A000430 -id:A000430 - cp's OEIS frontend

A064005 Erroneous version of A000430.

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 11, 12, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61

Offset: 1

dead

F. Smarandache, Definitions solved and unsolved problems, conjectures and theorems in number theory and geometry, edited by M. Perez, Xiquan Publishing House 2000

F. Smarandache, Definitions, Solved and Unsolved Problems, Conjectures, ...

A001248 Squares of primes.

Original entry on oeis.org

4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481

Offset: 1

N. J. A. Sloane

Also 4, together with numbers n such that Sum_{d|n}(-1)^d = -A048272(n) = -3. - Benoit Cloitre, Apr 14 2002
Also, all solutions to the equation sigma(x) + phi(x) = 2x + 1. - Farideh Firoozbakht, Feb 02 2005
Unique numbers having 3 divisors (1, their square root, themselves). - Alexandre Wajnberg, Jan 15 2006
Smallest (or first) new number deleted at the n-th step in an Eratosthenes sieve. - Lekraj Beedassy, Aug 17 2006
Subsequence of semiprimes A001358. - Lekraj Beedassy, Sep 06 2006
Integers having only 1 factor other than 1 and the number itself. Every number in the sequence is a multiple of 1 factor other than 1 and the number itself. 4 : 2 is the only factor other than 1 and 4; 9 : 3 is the only factor other than 1 and 9; and so on. - Rachit Agrawal (rachit_agrawal(AT)daiict.ac.in), Oct 23 2007
The n-th number with p divisors is equal to the n-th prime raised to power p-1, where p is prime. - Omar E. Pol, May 06 2008
There are 2 Abelian groups of order p^2 (C_p^2 and C_p x C_p) and no non-Abelian group. - Franz Vrabec, Sep 11 2008
Also numbers n such that phi(n) = n - sqrt(n). - Michel Lagneau, May 25 2012
For n > 1, n is the sum of numbers from A006254(n-1) to A168565(n-1). - Vicente Izquierdo Gomez, Dec 01 2012
A078898(a(n)) = 2. - Reinhard Zumkeller, Apr 06 2015
Let r(n) = (a(n) - 1)/(a(n) + 1); then Product_{n>=1} r(n) = (3/5) * (4/5) * (12/13) * (24/25) * (60/61) * ... = 2/5. - Dimitris Valianatos, Feb 26 2019
Numbers k such that A051709(k) = 1. - Jianing Song, Jun 27 2021

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 5000 terms from N. J. A. Sloane)
Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 143.
R. P. Boas and N. J. A. Sloane, Correspondence, 1974
Marius Coman, On the special relation between the numbers of the form 505+ 1008k and the squares of primes, 2015.
Brady Haran and Matt Parker, Squaring Primes, Numberphile video (2018).
Eric Weisstein's World of Mathematics, Prime Power.
OEIS Wiki, Index entries for number of divisors
Index to sequences related to prime signature

Cf. A000005, A000040, A001358, A001694, A002033, A005408, A005722, A006254.
Cf. A008864, A033273, A049001, A048272, A051709, A059956, A060800, A062799.
Cf. A078898, A082020, A085548, A087112, A095874, A168565, A192134, A211110.
Cf. A258599.
Subsequence of A000430, A001358, and of A190641.
Cf. A024450 (partial sums), A069482 (first differences).

a001248 n = a001248_list !! (n-1)
a001248_list = map (^ 2) a000040_list -- Reinhard Zumkeller, Sep 23 2011

[p^2: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 27 2014

A001248:=n->ithprime(n)^2; seq(A001248(k), k=1..50); # Wesley Ivan Hurt, Oct 11 2013

