A097974 -id:A097974 - cp's OEIS frontend

A108274 Sum of the first 10^n terms in A097974. a(n) = sum_{m=1..10^n} t(m), where t(m) is the sum of the prime divisors of m that are less than or equal to sqrt(m).

0, 11, 327, 7714, 184680, 4617253, 118697919, 3149768778, 85356405077, 2357169671137, 66097467843823, 1875931900135854, 53804720498131760, 1556256544987695973, 45343922927650954928, 1329347125287604758708, 39180941384720954859005

Offset: 0

Ryan Propper, Jul 24 2005

nonn

Does a(n+1)/a(n) converge?

			The first 10^2 terms in A097974 sum to 327, so a(2) = 327.

Hiroaki Yamanouchi, Table of n, a(n) for n = 0..19

Cf. A097974.

s = 0; k = 1; Do[s += Plus @@ Select[Select[Divisors[n], PrimeQ], #<=Sqrt[n] &]; If[n == k, Print[s]; s = 0; k *= 10], {n, 1, 10^7}]

a(n) = sum(m=1, 10^n, sumdiv(m, d, d*isprime(d)*(d<=sqrt(m)))); \\ Michel Marcus, Jul 07 2014

a(2)-a(7) and the example corrected and a(8)-a(16) from Hiroaki Yamanouchi, Jul 07 2014

A063962 Number of distinct prime divisors of n that are <= sqrt(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 0, 2, 1, 2, 1, 1, 0, 2, 1, 2, 1, 1, 0, 3, 0, 1, 2, 1, 1, 2, 0, 1, 1, 3, 0, 2, 0, 1, 2, 1, 1, 2, 0, 2, 1, 1, 0, 3, 1, 1, 1, 1, 0, 3, 1, 1, 1, 1, 1, 2, 0, 2, 1, 2, 0, 2, 0, 1, 3

Offset: 1

Reinhard Zumkeller, Sep 04 2001

nonn

For all primes p: a(p) = 0 (not marked) and for k > 1 a(p^k) = 1.
a(1) = 0 and for n > 0 a(n) is the number of marks when applying the sieve of Eratosthenes where a stage for prime p starts at p^2.
If we define a divisor d|n to be inferior if d <= n/d, then inferior divisors are counted by A038548 and listed by A161906. This sequence counts inferior prime divisors. - Gus Wiseman, Feb 25 2021

			a(33) = a(3*11) = 1, as 3^2 = 9 < 33 and 11^2 = 121 > 33.
From _Gus Wiseman_, Feb 25 2021: (Start)
The a(n) inferior prime divisors (columns) for selected n:
n =  3  8  24  3660  390  3570 87780
   ---------------------------------
    {}  2   2     2    2     2     2
            3     3    3     3     3
                  5    5     5     5
                      13     7     7
                            17    11
                                  19
(End)

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A055399, A001221.
Cf. A027748, A063962.
Zeros are at indices A008578.
The divisors are listed by A161906 and add up to A097974.
Dominates A333806 (the strictly inferior version).
The superior version is A341591.
The strictly superior version is A341642.
A001221 counts prime divisors, with sum A001414.
A033677 selects the smallest superior divisor.
A038548 counts inferior divisors.
A063538/A063539 have/lack a superior prime divisor.
A161908 lists superior divisors.
A207375 lists central divisors.
A217581 selects the greatest inferior prime divisor.
A341676 lists the unique superior prime divisors.
- Inferior: A033676, A066839, A069288, A072499, A333749, A333750.
- Superior: A051283, A059172, A070038, A072500, A116883, A341592, A341675.
- Strictly Inferior: A056924, A060775, A070039, A333805, A341596, A341674.
- Strictly Superior: A056924, A140271, A238535, A341594, A341595, A341673.
Cf. A000005, A001248, A005117, A006530, A020639, A048098, A064052, A341643.

a063962 n = length [p | p <- a027748_row n, p ^ 2 <= n]
-- Reinhard Zumkeller, Apr 05 2012

with(numtheory): a:=proc(n) local c,F,f,i: c:=0: F:=factorset(n): f:=nops(F): for i from 1 to f do if F[i]^2<=n then c:=c+1 else c:=c: fi od: c; end: seq(a(n),n=1..105); # Emeric Deutsch

Join[{0},Table[Count[Transpose[FactorInteger[n]][[1]],?(#<=Sqrt[n]&)],{n,2,110}]] (* _Harvey P. Dale, Mar 26 2015 *)

{ for (n=1, 1000, f=factor(n)~; a=0; for (i=1, length(f), if (f[1, i]^2<=n, a++, break)); write("b063962.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 04 2009

