howroyd, Jul 30 2018

a:=function(n) local total, perm, g, p, k;
  total:= 0; g:= SymmetricGroup(n);
  for p in Filtered([2..n], IsPrime) do for k in [1..QuoInt(n,p)] do
     perm:=PermList(List([0..p*k-1], i->i - (i mod p) + ((i + 1) mod p) + 1));
     total:=total + Size(Normalizer(g, perm)) * Factorial(n) / (p^k * (p-1) * Factorial(k) * Factorial(n-k*p));
  od; od;
  return total;
end; # Andrew Howroyd, Jul 30 2018

a(n)={n!*sum(p=2, n, if(isprime(p), n\p))} \\ Andrew Howroyd, Jul 30 2018

a(n) = n! * A013939(n). - Andrew Howroyd, Jul 30 2018

Terms a(8) and beyond from Andrew Howroyd, Jul 30 2018

A303482 Numbers k such that the average of all distinct prime factors of all positive integers <= k is an integer.

Original entry on oeis.org

2, 5, 81, 10742, 10130527, 1041972864, 23292549600

Offset: 1

Ilya Gutkovskiy, Apr 24 2018

Numbers k such that A013939(k)|A024924(k).

			5 is in the sequence because the distinct prime factors of 2, 3, 4, and 5 are 2, 3, 2 and 5 respectively and their average (2 + 3 + 2 + 5) / 4 = 3 is an integer. - _David A. Corneth_, Apr 26 2018

Eric Weisstein's World of Mathematics, Distinct Prime Factors

Cf. A001221, A008472, A013939, A024924, A078174, A078175, A226647, A284755, A303480.

s = t = 0; k = 2; lst = {}; While[k < 1000000000, p = #[[1]] & /@ FactorInteger@ k; s = s + Plus @@ p; t = t + Length@ p; If[ Mod[s, t] == 0, AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Apr 26 2018 *)

a(5) from Daniel Suteu, Apr 24 2018

a(6)-a(7) from Giovanni Resta, Apr 26 2018

A309912 a(n) = Product_{p prime, p <= n} floor(n/p).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 6, 6, 8, 12, 30, 30, 48, 48, 112, 210, 240, 240, 324, 324, 480, 840, 1848, 1848, 2304, 2880, 6240, 7020, 10080, 10080, 14400, 14400, 15360, 25344, 53856, 78540, 90720, 90720, 191520, 311220, 374400, 374400, 508032, 508032, 709632, 855360, 1788480, 1788480

Offset: 0

Ilya Gutkovskiy, Aug 22 2019

nonn

Product of exponents of prime factorization of A048803 (squarefree factorials).

			A048803(14) = 1816214400 = 2^7 * 3^4 * 5^2 * 7^2 * 11 * 13 so a(14) = 7 * 4 * 2 * 2 * 1 * 1 = 112.

Cf. A000040, A000720, A005361, A010786, A013939, A048803, A135291.

a:= n-> mul(floor(n/p), p=select(isprime, [$2..n])):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 23 2019

Table[Product[Floor[n/Prime[k]], {k, 1, PrimePi[n]}], {n, 0, 47}]

from math import prod
from sympy import primerange
def A309912(n): return prod(n//p for p in primerange(n)) # Chai Wah Wu, Jun 02 2025

a(n) = Product_{k=1..A000720(n)} floor(n/A000040(k)).

a(n) = A005361(A048803(n)).

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

A344672 a(n) = Sum_{primes p <=n} phi(floor(n/p)).

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 5, 5, 6, 8, 9, 7, 8, 12, 15, 13, 14, 14, 15, 13, 18, 24, 25, 17, 19, 27, 29, 23, 24, 22, 23, 23, 30, 38, 44, 28, 29, 41, 50, 38, 39, 35, 36, 34, 38, 50, 51, 37, 41, 51, 60, 52, 53, 49, 57, 49, 62, 78, 79, 43, 44, 66, 72, 58, 68, 68, 69, 65, 78, 78

Offset: 1

Michel Marcus, Jul 05 2021

nonn

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A000010, A013939 (with k instead of phi(k)), A346111 (with sigma instead).

Cf. A317625.

a[n_] := Plus @@ EulerPhi[Floor[n/Select[Range[n], PrimeQ]]]; Array[a, 100] (* Amiram Eldar, Jul 05 2021 *)

a(n) = my(s=0); forprime(p=2, n, s+=eulerphi(n\p)); s;

A380314 Numerator of sum of reciprocals of all prime divisors of all positive integers <= n.

Original entry on oeis.org

0, 1, 5, 4, 23, 71, 527, 316, 117, 283, 3183, 5737, 75736, 170777, 186793, 100904, 1730383, 1295397, 24782713, 13522987, 42878411, 91488457, 2113934201, 1149922463, 234446350, 494634185, 169835681, 89698402, 2608690087, 84946052281, 2639797313941, 1370038779503, 1412581913773

Offset: 1

Ilya Gutkovskiy, Jan 20 2025

Prime divisors counted without multiplicity.

			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, ...

Robert Israel, Table of n, a(n) for n = 1..2346

Cf. A000720, A007947, A013939, A024451, A024924, A028235, A284648, A379367, A380315 (denominators).

N:= 100: # for a(1) .. a(N)
P:= select(isprime,[$1..N]):
f:= proc(n) local k;
  numer(add(floor(n/P[k])/P[k],k=1..numtheory:-pi(n)))
end proc:
map(f, [$1..N]); # Robert Israel, Jan 26 2025

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

a(n) = my(vp=primes(primepi(n))); numerator(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 numerator of Sum_{k=1..pi(n)} floor(n/prime(k)) / prime(k).

