A013939 -id:A013939 - cp's OEIS frontend

A355538 Partial sum of A001221 (number of distinct prime factors) minus 1, ranging from 2 to n.

0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 5, 6, 6, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 14, 14, 14, 15, 16, 17, 18, 18, 19, 20, 21, 21, 23, 23, 24, 25, 26, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 35, 37, 37, 38, 39, 39, 40, 42, 42, 43, 44, 46, 46

Offset: 1

Gus Wiseman, Jul 23 2022

nonn

For initial terms up to 30 we have a(n) = Log_2 A355537(n).

The sum of the same range is A000096.
The product of the same range is A000142, Heinz number A070826.
For divisors (not just prime factors) we get A002541, also A006218, A077597.
A shifted variation is A013939.
The unshifted version is A022559, product A327486, w/o multiplicity A355537.
The ranges themselves are the rows of A131818 (shifted).
Partial sums of A297155 (shifted).
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A066843 gives partial sums of A000005.
Cf. A000720, A002110, A076610, A305054, A355538, A355731, A355733, A355741, A355744, A355745, A355746, A355747.

Table[Total[(PrimeNu[#]-1)&/@Range[2,n]],{n,1,100}]

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

A368613 a(n) = Sum_{k=2..n} pi(k-1) * floor(n/k).

Original entry on oeis.org

0, 0, 1, 3, 5, 9, 12, 18, 23, 29, 33, 44, 49, 58, 67, 79, 85, 100, 107, 123, 135, 147, 155, 179, 190, 204, 218, 238, 247, 273, 283, 306, 322, 339, 355, 388, 399, 418, 436, 468, 480, 514, 527, 555, 582, 604, 618, 663, 681, 711, 733, 764, 779, 819, 841, 881, 905, 930

Offset: 1

Wesley Ivan Hurt, Dec 31 2023

Cf. A000720 (pi), A013939, A046992, A368610, A368611, A368612.

Table[Sum[PrimePi[k - 1] Floor[n/k], {k, 2, n}], {n, 100}]

a(n) = A368611(n) - A013939(n).

A082287 a(1) = 1; for n > 1, n appears omega(n) times, where omega(n)=A001221(n) is the number of distinct prime factors of n, a(1)=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 15, 16, 17, 18, 18, 19, 20, 20, 21, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 30, 30, 31, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 38, 38, 39, 39, 40, 40, 41, 42, 42, 42, 43, 44, 44, 45, 45, 46, 46, 47

Offset: 1

Reinhard Zumkeller, Apr 07 2003

nonn

A027748(n) divides a(n) and a(n)=A027748(n) iff a(n) is prime; a(A013939(n)+1)=n.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A082288.

Join[{1},Flatten[Table[PadRight[{},PrimeNu[n],n],{n,2,50}]]] (* Harvey P. Dale, Jan 08 2020 *)

a(n)=if(n<0,0,t=1;while(sum(k=1,t,floor(t/prime(k)))Benoit Cloitre, Nov 08 2009

a(n) is the least k such that Sum_{p<=k} floor(k/p) >= n where p runs through the primes. - Benoit Cloitre, Nov 08 2009

A181967 Sum of the sizes of the normalizers of all prime order cyclic subgroups of the alternating group A_n.

Original entry on oeis.org

0, 0, 3, 24, 180, 1440, 12600, 120960, 1270080, 14515200, 179625600, 2634508800, 37362124800, 566658892800, 9807557760000, 167382319104000, 3023343138816000, 57621363351552000, 1155628453883904000, 25545471085854720000, 587545834974658560000, 13488008733331292160000

Offset: 1

Olivier Gérard, Apr 04 2012

nonn

The first 11 terms of this sequence are the same as A317527. - Andrew Howroyd, Jul 30 2018

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A181951 for the number of such subgroups.
Cf. A181966 is the symmetric group case.
Cf. A013939, A317527.

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

a:=function(n) local total, perm, g, p, k;
  total:= 0; g:= AlternatingGroup(n);
  for p in Filtered([2..n], IsPrime) do for k in [1..QuoInt(n,p)] do
     if p>2 or IsEvenInt(k) then
       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));
     fi;
  od; od;
  return total;
end; # Andrew Howroyd, Jul 30 2018

a(n)={n!*sum(p=2, n, if(isprime(p), if(p==2, n\4, n\p)))/2} \\ Andrew Howroyd, Jul 30 2018

a(n) = n! * (A013939(n) - floor((n + 2)/4)) / 2. - Andrew Howroyd, Jul 30 2018

Some incorrect conjectures removed by Andrew Howroyd, Jul 30 2018

Terms a(9) and beyond from Andrew Howroyd, Jul 30 2018

A241419 Number of numbers m <= n that have a prime divisor greater than sqrt(n) (i.e., A006530(m)>sqrt(n)).