G.f.: Sum_{k>=1} x^(prime(k)^2) / (1 - x^prime(k)). - Ilya Gutkovskiy, Apr 04 2020

a(A002110(n)) = n for n > 2. - Gus Wiseman, Feb 25 2021

Revised definition from Emeric Deutsch, Jan 31 2006

A217581 Largest prime divisor of n <= sqrt(n), 1 if n is prime or 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 3, 5, 2, 3, 2, 1, 5, 1, 2, 3, 2, 5, 3, 1, 2, 3, 5, 1, 3, 1, 2, 5, 2, 1, 3, 7, 5, 3, 2, 1, 3, 5, 7, 3, 2, 1, 5, 1, 2, 7, 2, 5, 3, 1, 2, 3, 7, 1, 3, 1, 2, 5, 2, 7, 3, 1, 5, 3, 2, 1, 7, 5, 2, 3

Offset: 1

Peter Luschny, Mar 21 2013

nonn

If we define a divisor d|n to be inferior if d <= n/d, then inferior divisors are counted by A038548 and listed by A161906. This sequence selects the greatest inferior prime divisor of n. - Gus Wiseman, Apr 06 2021

			From _Gus Wiseman_, Apr 06 2021: (Start)
The sequence selects the greatest element (or 1 if empty) of each of the following sets of strictly superior divisors:
   1:{}     16:{2}      31:{}     46:{2}
   2:{}     17:{}       32:{2}    47:{}
   3:{}     18:{2,3}    33:{3}    48:{2,3}
   4:{2}    19:{}       34:{2}    49:{7}
   5:{}     20:{2}      35:{5}    50:{2,5}
   6:{2}    21:{3}      36:{2,3}  51:{3}
   7:{}     22:{2}      37:{}     52:{2}
   8:{2}    23:{}       38:{2}    53:{}
   9:{3}    24:{2,3}    39:{3}    54:{2,3}
  10:{2}    25:{5}      40:{2,5}  55:{5}
  11:{}     26:{2}      41:{}     56:{2,7}
  12:{2,3}  27:{3}      42:{2,3}  57:{3}
  13:{}     28:{2}      43:{}     58:{2}
  14:{2}    29:{}       44:{2}    59:{}
  15:{3}    30:{2,3,5}  45:{3,5}  60:{2,3,5}
(End)

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A033676.
Positions of first appearances are 1 and A001248.
These divisors are counted by A063962.
These divisors add up to A097974.
The smallest prime factor of the same type is A107286.
A strictly superior version is A341643.
A superior version is A341676.
A038548 counts superior (or inferior) divisors.
A048098 lists numbers without a strictly superior prime divisor.
A056924 counts strictly superior (or strictly inferior) divisors.
A063538/A063539 have/lack a superior prime divisor.
A140271 selects the smallest strictly superior divisor.
A161906 lists inferior divisors.
A207375 lists central divisors.
A341591 counts superior prime divisors.
A341642 counts strictly superior prime divisors.
A341673 lists strictly superior divisors.
- Inferior: A066839, A069288, A333749, A333750.
- Superior: A033677, A051283, A059172, A070038, A116882, A116883, A161908, A341592, A341593, A341675.
- Strictly Inferior: A060775, A333805, A333806, A341596, A341674.
- Strictly Superior: A238535, A341594, A341595, A341644, A341645, A341646.
Cf. A000005, A001055, A001221, A001222, A001248, A001414, A006530, A020639, A064052.

A217581 := n -> `if`(isprime(n) or n=1, 1, max(op(select(i->i^2<=n, numtheory[factorset](n)))));

Table[If[n == 1 || PrimeQ[n], 1, Select[Transpose[FactorInteger[n]][[1]], # <= Sqrt[n] &][[-1]]], {n, 100}] (* T. D. Noe, Mar 25 2013 *)

a(n) = {my(m=1); foreach(factor(n)[,1], d, if(d^2 <= n, m=max(m,d))); m} \\ Andrew Howroyd, Oct 11 2023

A098002 Sum of squares of distinct prime divisors p of n, where each p <= sqrt(n).

Original entry on oeis.org

0, 0, 0, 4, 0, 4, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 25, 4, 9, 4, 0, 38, 0, 4, 9, 4, 25, 13, 0, 4, 9, 29, 0, 13, 0, 4, 34, 4, 0, 13, 49, 29, 9, 4, 0, 13, 25, 53, 9, 4, 0, 38, 0, 4, 58, 4, 25, 13, 0, 4, 9, 78, 0, 13, 0, 4, 34, 4, 49, 13, 0, 29, 9, 4, 0, 62, 25, 4, 9, 4, 0, 38, 49

Offset: 1

Leroy Quet, Sep 08 2004

nonn

			2 and 3 are the distinct prime divisors of 12 and both 2 and 3 are <= sqrt(12), so a(12) = 2^2 + 3^2 = 13.