Prime[Range[30]]^2 (* Zak Seidov, Dec 07 2011 *)
Select[Range[40000], DivisorSigma[0, #] == 3 &] (* Carlos Eduardo Olivieri, Jun 01 2015 *)

forprime(p=2,1e3,print1(p^2", ")) \\ Charles R Greathouse IV, Jun 10 2011

A001248(n)=prime(n)^2  \\ M. F. Hasler, Sep 16 2012

from sympy import prime
def A001248(n): return prime(n)**2 # Chai Wah Wu, Aug 09 2024

[n^2 for n in prime_range(1,301)] # G. C. Greubel, May 02 2024

n such that A062799(n) = 2. - Benoit Cloitre, Apr 06 2002
A000005(a(n)^(k-1)) = A005408(k) for all k>0. - Reinhard Zumkeller, Mar 04 2007
a(n) = A000040(n)^(3-1)=A000040(n)^2, where 3 is the number of divisors of a(n). - Omar E. Pol, May 06 2008
A000005(a(n)) = 3 or A002033(a(n)) = 2. - Juri-Stepan Gerasimov, Oct 10 2009
A033273(a(n)) = 3. - Juri-Stepan Gerasimov, Dec 07 2009
For n > 2: (a(n) + 17) mod 12 = 6. - Reinhard Zumkeller, May 12 2010
A192134(A095874(a(n))) = A005722(n) + 1. - Reinhard Zumkeller, Jun 26 2011
For n > 2: a(n) = 1 (mod 24). - Zak Seidov, Dec 07 2011
A211110(a(n)) = 2. - Reinhard Zumkeller, Apr 02 2012
a(n) = A087112(n,n). - Reinhard Zumkeller, Nov 25 2012
a(n) = prime(n)^2. - Jon E. Schoenfield, Mar 29 2015
Product_{n>=1} a(n)/(a(n)-1) = Pi^2/6. - Daniel Suteu, Feb 06 2017
Sum_{n>=1} 1/a(n) = P(2) = 0.4522474200... (A085548). - Amiram Eldar, Jul 27 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(2)/zeta(4) = 15/Pi^2 (A082020).
Product_{n>=1} (1 - 1/a(n)) = 1/zeta(2) = 6/Pi^2 (A059956). (End)

A034785 a(n) = 2^(n-th prime).

Original entry on oeis.org

4, 8, 32, 128, 2048, 8192, 131072, 524288, 8388608, 536870912, 2147483648, 137438953472, 2199023255552, 8796093022208, 140737488355328, 9007199254740992, 576460752303423488, 2305843009213693952

Offset: 1

Asher Auel

These are the "outputs" in Conway's PRIMEGAME (see A007542). - Alonso del Arte, Jan 03 2011

Multiplicative encoding of the n-th prime. - Daniel Forgues, Feb 26 2017

			a(4) = 128 because the fourth prime number is 7 and 2^7 = 128.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index to divisibility sequences.

Cf. A000040, A000430, A051006, A073718 (2^(n-th composite)), A074736.

Cf. A184082, A184083.

a034785 = (2 ^) . a000040
-- Reinhard Zumkeller, Feb 07 2015, Jan 24 2012

[2^p: p in PrimesUpTo(100)]; // Vincenzo Librandi, Apr 29 2014

2^Prime@Range@40  (* Vladimir Joseph Stephan Orlovsky, Apr 11 2011 *)

a(n)=1<Charles R Greathouse IV, Apr 07 2012

from sympy import prime
def A034785(n): return 1<Chai Wah Wu, Aug 09 2024

From Amiram Eldar, Aug 11 2020: (Start)
a(n) = 2^A000040(n).
Sum_{n>=1} 1/a(n) = A051006. (End)
From Amiram Eldar, Nov 22 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = A184083.
Product_{n>=1} (1 - 1/a(n)) = A184082. (End)

More terms from James Sellers, Feb 04 2000

A341596 Number of strictly inferior squarefree divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 4, 1, 2, 3, 2, 1, 4, 1, 3, 2, 2, 1, 4, 2, 3, 2, 2, 1, 5, 1, 2, 3, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 2, 3, 2, 2, 4, 1, 3, 2, 2, 1, 5, 2, 2, 2

Offset: 1

Gus Wiseman, Feb 23 2021

nonn

We define a divisor d|n to be strictly inferior if d < n/d. Strictly inferior divisors are counted by A056924 and listed by A341674.