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 4, 4, 2, 3, 4, 4, 5, 6, 7, 7, 8, 8, 9, 10, 11, 12, 13, 13, 9, 10, 10, 11, 12, 12, 13, 13, 14, 15, 16, 16, 17, 18, 19, 19, 20, 21, 22, 23, 23, 24, 25, 25, 19, 19, 20, 21, 22, 22, 23, 23, 24, 25, 26, 26, 27, 28, 28, 28, 29, 30, 31, 32, 33, 33, 34, 34, 35, 36, 36, 37, 38

Offset: 1

Michael De Vlieger, Aug 08 2014

nonn

Values of n that are squares of primes p^2 seem to reduce the value of a(p^2) from the value a(p^2 - 1). Example, a(24) = 13, a(25) = 9; a(120) = 70, a(121) = 60.

a(p^2) = a(p^2-1) - p + 1 if p is prime. If n is not the square of a prime, a(n) >= a(n-1).- Robert Israel, Aug 11 2014

			a(12) = 4, because there are four values of m = {5, 7, 10, 11} that have prime divisors that exceed sqrt(12) = 3.464... These prime divisors are {5, 7, 5, 11} respectively.

Michael De Vlieger, Table of n, a(n) for n = 1..5000
E. Naslund, The Average Largest Prime Factor, Integers, Vol. 13 (2013), A81. See "2. The Main Theorem."

Cf. A013939, A295084.

N:= 1000: # to get a(1) to a(N)
MF:= map(m -> max(numtheory:-factorset(m))^2,<($1..N)>): MF[1]:= 0:
seq(nops(select(m -> MF[m]>n, [$1..n])),n=1..N); # Robert Israel, Aug 11 2014

a241419[n_Integer] :=
Module[{f},
  f = Reap[For[m = 1, m <= n, m++,
     If[Max[First[Transpose[FactorInteger[m]]]] > Sqrt[n], Sow[m],
      False]]];
  If[Length[f[[2]]] == 0, Length[f[[2]]], Length[f[[2, 1]]]]]; a241419[120]

isok(i, n) = {my(f = factor(i)); my(sqrn = sqrt(n)); for (k=1, #f~, if ((p=f[k, 1]) && (p>sqrn) , return (1)););}
a(n) = sum(i=1, n, isok(i, n)); \\ Michel Marcus, Aug 11 2014

A241419(n) = my(r=0); forprime(p=sqrtint(n)+1,n, r+=n\p); r; \\ Max Alekseyev, Nov 14 2017

from math import isqrt
from sympy import primerange
def A241419(n): return int(sum(n//p for p in primerange(isqrt(n)+1,n+1))) # Chai Wah Wu, Oct 06 2024

a(n) = Sum_{prime p > sqrt(n)} floor(n/p). - Max Alekseyev, Nov 14 2017

A300671 Expansion of 1/(1 - Sum_{k>=1} x^prime(k)/(1 - x^prime(k))).

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 8, 15, 23, 40, 63, 108, 172, 290, 471, 782, 1280, 2119, 3474, 5741, 9432, 15557, 25590, 42180, 69413, 114371, 188276, 310136, 510637, 841045, 1384883, 2280831, 3755862, 6185457, 10185941, 16774695, 27624215, 45492412, 74916559, 123374127, 203172520, 334587577

Offset: 0

Ilya Gutkovskiy, Mar 11 2018

nonn

Invert transform of A001221.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
N. J. A. Sloane, Transforms

Cf. A001221, A013939, A112965, A129921, A180305, A293548, A300672.

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*nops(ifactors(i)[2]), i=1..n))
    end:
seq(a(n), n=0..42);  # Alois P. Heinz, Feb 11 2021

nmax = 42; CoefficientList[Series[1/(1 - Sum[x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 42; CoefficientList[Series[1/(1 - Sum[PrimeNu[k] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[PrimeNu[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 42}]

G.f.: 1/(1 - Sum_{k>=2} A001221(k)*x^k).

A308495 a(n) is the position of the first occurrence of prime(n) in A027748.

Original entry on oeis.org

2, 3, 5, 8, 13, 16, 22, 25, 32, 41, 45, 55, 62, 66, 73, 83, 94, 98, 109, 117, 120, 132, 138, 150, 166, 173, 177, 185, 188, 196, 224, 231, 243, 247, 267, 271, 284, 295, 303, 315, 327, 331, 353, 356, 364, 368, 394, 419, 426, 430, 439, 452, 456, 475, 487, 500

Offset: 1

Peter Dolland, Jun 01 2019

nonn

			For n = 5: a(5) = 13, A027748(13) = A000040(5) = 11.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000040, A001221, A013939, A027748, A082186.

-- expected to be part of A027748
a027748_list = concat (map a027748_row [1..])
minIdx [] _ = []
minIdx _ [] = []
minIdx (a:as) (b:bs)
    | a == b = 1 : (map succ (minIdx as bs))
    | otherwise = map succ (minIdx as (b:bs))
a308495_list = minIdx a027748_list a000040_list
a308495 n = a308495_list !! (n-1)

b:= proc(n) option remember; `if`(n=1, 1,
      b(n-1) +nops(ifactors(n)[2]))
    end:
a:= n-> b(ithprime(n)):
seq(a(n), n=1..60);  # Alois P. Heinz, Jun 06 2019

b[n_] := b[n] = If[n == 1, 1, b[n-1] + PrimeNu[n]];
a[n_] := b[Prime[n]];
Array[a, 60] (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)

a(n) = 1 + sum(k=1, prime(n), omega(k)); \\ Michel Marcus, Jun 05 2019

A027748(a(n)) = A000040(n).
a(n) = 1 + A013939(A000040(n)). - Charlie Neder, Jun 04 2019
a(n) = A082186(A000040(n)). - Alois P. Heinz, Jun 06 2019
a(n) = 1 + Sum_{k=1..n} floor(prime(n)/prime(k)). - Benoit Cloitre, Jan 11 2025

A342173 a(n) = Sum_{j=1..n-1} floor(prime(n)/prime(j)).

Original entry on oeis.org

0, 1, 3, 6, 11, 14, 20, 23, 30, 39, 43, 53, 60, 64, 71, 81, 92, 96, 107, 115, 118, 130, 136, 148, 164, 171, 175, 183, 186, 194, 222, 229, 241, 245, 265, 269, 282, 293, 301, 313, 325, 329, 351, 354, 362, 366, 392, 417, 424, 428, 437, 450, 454, 473, 485, 498, 511

Offset: 1

J. M. Bergot and Robert Israel, Mar 03 2021

nonn

a(n) is the sum of the quotients in integer division of prime(n) by all smaller primes.

			a(4) = floor(7/2) + floor(7/3) + floor(7/5) = 6.

Robert Israel, Table of n, a(n) for n = 1..10000
Lorenzo Sauras-Altuzarra, Some properties of the factors of Fermat numbers, Art Discrete Appl. Math. (2022).

Cf. A033955, A006093, A013939, A308495.

f:= proc(n) local t,i,s;
  t:= ithprime(n);
  add(floor(t/ ithprime(i)),i=1..n-1)
end proc:
map(f, [$1..100]);

Table[Sum[Floor[Prime[n]/Prime[j]],{j,n-1}],{n,64}] (* Stefano Spezia, Mar 04 2021 *)

a(n) = sum(j=1, n-1, prime(n)\prime(j)); \\ Michel Marcus, Mar 04 2021

a(n) = A308495(n) - 2. - Hugo Pfoertner, Mar 04 2021

a(n) = A013939(A006093(n)). - Flávio V. Fernandes, Jan 03 2025

A346111 a(n) = Sum_{primes p <=n} sigma(floor(n/p)).

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 12, 13, 14, 15, 24, 25, 23, 23, 30, 31, 35, 36, 44, 41, 37, 38, 61, 60, 48, 46, 59, 60, 71, 72, 79, 74, 63, 58, 95, 96, 79, 66, 95, 96, 102, 103, 110, 108, 98, 99, 142, 138, 114, 102, 116, 117, 136, 129, 152, 134, 110, 111, 191, 192, 154, 142

Offset: 1

Michel Marcus, Jul 05 2021

nonn

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

Cf. A000203, A013939 (with k instead of sigma(k)), A344672 (with phi instead).

a[n_] := Plus @@ DivisorSigma[1, Floor[n/Select[Range[n], PrimeQ]]]; Array[a, 100] (* Amiram Eldar, Jul 05 2021 *)

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

A082186 1 + sum of first n terms of A001221.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 15, 16, 18, 20, 21, 22, 24, 25, 27, 29, 31, 32, 34, 35, 37, 38, 40, 41, 44, 45, 46, 48, 50, 52, 54, 55, 57, 59, 61, 62, 65, 66, 68, 70, 72, 73, 75, 76, 78, 80, 82, 83, 85, 87, 89, 91, 93, 94, 97, 98, 100, 102, 103, 105, 108, 109, 111, 113

Offset: 1

Jon Perry, May 09 2003

Michael De Vlieger, Table of n, a(n) for n = 1..2000

Equals 1 + A013939.

with(numtheory): s := 1: for n from 1 to 150 do it := nops(ifactors(n)[2]): s := s+it: printf(`%d,`,s): od: # James Sellers, May 19 2003

a082186[n_] := Block[{t = Table[0, {n}], i},
For[i = 0, i < n, t[[i]] = 1 + Plus @@ PrimeNu /@ Range[i], i++]; t]; a082186[69] (* Michael De Vlieger, Dec 24 2014 *)

More terms from James Sellers, May 19 2003

cp's OEIS Frontend

A355538 Partial sum of A001221 (number of distinct prime factors) minus 1, ranging from 2 to n.

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A368613 a(n) = Sum_{k=2..n} pi(k-1) * floor(n/k).

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A082287 a(1) = 1; for n > 1, n appears omega(n) times, where omega(n)=A001221(n) is the number of distinct prime factors of n, a(1)=1.

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A181967 Sum of the sizes of the normalizers of all prime order cyclic subgroups of the alternating group A_n.

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A241419 Number of numbers m <= n that have a prime divisor greater than sqrt(n) (i.e., A006530(m)>sqrt(n)).

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A300671 Expansion of 1/(1 - Sum_{k>=1} x^prime(k)/(1 - x^prime(k))).

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A308495 a(n) is the position of the first occurrence of prime(n) in A027748.

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