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

Cf. A097974.

ssdpd[n_]:=Total[Select[Transpose[FactorInteger[n]][[1]],#<=Sqrt[n]&]^2]; Join[{0},Array[ssdpd,90,2]] (* Harvey P. Dale, Mar 11 2013 *)

a(n) = sumdiv(n, d, isprime(d)*(d^2<=n)*d^2); \\ Michel Marcus, Dec 22 2017

G.f.: Sum_{k>=1} prime(k)^2 * x^(prime(k)^2) / (1 - x^prime(k)). - Ilya Gutkovskiy, Aug 19 2021

More terms from John W. Layman, Sep 14 2004

A333808 Sum of distinct prime divisors of n that are < sqrt(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0, 5, 0, 2, 3, 2, 0, 5, 0, 2, 3, 2, 0, 5, 0, 2, 3, 2, 0, 10, 0, 2, 3, 2, 5, 5, 0, 2, 3, 7, 0, 5, 0, 2, 8, 2, 0, 5, 0, 7, 3, 2, 0, 5, 5, 9, 3, 2, 0, 10, 0, 2, 10, 2, 5, 5, 0, 2, 3, 14, 0, 5, 0, 2, 8, 2, 7, 5, 0, 7, 3, 2, 0, 12, 5, 2, 3, 2, 0, 10

Offset: 1

Ilya Gutkovskiy, Apr 05 2020

nonn

Cf. A008472, A070039, A097974, A333806, A333807.

Table[DivisorSum[n, # &, # < Sqrt[n] && PrimeQ[#] &], {n, 1, 90}]
nmax = 90; CoefficientList[Series[Sum[Prime[k] x^(Prime[k] (Prime[k] + 1))/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

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

A333751 Sum of nonprime divisors of n that are <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 11, 1, 1, 1, 5, 1, 7, 1, 5, 1, 1, 1, 11, 1, 1, 1, 5, 1, 7, 1, 5, 1, 1, 1, 11, 1, 1, 1, 13, 1, 7, 1, 5, 1, 1, 1, 19, 1, 1, 1, 5, 1, 7, 1, 13, 10, 1, 1, 11, 1, 1, 1, 13, 1, 16

Offset: 1

Ilya Gutkovskiy, Apr 03 2020

nonn

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

Cf. A018252, A023890, A066839, A069289, A069293, A097974, A333748, A333752, A333753.

f:= proc(n) convert(select(t -> not isprime(t) and t^2 <= n, numtheory:-divisors(n)),`+`) end proc:
map(f, [$1..100]); # Robert Israel, Sep 12 2024

Table[DivisorSum[n, # &, # <= Sqrt[n] && !PrimeQ[#] &], {n, 1, 90}]
nmax = 90; CoefficientList[Series[Sum[Boole[!PrimeQ[k]] k x^(k^2)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, if ((d^2<=n) && !isprime(d), d)); \\ Michel Marcus, Apr 03 2020

G.f.: Sum_{k>=1} A018252(k) * x^(A018252(k)^2) / (1 - x^A018252(k)).

A333753 Sum of prime power divisors of n that are <= sqrt(n).

Original entry on oeis.org

0, 0, 0, 2, 0, 2, 0, 2, 3, 2, 0, 5, 0, 2, 3, 6, 0, 5, 0, 6, 3, 2, 0, 9, 5, 2, 3, 6, 0, 10, 0, 6, 3, 2, 5, 9, 0, 2, 3, 11, 0, 5, 0, 6, 8, 2, 0, 9, 7, 7, 3, 6, 0, 5, 5, 13, 3, 2, 0, 14, 0, 2, 10, 14, 5, 5, 0, 6, 3, 14, 0, 17, 0, 2, 8, 6, 7, 5, 0, 19, 12, 2, 0, 16, 5, 2, 3, 14, 0, 19

Offset: 1

Ilya Gutkovskiy, Apr 03 2020

nonn

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

Cf. A023889, A066839, A069289, A069293, A097974, A246655, A333750, A333751, A333752.

f:= proc(n) local F,i,j,t;
  F:= ifactors(n)[2];
  t:= 0;
  for i from 1 to nops(F) do
    j:= min(F[i,2],ilog[F[i,1]^2](n));
    t:= t + (F[i,1]^j-1)*F[i,1]/(F[i,1]-1)
  od;
  t
end proc:
map(f, [$1..100]); # Robert Israel, Feb 15 2023

Table[DivisorSum[n, # &, # <= Sqrt[n] && PrimePowerQ[#] &], {n, 1, 90}]
nmax = 90; CoefficientList[Series[Sum[Boole[PrimePowerQ[k]] k x^(k^2)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, if ((d^2<=n) && isprimepower(d), d)); \\ Michel Marcus, Apr 03 2020