			The strictly inferior squarefree divisors of selected n:
n = 1  2  6  12  30  60  120  210  240  420  630  1050  1260
    --------------------------------------------------------
    {} 1  1  1   1   1   1    1    1    1    1    1     1
          2  2   2   2   2    2    2    2    2    2     2
             3   3   3   3    3    3    3    3    3     3
                 5   5   5    5    5    5    5    5     5
                     6   6    6    6    6    6    6     6
                         10   7    10   7    7    7     7
                              10   15   10   10   10    10
                              14        14   14   14    14
                                        15   15   15    15
                                             21   21    21
                                                  30    30
                                                        35

Amiram Eldar, Table of n, a(n) for n = 1..10000

Positions of ones are A000430.
The weakly inferior version is A333749.
The version counting odd instead of squarefree divisors is A333805.
The version counting prime instead of squarefree divisors is A333806.
The weakly superior version is A341592.
The strictly superior version is A341595.
The version counting prime-power instead of squarefree divisors is A341677.
A001221 counts prime divisors, with sum A001414.
A001222 counts prime power divisors.
A005117 lists squarefree numbers.
A033676 selects the greatest inferior divisor.
A033677 selects the smallest superior divisor.
A038548 counts superior (or inferior) divisors.
A056924 counts strictly superior (or strictly inferior) divisors.
A207375 lists central divisors.
- Inferior: A063962, A066839, A069288, A161906, A217581, A333750.
- Superior: A051283, A059172, A063538, A063539, A070038, A116882, A116883, A161908, A341591, A341593, A341675, A341676.
- Strictly Inferior: A060775, A070039, A341674.
- Strictly Superior: A048098, A064052, A140271, A238535, A341594, A341642, A341643, A341644, A341645, A341646, A341673.
Cf. A000005, A000203, A001055, A001248, A050320.

Table[Length[Select[Divisors[n],SquareFreeQ[#]&&#

a(n) = sumdiv(n, d, d^2 < n && issquarefree(d)); \\ Amiram Eldar, Nov 01 2024

A080257 Numbers having at least two distinct or a total of at least three prime factors.

Original entry on oeis.org

6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100

Offset: 1

Reinhard Zumkeller, Feb 10 2003

Complement of A000430; A080256(a(n)) > 3.
A084114(a(n)) > 0, see also A084110.
Also numbers greater than the square of their smallest prime-factor: a(n)>A020639(a(n))^2=A088377(a(n));
a(n)>A000430(k) for n<=13, a(n) < A000430(k) for n>13.
Numbers with at least 4 divisors. - Franklin T. Adams-Watters, Jul 28 2006
Union of A024619 and A033942; A211110(a(n)) > 2. - Reinhard Zumkeller, Apr 02 2012
Also numbers > 1 that are neither prime nor a square of a prime. Also numbers whose omega-sequence (A323023) has sum > 3. Numbers with omega-sequence summing to m are: A000040 (m = 1), A001248 (m = 3), A030078 (m = 4), A068993 (m = 5), A050997 (m = 6), A325264 (m = 7). - Gus Wiseman, Jul 03 2019
Numbers n such that sigma_2(n)*tau(n) = A001157(n)*A000005(n) >= 4*n^2. Note that sigma_2(n)*tau(n) >= sigma(n)^2 = A072861 for all n. - Joshua Zelinsky, Jan 23 2025

			8=2*2*2 and 10=2*5 are terms; 4=2*2 is not a term.
From _Gus Wiseman_, Jul 03 2019: (Start)
The sequence of terms together with their prime indices begins:
   6: {1,2}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  27: {2,2,2}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001248, A001221, A001222, A006881, A030078, A088381, A088383, A000005.

Cf. A060687, A118914, A323014, A323023, A325249.

a080257 n = a080257_list !! (n-1)
a080257_list = m a024619_list a033942_list where
   m xs'@(x:xs) ys'@(y:ys) | x < y  = x : m xs ys'
                           | x == y = x : m xs ys
                           | x > y  = y : m xs' ys
-- Reinhard Zumkeller, Apr 02 2012

Select[Range[100],PrimeNu[#]>1||PrimeOmega[#]>2&] (* Harvey P. Dale, Jul 23 2013 *)

PARI
```
is(n)=omega(n)>1 || isprimepower(n)>2
```

is(n)=my(k=isprimepower(n)); if(k, k>2, !isprime(n)) \\ Charles R Greathouse IV, Jan 23 2025

a(n) = n + O(n/log n). - Charles R Greathouse IV, Sep 14 2015

Definition clarified by Harvey P. Dale, Jul 23 2013

A347440 Number of factorizations of n with alternating product < 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 4, 0, 1, 1, 2, 0, 3, 0, 3, 1, 1, 1, 3, 0, 1, 1, 4, 0, 3, 0, 2, 2, 1, 0, 6, 0, 2, 1, 2, 0, 4, 1, 4, 1, 1, 0, 6, 0, 1, 2, 3, 1, 3, 0, 2, 1, 3, 0, 8, 0, 1, 2, 2, 1, 3, 0, 6, 1, 1, 0, 6, 1, 1, 1

Offset: 1

Gus Wiseman, Sep 07 2021

nonn

All such factorizations have even length and alternating sum < 0, so partitions of this type are counted by A344608.
Also the number of factorizations of n with alternating sum < 0.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

			The a(n) factorizations for n = 6, 12, 24, 30, 48, 72, 96, 120:
  2*3  2*6  3*8      5*6   6*8      8*9      2*48         2*60
       3*4  4*6      2*15  2*24     2*36     3*32         3*40
            2*12     3*10  3*16     3*24     4*24         4*30
            2*2*2*3        4*12     4*18     6*16         5*24
                           2*2*2*6  6*12     8*12         6*20
                           2*2*3*4  2*2*2*9  2*2*3*8      8*15
                                    2*2*3*6  2*2*4*6      10*12
                                    2*3*3*4  2*3*4*4      2*2*5*6
                                             2*2*2*12     2*3*4*5
                                             2*2*2*2*2*3  2*2*2*15
                                                          2*2*3*10

PlanetMath, alternating sum

Positions of 0's are A000430.
Positions of 2's are A054753.
Positions of non-0's are A080257.
Positions of 1's are A332269.
The weak version (<= 1 instead of < 1) is A339846, ranked by A028982.
The reciprocal version is A339890.
The additive version is A344608, ranked by A119899.
The even-sum additive version is A344743, ranked by A119899 /\ A300061.
Allowing any integer alternating product gives A347437, additive A347446.
The equal version (= 1 instead of < 1) is A347438.
Allowing any integer reciprocal alternating product gives A347439.
The complement (>= 1 instead of < 1) is counted by A347456.
A038548 counts possible reverse-alternating products of factorizations.
A046099 counts factorizations with no alternating permutations.
A071321 gives the alternating sum of prime factors (reverse: A071322).
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A273013 counts ordered factorizations of n^2 with alternating product 1.
A347460 counts possible alternating products of factorizations.
Cf. A008549, A058622, A119620, A294175, A330972, A347442, A347447, A347454.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[facs[n],altprod[#]<1&]],{n,100}]

a(2^n) = A344608(n).

a(n) = A339846(n) - A347438(n).

A056595 Number of nonsquare divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 2, 1, 3, 1, 4, 1, 3, 3, 2, 1, 4, 1, 4, 3, 3, 1, 6, 1, 3, 2, 4, 1, 7, 1, 3, 3, 3, 3, 5, 1, 3, 3, 6, 1, 7, 1, 4, 4, 3, 1, 7, 1, 4, 3, 4, 1, 6, 3, 6, 3, 3, 1, 10, 1, 3, 4, 3, 3, 7, 1, 4, 3, 7, 1, 8, 1, 3, 4, 4, 3, 7, 1, 7, 2, 3, 1, 10, 3, 3, 3, 6, 1, 10, 3, 4, 3, 3, 3, 9, 1, 4, 4, 5, 1, 7, 1

Offset: 1

Labos Elemer, Jul 21 2000

nonn

a(A000430(n))=1; a(A030078(n))=2; a(A030514(n))=2; a(A006881(n))=3; a(A050997(n))=3; a(A030516(n))=3; a(A054753(n))=4; a(A000290(n))=A055205(n). - Reinhard Zumkeller, Aug 15 2011

			a(36)=5 because the set of divisors of 36 has tau(36)=nine elements, {1, 2, 3, 4, 6, 9, 12, 18, 36}, five of which, that is {2, 3, 6, 12, 18}, are not perfect squares.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A000188, A046951.
See A194095 and A194096 for record values and where they occur.
Cf. A001620, A013661.

a056595 n = length [d | d <- [1..n], mod n d == 0, a010052 d == 0]
-- Reinhard Zumkeller, Aug 15 2011

A056595 := proc(n)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if not issqr(d) then
            a := a+1 ;
        end if;
    end do:
    a;
end proc:
seq(A056595(n),n=1..40) ; # R. J. Mathar, Aug 18 2024

Table[Count[Divisors[n],?(#!=Floor[Sqrt[#]]^2&)],{n,110}] (* _Harvey P. Dale, Jul 10 2013 *)
a[1] = 0; a[n_] := Times @@ (1 + (e = Last /@ FactorInteger[n])) - Times @@ (1 + Floor[e/2]); Array[a, 100] (* Amiram Eldar, Jul 22 2019 *)

a(n)=sumdiv(n,d,!issquare(d)) \\ Charles R Greathouse IV, Aug 28 2016

a(n) = A000005(n) - A046951(n) = tau(n) - tau(A000188(n)).

Sum_{k=1..n} a(k) ~ n*log(n) + (2*gamma - zeta(2) - 1)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 01 2023

A050216 Number of primes between (prime(n))^2 and (prime(n+1))^2, with a(0) = 2 by convention.

Original entry on oeis.org

2, 2, 5, 6, 15, 9, 22, 11, 27, 47, 16, 57, 44, 20, 46, 80, 78, 32, 90, 66, 30, 106, 75, 114, 163, 89, 42, 87, 42, 100, 354, 99, 165, 49, 299, 58, 182, 186, 128, 198, 195, 76, 356, 77, 144, 75, 463, 479, 168, 82, 166, 270, 90, 438, 275, 274, 292, 91, 292, 199, 99

Offset: 0

Eric W. Weisstein

The function in Brocard's Conjecture, which states that for n >= 2, a(n) >= 4.
The lines in the graph correspond to prime gaps of 2, 4, 6, ... . - T. D. Noe, Feb 04 2008
Lengths of blocks of consecutive primes in A000430 (union of primes and squares of primes). - Reinhard Zumkeller, Sep 23 2011
In the n-th step of the sieve of Eratosthenes, all multiples of prime(n) are removed. Then a(n) gives the number of new primes obtained after the n-th step. - Jean-Christophe Hervé, Oct 27 2013
More precisely, after the n-th step, one is sure to have eliminated all composites less than prime(n+1)^2, since any composite N has a prime factor <= sqrt(N). It is in exactly this (restricted) sense that a(n) yields the number of "new primes" (additional numbers known to be prime) after the n-th step. But one knows after the n-th step also that all remaining numbers between prime(n+1)^2 and prime(n+1)*(prime(n+1)+2) are prime: By construction they don't have a factor less than prime(n+1) and they don't have a factor prime(n+1) so the least prime factor could be prime(n+2) >= prime(n+1)+2. For example, after eliminating multiples of 3 in the 2nd step, one has (2, 3, 5, 7, 11, 13, 17, 23, 25, 29, 31, 35, ...) and one knows that all remaining numbers strictly in between 5^2=25 and 5*(5+2)=35 are prime, too. - M. F. Hasler, Dec 31 2014
Numerically, the slope of the lowest "ray" m(n) = min {a(k); k>n}, seems to converge to a value somewhere in the range 1.75 < m(n)/n < 1.8; with m(n)/n > 1.7 for n > 900, m(n)/n > 1.75 for n > 2700. - M. F. Hasler, Dec 31 2014
Legendre's conjecture (see A014085) would imply that a(n) >= 2 for all n and that sequences A054272, A250473 and A250474 were thus strictly increasing (see the Wikipedia article about Brocard's conjecture). - Antti Karttunen, Jan 01 2015
a(n) >= 4 up to at least n = 4*10^5. - Eric W. Weisstein, Jan 13 2025

			There are 2 primes less than 2^2, there are 2 primes between 2^2 and 3^2, 5 primes between 3^2 and 5^2, etc. [corrected by Jonathan Sperry, Aug 30 2013]

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 183.

T. D. Noe, Table of n, a(n) for n = 0..10000
Joel E. Cohen, Conjectures about Primes and Cyclic Numbers, arXiv:2508.08335 [math.NT], 2025. See p. 16.
Anthony G. Shannon and Jean V. Leyendekkers, On Legendre's Conjecture, Notes on Number Theory and Discrete Mathematics, Vol. 23, No. 2 (2017): 117-125.
Nicolas Vaillant, Graph of A050216(n) / (n * A001223(n))
Nicolas Vaillant, Average density of primes between prime(n)^2 and prime(n+1)^2
Eric Weisstein's World of Mathematics, Brocard's Conjecture.
Wikipedia, Brocard's Conjecture

First differences of A000879.
One more than A251723.
Cf. A010051, A001248, A014085, A089609, A251719, A256468, A256469, A256470, A029707, A137859.
Cf. A380135 (High water marks for number of primes between prime(n)^2 and prime(n+1)^2).
Cf. A380136 (Positions of the high water marks for number of primes between prime(n)^2 and prime(n+1)^2).

import Data.List (group)
a050216 n = a050216_list !! (n-1)
a050216_list =
   map length $ filter (/= [0]) $ group $ map a010051 a000430_list
-- Reinhard Zumkeller, Sep 23 2011

A050216 := proc(n)
    local p,pn ;
    if n = 0 then
        2;
    else
        p := ithprime(n) ;
        pn := nextprime(p) ;
        numtheory[pi](pn^2)-numtheory[pi](p^2) ;
    end if;
end proc:
seq(A050216(n),n=0..40) ; # R. J. Mathar, Jan 27 2025

-Subtract @@@ Partition[PrimePi[Prime[Range[20]]^2], 2, 1] (* Eric W. Weisstein, Jan 10 2025 *)

a(n)={n||return(2);primepi(prime(n+1)^2)-primepi(prime(n)^2)} \\ M. F. Hasler, Dec 31 2014

For all n >= 1, a(n) = A256468(n) + A256469(n). - Antti Karttunen, Mar 30 2015

Limit_{N->oo} (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} prime(n)) = 1. - Alain Rocchelli, Sep 30 2023

Edited by N. J. A. Sloane, Nov 15 2009

A166684 Numbers n such that d(n)<4.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251

Offset: 1

Juri-Stepan Gerasimov, Oct 18 2009

1 together with primes and squares of primes.
Numbers n such that A229964(n) = 0. - Eric M. Schmidt, Oct 05 2013
Numbers that cannot be written as a product of 2 distinct nonunits. - Peter Munn, May 26 2023

G. C. Greubel, Table of n, a(n) for n = 1..10000

A000430 is the main entry for this sequence.

Cf. A000005, A229964.

Select[Range[300],DivisorSigma[0,#]<4&] (* or *) Select[With[ {prs = Prime[Range[200]]},Union[Join[{1},prs,prs^2]]],#<301&] (* Harvey P. Dale, Jan 04 2012 *)

is(n)=isprime(n) || (issquare(n,&n) && isprime(n)) || n==1 \\ Charles R Greathouse IV, Dec 23 2022

from math import isqrt
from sympy import primepi
def A166684(n):
    def f(x): return n-1+x-primepi(x)-primepi(isqrt(x))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return int(m) # Chai Wah Wu, Aug 09 2024

a(n) = A000430(n-1), n>1. - R. J. Mathar, May 21 2010

Corrected (193 inserted) by R. J. Mathar, May 21 2010

A341677 Number of strictly inferior prime-power divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 1, 2, 0, 3, 0, 2, 1, 1, 1, 3, 0, 1, 1, 3, 0, 2, 0, 2, 2, 1, 0, 3, 0, 2, 1, 2, 0, 2, 1, 3, 1, 1, 0, 4, 0, 1, 2, 2, 1, 2, 0, 2, 1, 3, 0, 4, 0, 1, 2, 2, 1, 2, 0, 4, 1, 1, 0, 4, 1, 1, 1

Offset: 1

Gus Wiseman, Feb 23 2021

nonn

We define a divisor d|n to be strictly inferior if d < n/d. Strictly inferior divisors are counted by A056924 and listed by A341674.

			The strictly inferior prime-power divisors of n!:
n = 1  2  6  24  120  720  5040  40320
    ----------------------------------
    .  .  2   2    2    2     2      2
              3    3    3     3      3
              4    4    4     4      4
                   5    5     5      5
                   8    8     7      7
                        9     8      8
                       16     9      9
                             16     16
                                    32
                                    64
                                   128

Amiram Eldar, Table of n, a(n) for n = 1..10000

Positions of zeros are A166684.
The weakly inferior version is A333750.
The version for odd instead of prime-power divisors is A333805.
The version for prime instead of prime-power divisors is A333806.
The weakly superior version is A341593.
The version for squarefree instead of prime-power divisors is A341596.
The strictly superior version is A341644.
A000961 lists prime powers.
A001221 counts prime divisors, with sum A001414.
A001222 counts prime-power divisors.
A005117 lists squarefree numbers.
A038548 counts superior (or inferior) divisors.
A056924 counts strictly superior (or strictly inferior) divisors.
A207375 lists central divisors.
- Inferior: A033676, A063962, A066839, A069288, A161906, A217581, A333749.
- Superior: A033677, A051283, A059172, A063538, A063539, A070038, A116882, A116883, A161908, A341591, A341592, A341676.
- Strictly Inferior: A060775, A070039, A341674.
- Strictly Superior: A048098, A064052, A140271, A238535, A341594, A341595, A341642, A341643, A341645, A341646, A341673.
Cf. A000005, A000203, A000430, A001248, A006530, A020639.

Table[Length[Select[Divisors[n],PrimePowerQ[#]&&#

a(n) = sumdiv(n, d, d^2 < n && isprimepower(d)); \\ Amiram Eldar, Nov 01 2024

cp's OEIS Frontend

A064005 Erroneous version of A000430.

Views

Author

Keywords

References

Links

A001248 Squares of primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Python

SageMath

Formula

A034785 a(n) = 2^(n-th prime).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Python

Formula

Extensions

A341596 Number of strictly inferior squarefree divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A080257 Numbers having at least two distinct or a total of at least three prime factors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Formula

Extensions

A347440 Number of factorizations of n with alternating product < 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A056595 Number of nonsquare divisors of n.

Views

Author