A064182 Sum_{k <= 10^n} number of distinct primes dividing k (A001221).

Original entry on oeis.org

0, 11, 171, 2126, 24300, 266400, 2853708, 30130317, 315037281, 3271067968, 33787242719, 347589015681, 3564432632541, 36457601891708, 372096179850464, 3790896863469849, 38562555830676602, 391760068087338367, 3975397006170581823, 40066272402579605194

Offset: 0

Robert G. Wilson v, Sep 20 2001

nonn

This is just a subsequence of A013939. - N. J. A. Sloane, Jul 16 2011

Cf. A001221, A013939.

s = 0; k = 2; Do[ While[ k <= 10^n, s = s + PrimeNu@ k; k++ ]; Print[ s], {n, 8}]

a(n)=sum(k=1,10^n,omega(k)) \\ Charles R Greathouse IV, Jul 13 2011

a(n)=sum(k=1,10^n\2,primepi(10^n\k)) \\ Charles R Greathouse IV, Jul 13 2011

a(n) = Sum_{k, 1, limit}, PrimePi(10^n/k); which seems to be about 10^n/2.
On the contrary, I guess that a(n) ~ 10^n * log n. - Charles R Greathouse IV, Jul 13 2011
a(n) ~ 10^n * (0.26149721284764278375 + log(log(10^n))). - Hiroaki Yamanouchi, Jul 10 2014

a(11)-a(13) from Giovanni Resta, Oct 26 2012
a(14)-a(16) from Hiroaki Yamanouchi, Jun 29 2014
a(17) from Hiroaki Yamanouchi, Jul 10 2014
a(18) from Henri Lifchitz, Aug 25 2014
a(19) from Henri Lifchitz, Dec 17 2017

A069470 a(n) = Sum_{k>=1} floor(n/(k*(k+1)/2)).

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 9, 10, 11, 13, 15, 16, 19, 20, 21, 24, 25, 26, 29, 30, 32, 35, 36, 37, 40, 41, 42, 44, 46, 47, 52, 53, 54, 56, 57, 58, 62, 63, 64, 66, 68, 69, 73, 74, 75, 79, 80, 81, 84, 85, 87, 89, 90, 91, 94, 96, 98, 100, 101, 102, 107, 108, 109, 112, 113, 114, 118

Offset: 0

Henry Bottomley, Mar 25 2002

nonn

The summation has floor(1/2 + sqrt(2*n)) = A002024(n) nonzero terms. - Enrique Pérez Herrero, Apr 05 2010

			a(11) = floor(11/1) + floor(11/3) + floor(11/6) + floor(11/10) + floor(11/15) + ... = 11 + 3 + 1 + 1 + 0 + ... = 16.

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A000217, A006218, A007862, A013936, A013937, A013938, A013939.

Cf. A002024, A004275. - Enrique Pérez Herrero, Apr 05 2010

[(&+[Floor(n/(k*(k+1)/2)): k in [1..100]]): n in [0..30]]; // G. C. Greubel, May 23 2018

A069470[n_]:=Sum[Floor[(2*n)/(k*(1 + k))], {k, 1, Floor[1/2 + Sqrt[2*n]]}] (* Enrique Pérez Herrero, Apr 05 2010 *)

for(n=0, 30, print1(sum(k=1, 100, floor(n/(k*(k+1)/2))), ", ")) \\ G. C. Greubel, May 23 2018

a(n) = a(n-1) + A007862(n).
It appears that limit((sum(floor((1/2)*n/(k*(k+1))), k=1..n))/n, n=infinity) = 1/2. - Stephen Crowley, Aug 12 2009
From Enrique Pérez Herrero, Apr 05 2010: (Start)
a(n) <= floor((2*n^2)/(1 + n)) = A004275(n).
a(n) <= floor((2*n*floor((1 + 2*sqrt(2*n))/2))/(1+floor((1+2*sqrt(2*n))/2))). (End)
G.f.: (1/(1 - x)) * Sum_{k>=1} x^(k*(k+1)/2)/(1 - x^(k*(k+1)/2)). - Ilya Gutkovskiy, Jul 11 2019

A095233 a(n) = a(n-1) + Sum(floor(n/p): p prime), a(1) = 1.

Original entry on oeis.org

1, 2, 4, 7, 11, 17, 24, 32, 41, 52, 64, 78, 93, 110, 129, 149, 170, 193, 217, 243, 271, 301, 332, 365, 399, 435, 472, 511, 551, 594, 638, 683, 730, 779, 830, 883, 937, 993, 1051, 1111, 1172, 1236, 1301, 1368, 1437, 1508, 1580, 1654, 1729, 1806, 1885

Offset: 1

Reinhard Zumkeller, Jun 22 2004

nonn

a(n+1) - a(n) = A013939(n+1).

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001221.

A013939[n_] := Sum[Floor[n/Prime[k]], {k, 1, n}]; RecurrenceTable[{a[n] == a[n - 1] + A013939[n], a[1] == 1}, a, {n, 1, 50}] (* G. C. Greubel, May 20 2017 *)

cp's OEIS Frontend

A115228 Nonsquarefree numbers n such that 2n+1 is also nonsquarefree (A013929).

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A181966 Sum of the sizes of normalizers of all prime order cyclic subgroups of the symmetric group S_n.

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A303482 Numbers k such that the average of all distinct prime factors of all positive integers <= k is an integer.

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A309912 a(n) = Product_{p prime, p <= n} floor(n/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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A344672 a(n) = Sum_{primes p <=n} phi(floor(n/p)).

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A380314 Numerator of sum of reciprocals of all prime divisors of all positive integers <= n.

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