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

A333752 Sum of squarefree divisors of n that are <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 3, 4, 3, 1, 6, 1, 3, 4, 3, 1, 6, 1, 3, 4, 3, 1, 6, 6, 3, 4, 3, 1, 11, 1, 3, 4, 3, 6, 12, 1, 3, 4, 8, 1, 12, 1, 3, 9, 3, 1, 12, 8, 8, 4, 3, 1, 12, 6, 10, 4, 3, 1, 17, 1, 3, 11, 3, 6, 12, 1, 3, 4, 15, 1, 12, 1, 3, 9, 3, 8, 12, 1, 8, 4, 3, 1, 19, 6, 3, 4, 3, 1, 17

Offset: 1

Ilya Gutkovskiy, Apr 03 2020

nonn

Cf. A005117, A048250, A066839, A069289, A069293, A097974, A333749, A333751, A333753.

Table[DivisorSum[n, # &, # <= Sqrt[n] && SquareFreeQ[#] &], {n, 1, 90}]
nmax = 90; CoefficientList[Series[Sum[MoebiusMu[k]^2 k x^(k^2)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, if ((d^2<=n) && issquarefree(d), d)); \\ Michel Marcus, Apr 03 2020

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

A097975 a(n) is the prime divisor of n which is >= sqrt(n), or 0 if there is no such prime divisor.

Original entry on oeis.org

0, 2, 3, 2, 5, 3, 7, 0, 3, 5, 11, 0, 13, 7, 5, 0, 17, 0, 19, 5, 7, 11, 23, 0, 5, 13, 0, 7, 29, 0, 31, 0, 11, 17, 7, 0, 37, 19, 13, 0, 41, 7, 43, 11, 0, 23, 47, 0, 7, 0, 17, 13, 53, 0, 11, 0, 19, 29, 59, 0, 61, 31, 0, 0, 13, 11, 67, 17, 23, 0, 71, 0, 73, 37, 0, 19, 11, 13, 79, 0, 0, 41, 83

Offset: 1

Leroy Quet, Sep 07 2004

nonn

Sequence also is the sum of distinct prime divisors of n which are >= sqrt(n). At most one prime divisor of n is >= square root of n.

Diana Mecum, Table of n, a(n) for n = 1..1000

Cf. A097974.

Do[l = Select[Select[Divisors[n], PrimeQ], # >= Sqrt[n]&]; If[Length[l] == 0, Print[0], Print[l[[1]]]], {n, 1, 50}] (* Ryan Propper, Jul 24 2005 *)
Array[Select[FactorInteger[#][[All, 1]], Function[p, p >= Sqrt@ #]] /. {{} -> {0}, {1} -> {0}} &, 83][[All, 1]] (* Michael De Vlieger, Dec 22 2017 *)

a(n) = sumdiv(n, d, if (isprime(d) && (d^2 >= n), d)); \\ Michel Marcus, Dec 23 2017

More terms from Ryan Propper, Jul 24 2005
More terms from Stefan Steinerberger, Jan 21 2006
Further terms from Diana L. Mecum, Jun 15 2007

A347159 Sum of cubes of distinct prime divisors of n that are <= sqrt(n).

Original entry on oeis.org

0, 0, 0, 8, 0, 8, 0, 8, 27, 8, 0, 35, 0, 8, 27, 8, 0, 35, 0, 8, 27, 8, 0, 35, 125, 8, 27, 8, 0, 160, 0, 8, 27, 8, 125, 35, 0, 8, 27, 133, 0, 35, 0, 8, 152, 8, 0, 35, 343, 133, 27, 8, 0, 35, 125, 351, 27, 8, 0, 160, 0, 8, 370, 8, 125, 35, 0, 8, 27, 476, 0, 35, 0, 8, 152

Offset: 1

Ilya Gutkovskiy, Aug 20 2021

nonn

Cf. A001158, A005064, A005067, A063962, A097974, A098002, A280375, A347157, A347160.

Table[DivisorSum[n, #^3 &, # <= Sqrt[n] && PrimeQ[#] &], {n, 1, 75}]
nmax = 75; CoefficientList[Series[Sum[Prime[k]^3 x^(Prime[k]^2)/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=1} prime(k)^3 * x^(prime(k)^2) / (1 - x^prime(k)).

cp's OEIS Frontend

A108274 Sum of the first 10^n terms in A097974. a(n) = sum_{m=1..10^n} t(m), where t(m) is the sum of the prime divisors of m that are less than or equal to sqrt(m).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A063962 Number of distinct prime divisors of n that are <= sqrt(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A217581 Largest prime divisor of n <= sqrt(n), 1 if n is prime or 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A098002 Sum of squares of distinct prime divisors p of n, where each p <= sqrt(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A333808 Sum of distinct prime divisors of n that are < sqrt(n).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A333751 Sum of nonprime divisors of n that are <= sqrt(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A333753 Sum of prime power divisors of n that are <= sqrt(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica