A008479 -id:A008479 - cp's OEIS frontend

A372720 a(n) = A000005(n) - A008479(n).

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 3, 1, 1, 3, 1, 4, 3, 3, 1, 4, 1, 3, 1, 4, 1, 7, 1, 1, 3, 3, 3, 4, 1, 3, 3, 5, 1, 7, 1, 4, 4, 3, 1, 4, 1, 2, 3, 4, 1, 1, 3, 5, 3, 3, 1, 10, 1, 3, 4, 1, 3, 7, 1, 4, 3, 7, 1, 4, 1, 3, 3, 4, 3, 7, 1, 5, 1, 3, 1, 10, 3, 3, 3

Offset: 1

Michael De Vlieger, May 13 2024

sign

A095960(50) = 3, a(50) = 2.

a(162) = -2 is the first negative term.

			Table of a(n), b(n) = A000005(n), and c(n) = A008479(n) for n <= 12:
  n  b(n) c(n) a(n)
 ------------------
  1    1    1    0
  2    2    1    1
  3    2    1    1
  4    3    2    1
  5    2    1    1
  6    4    1    3
  7    2    1    1
  8    4    3    1
  9    3    2    1
 10    4    1    3
 11    2    1    1
 12    6    2    4
a(12) = 4 since 12 has 6 divisors {1, 2, 3, 4, 6, 12}, and row 12 of A369609 has 2 terms {6, 12}.
a(18) = 3 since 18 has 6 divisors {1, 2, 3, 6, 9, 18}, and row 18 of A369609 has 3 terms {6, 12, 18}.
a(50) = 2 since 50 has 6 divisors {1, 2, 5, 10, 25, 50}, and row 50 of A369609 has 4 terms {10, 20, 40, 50}
a(162) = -2 since 162 has 10 divisors {1,2,3,6,9,18,27,54,81,162} but row 162 of A369609 has 12 terms {6,12,18,24,36,48,54,72,96,108,144,162}.
a(500) = 0 since 500 has as many divisors {1,2,4,5,10,20,25,50,100,125,250,500} as terms in row 500 of A369609 {10,20,40,50,80,100,160,200,250,320,400,500}.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005, A003557, A008479, A095960, A119288, A162306, A183093, A303554, A355432, A360767, A369609.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; DivisorSigma[0, n] - Count[Range[n/r], _?(Divisible[r, rad[#]] &)], {n, 120}]

a(n) = my(f=factor(n)[, 1], s); forvec(v=vector(#f, i, [1, logint(n, f[i])]), if(prod(i=1, #f, f[i]^v[i])<=n, s++)); numdiv(n) - s; \\ after A008479 \\ Michel Marcus, Jun 03 2024

a(n) = A095960(n) for n in A303554, i.e., for squarefree n or prime powers n.
a(n) = A095960(n) for n in A360767, i.e., for nonsquarefree composite n such that omega(n) > 1 and A003557(n) < A119288(n), since A008479(n) is the number of terms k in row n of A010846 such that k <= A003557(n).
a(n) = A183093(n) - A355432(n).

A293780 Where records occur in A008479.

Original entry on oeis.org

1, 4, 8, 16, 32, 48, 54, 72, 96, 108, 144, 162, 192, 216, 288, 324, 384, 432, 486, 576, 648, 768, 864, 972, 1152, 1296, 1458, 1536, 1728, 1944, 2304, 2592, 2880, 3000, 3240, 3600, 3750, 3840, 4050, 4320, 4500, 4800, 4860, 5400, 5760, 6000, 6480, 6750, 7200, 7290, 7500

Offset: 1

David A. Corneth, Oct 19 2017

nonn

Also values m such that A008479(k) < A008479(m) for all k < m.
Is the squarefree part of a(n); A007913(a(n)) a product of the first primes?
The sequence omega(a(n)) is not nondecreasing; a(1641) and a(1651) are both 7-smooth but a(1640) and a(1650) are divisible by 11.
Is A008479(a(n)) = n?
The least a(n) such that omega(a(n)) gives a record is 4, 48, 2880, 504210, 616319550, 317433916800.

			8 is in the sequence as A008479(k) < A008479(8) for k < 8.

Rémy Sigrist, Table of n, a(n) for n = 1..2261
Rémy Sigrist, C++ program for A293780

Cf. A007913, A008479, A055932, A293781.

Block[{s = {1}~Join~Array[Times @@ FactorInteger[#][[All, 1]] &, 10^4, 2], t}, t = 1 + Array[Count[Take[s, # - 1], s[[#]]] &, Length@ s]; Map[FirstPosition[t, #][[1]] &, Union@ FoldList[Max, t]] ] (* Michael De Vlieger, Oct 21 2017 *)

lista(nn) = {nbm = 0; v = vector(nn, k, factor(k)[,1]); for (n=1, nn, vn = vector(n, k, v[k]); nbn = #select(x->(x==v[n]), vn); if (nbn > nbm, print1(n, ", "); nbm = nbn););} \\ Michel Marcus, Oct 30 2017

A373737 a(n) is the smallest number k in the sorted sequence S(q) = {k : rad(k) = q}, q = A120944(n), such that tau(k) - A008479(k) is not positive, where rad = A007947 and tau = A000005.

Original entry on oeis.org

162, 250, 686, 1875, 7203, 2662, 4394, 750, 3993, 578, 12005, 722, 6591, 2058, 1058, 14739, 73205, 20577, 1682, 1922, 142805, 5346, 36501, 3430, 2738, 102487, 6318, 3362, 417605, 3698, 73167, 199927, 89373, 4418, 651605, 5202, 25725, 5618, 13310, 151959, 6498

Offset: 1

Michael De Vlieger, Jun 24 2024

Numbers k whose position i in S(n) is such that tau(k) <= i, i.e., that A372720(k) is not positive.
For k = p^m, m > 0, in S(p), p prime, tau(p^m) > A008479(p^m) since tau(p^m) = m + 1 and A008479(p^m) = m. Therefore we consider only composite squarefree q in this sequence.
a(n) is in A126706.
Conjecture: a(n) <= s*gpf(s)^floor(log_gpf(s) s^2), where gpf = A006530.

			a(1) = 162 since the 12th term in S(6) = A033845 = {6, 12, 18, 24, 36, 48, 54, ..., 162, ...} is the smallest k = S(6, i) such that tau(S(6, i)) <= i: tau(162) = 10 while i = 12.
a(2) = 250 since S(10, 9) = 250 gives tau(250) = 8, and 8 < 9.
a(3) = 686 since S(14, 10) = 686 is such that A372720(686) <= 0, etc.
Table of first and some notable terms:
       n        q     i         a(n) a(n)/q  A372720(a(n))
  --------------------------------------------------------
       1        6    12         162   3^3         -2
       2       10     9         250   5^2         -1
       3       14    10         686   7^2         -2
       4       15    11        1875   5^3         -1
       5       21    13        7203   7^3         -3
       6       22    12        2662   11^2        -4
       7       26    13        4394   13^2        -5
       8       30    16         750   5^2          0
      82      210    51       26250   5^3        -11
    1061     2310    99      635250   5^2 * 11    -3
   15013    30030   222    25375350   5 * 13^2   -30
  268015   510510   338   679488810   11^3       -18

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..268015 in blue, and A120944(n)^3 in red.
Michael De Vlieger, Diagram of S(6) = A033845 arranged such that k appears vertically in order of magnitude, with smallest at the bottom. Color function relates to A372720(k), with positive values from largest in light yellow grading to A372720(k) = 1 in orange, and negative values with smallest absolute value in dark blue to greatest in light blue. a(1) = 162 appears at right.
Michael De Vlieger, Diagram of S(30) = A143207 arranged such that k appears vertically in order of magnitude, with smallest at the bottom. Color function is as above, but with A372720(k) = 0 in red. a(2) = 750 appears at left in red.

Cf. A000005, A008479, A120944, A126706, A372720.

(* First, load function f from A162306 *)
Table[k = 1; s = f[n, n^3]; While[DivisorSigma[0, n*s[[k]]] - k > 0, k++]; s[[k]], {n, Select[Range[6, 120], And[SquareFreeQ[#], CompositeQ[#]] &]}]

A375011 a(n) = A373738(n) - A008479(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 5, 1, 4, 3, 1, 1, 6, 1, 5, 3, 5, 1, 6, 1, 5, 1, 5, 1, 11, 1, 1, 4, 5, 3, 8, 1, 5, 4, 7, 1, 12, 1, 6, 5, 6, 1, 8, 1, 7, 4, 6, 1, 8, 3, 7, 4, 6, 1, 17, 1, 6, 5, 1, 3, 14, 1, 6, 4, 12, 1, 9, 1, 6, 6, 6, 3, 15, 1, 8, 1, 7, 1, 18, 3

Offset: 1

Michael De Vlieger, Aug 14 2024

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 2..2^20, ignoring a(1) = 0, with a color code where red indicates omega(n) = 2, gold omega(n) = 3, lime green = omega(n) = 4, spring green = omega(n) = 5, and blue = omega(n) = 6, where omega = A001221.

Cf. A001221, A008479, A246655, A372192, A373738.

{0}~Join~Table[facs = FactorInteger[n][[All, 1]]; r = Times @@ facs;
  Floor[(1/PrimeNu[n]!)*Times @@ Map[Log[#, n] + 1 &, facs]] -
    Count[Range[n], _?(Times @@ FactorInteger[#][[All, 1]] == r &)], {n, 2, 120}]

a(1) = 0; a(k) = 1 for k in A246655.

A051903 Maximum exponent in the prime factorization of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 2, 1, 1, 1, 3, 1

Offset: 1

Labos Elemer, Dec 16 1999

Smallest number of factors of all factorizations of n into squarefree numbers, see also A128651, A001055. - Reinhard Zumkeller, Mar 30 2007
Maximum number of invariant factors among abelian groups of order n. - Álvar Ibeas, Nov 01 2014
a(n) is the highest of the frequencies of the parts of the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1..r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(24) = 3; indeed, the partition having Heinz number 24 = 2*2*2*3 is [1,1,1,2], where the distinct parts 1 and 2 have frequencies 3 and 1, respectively. - Emeric Deutsch, Jun 04 2015
From Thomas Ordowski, Dec 02 2019: (Start)
a(n) is the smallest k such that b^(phi(n)+k) == b^k (mod n) for all b.
The Euler phi function can be replaced by the Carmichael lambda function.
Problems:
(*) Are there composite numbers n > 4 such that n == a(n) (mod phi(n))? By Lehmer's totient conjecture, there are no such squarefree numbers.
(**) Are there odd numbers n such that a(n) > 1 and n == a(n) (mod lambda(n))? These are odd numbers n such that a(n) > 1 and b^n == b^a(n) (mod n) for all b.
(***) Are there odd numbers n such that a(n) > 1 and n == a(n) (mod ord_{n}(2))? These are odd numbers n such that a(n) > 1 and 2^n == 2^a(n) (mod n).
Note: if (***) do not exist, then (**) do not exist. (End)
Niven (1969) proved that the asymptotic mean of this sequence is 1 + Sum_{j>=2} 1 - (1/zeta(j)) (A033150). - Amiram Eldar, Jul 10 2020

			For n = 72 = 2^3*3^2, a(72) = max(exponents) = max(3,2) = 3.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
Benjamin Merlin Bumpus and Zoltan A. Kocsis, Spined categories: generalizing tree-width beyond graphs, arXiv:2104.01841 [math.CO], 2021.
Cao Hui-Zhong, The Asymptotic Formulas Related to Exponents in Factoring Integers, Math. Balkanica, Vol. 5 (1991), Fasc. 2.
Ivan Niven, Averages of Exponents in Factoring Integers, Proc. Amer. Math. Soc., Vol. 22, No. 2 (1969), pp. 356-360.
Eric Weisstein's World of Mathematics, Niven's Constant.
Index entries for sequences computed from exponents in factorization of n

Average value is A033150 = 1.7052....

Cf. A002322, A005361, A008479, A028234, A051904, A052409, A067029, A091050, A129132, A327295, A328310, A329885.

a051903 1 = 0
a051903 n = maximum $ a124010_row n -- Reinhard Zumkeller, May 27 2012

A051903 := proc(n)
        a := 0 ;
        for f in ifactors(n)[2] do
                a := max(a,op(2,f)) ;
        end do:
        a ;
end proc: # R. J. Mathar, Apr 03 2012
# second Maple program:
a:= n-> max(0, seq(i[2], i=ifactors(n)[2])):
seq(a(n), n=1..120);  # Alois P. Heinz, May 09 2020

Table[If[n == 1, 0, Max @@ Last /@ FactorInteger[n]], {n, 100}] (* Ray Chandler, Jan 24 2006 *)

a(n)=if(n>1,vecmax(factor(n)[,2]),0) \\ Charles R Greathouse IV, Oct 30 2012

from sympy import factorint
def A051903(n):
    return max(factorint(n).values()) if n > 1 else 0
# Chai Wah Wu, Jan 03 2015

;; With memoization-macro definec.
(definec (A051903 n) (if (= 1 n) 0 (max (A067029 n) (A051903 (A028234 n))))) ;; Antti Karttunen, Aug 08 2016

a(n) = max_{k=1..A001221(n)} A124010(n,k). - Reinhard Zumkeller, Aug 27 2011
a(1) = 0; for n > 1, a(n) = max(A067029(n), a(A028234(n))). - Antti Karttunen, Aug 08 2016
Conjecture: a(n) = a(A003557(n)) + 1. This relation together with a(1) = 0 defines the sequence. - Velin Yanev, Sep 02 2017
Comment from David J. Seal, Sep 18 2017: (Start)
This conjecture seems very easily provable to me: if the factorization of n is p1^k1 * p2^k2 * ... * pm^km, then the factorization of the largest squarefree divisor of n is p1 * p2 * ... * pm. So the factorization of A003557(n) is p1^(k1-1) * p2^(k2-1) * ... * pm^(km-1) if exponents of zero are allowed, or with the product terms that have an exponent of zero removed if they're not (if that results in an empty product, consider it to be 1 as usual).
The formula then follows from the fact that provided all ki >= 1, Max(k1, k2, ..., km) = Max(k1-1, k2-1, ..., km-1) + 1, and Max(k1-1, k2-1, ..., km-1) is not altered by removing the ki-1 values that are 0, provided we treat the empty Max() as being 0. That proves the formula and the provisos about empty products and Max() correspond to a(1) = 0.
Also, for any n, applying the formula Max(k1, k2, ..., km) times to n = p1^k1 * p2^k2 * ... * pm^km reduces all the exponents to zero, i.e., to the case a(1) = 0, so that case and the formula generate the sequence. (End)
Sum_{k=1..n} (-1)^k * a(k) ~ c * n, where c = Sum_{k>=2} 1/((2^k-1)*zeta(k)) = 0.44541445377638761933... . - Amiram Eldar, Jul 28 2024
a(n) <= log(n)/log(2). - Hal M. Switkay, Jul 03 2025

A051904 Minimal exponent in prime factorization of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Labos Elemer, Dec 16 1999

The asymptotic mean of this sequence is 1 (Niven, 1969). - Amiram Eldar, Jul 10 2020

Let k = A007947(n), then for n > 1 k^a(n) is the greatest power of k which divides n; see example. - David James Sycamore, Sep 07 2023

			For n = 72 = 2^3*3^2, a(72) = min(exponents) = min(3,2) = 2.
For n = 72, using alternative definition: rad(72) = 6; and 6^2 = 36 divides 72 but no higher power of 6 divides 72, so a(72) = 2.
For n = 432, rad(432) = 6 and 6^3 = 216 divides 432 but no higher power of 6 divides 432, therefore a(432) = 3. - _David James Sycamore_, Sep 08 2023

Daniel Forgues, Table of n, a(n) for n = 1..100000
Cao Hui-Zhong, The Asymptotic Formulas Related to Exponents in Factoring Integers, Math. Balkanica, Vol. 5 (1991), Fasc. 2.
Ivan Niven, Averages of Exponents in Factoring Integers, Proc. Amer. Math. Soc., Vol. 22, No. 2 (1969), pp. 356-360.
Eric Weisstein's World of Mathematics, Niven's Constant
Index entries for sequences computed from exponents in factorization of n

Cf. A000961, A001221, A005361, A008479, A051903, A052409, A076558, A124010, A362974.

Cf. A007947.

a051904 1 = 0
a051904 n = minimum $ a124010_row n  -- Reinhard Zumkeller, Jul 15 2012

a := proc (n) if n = 1 then 0 else min(seq(op(2, op(j, op(2, ifactors(n)))), j = 1 .. nops(op(2, ifactors(n))))) end if end proc: seq(a(n), n = 1 .. 100); # Emeric Deutsch, May 20 2015

Table[If[n == 1, 0, Min @@ Last /@ FactorInteger[n]], {n, 100}] (* Ray Chandler, Jan 24 2006 *)

a(n)=vecmin(factor(n)[,2]) \\ Charles R Greathouse IV, Nov 19 2012

from sympy import factorint
def a(n):
    f = factorint(n)
    l = [f[p] for p in f]
    return 0 if n == 1 else min(l)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 13 2017

(define (A051904 n) (cond ((= 1 n) 0) ((= 1 (A001221 n)) (A001222 n)) (else (min (A067029 n) (A051904 (A028234 n)))))) ;; Antti Karttunen, Jul 12 2017

a(n) = min_{k=1..A001221(n)} A124010(n,k). - Reinhard Zumkeller, Aug 27 2011
a(1) = 0, for n > 1, if A001221(n) = 1 (when n is in A000961), a(n) = A001222(n), otherwise a(n) = min(A067029(n), a(A028234(n))). - Antti Karttunen, Jul 12 2017
Sum_{k=1..n} a(k) ~ n + zeta(3/2)*n^(1/2)/zeta(3) + (zeta(2/3)/zeta(2) + c0)*n^(1/3), where c0 = A362974 = Product_{p prime} (1 + 1/p^(4/3) + 1/p^(5/3)) [Cao Hui-Zhong, 1991]. - Vaclav Kotesovec, Mar 24 2025

A087207 A binary representation of the primes that divide a number, shown in decimal.

Original entry on oeis.org

0, 1, 2, 1, 4, 3, 8, 1, 2, 5, 16, 3, 32, 9, 6, 1, 64, 3, 128, 5, 10, 17, 256, 3, 4, 33, 2, 9, 512, 7, 1024, 1, 18, 65, 12, 3, 2048, 129, 34, 5, 4096, 11, 8192, 17, 6, 257, 16384, 3, 8, 5, 66, 33, 32768, 3, 20, 9, 130, 513, 65536, 7, 131072, 1025, 10, 1, 36, 19, 262144, 65, 258

Offset: 1

Mitch Cervinka (puritan(AT)planetkc.com), Oct 26 2003

The binary representation of a(n) shows which prime numbers divide n, but not the multiplicities. a(2)=1, a(3)=10, a(4)=1, a(5)=100, a(6)=11, a(10)=101, a(30)=111, etc.
For n > 1, a(n) gives the (one-based) index of the column where n is located in array A285321. A008479 gives the other index. - Antti Karttunen, Apr 17 2017
From Antti Karttunen, Jun 18 & 20 2017: (Start)
A268335 gives all n such that a(n) = A248663(n); the squarefree numbers (A005117) are all the n such that a(n) = A285330(n) = A048675(n).
For all n > 1 for which the value of A285331(n) is well-defined, we have A285331(a(n)) <= floor(A285331(n)/2), because then n is included in the binary tree A285332 and a(n) is one of its ancestors (in that tree), and thus must be at least one step nearer to its root than n itself.
Conjecture: Starting at any n and iterating the map n -> a(n), we will always reach 0 (see A288569). This conjecture is equivalent to the conjecture that at any n that is neither a prime nor a power of two, we will eventually hit a prime number (which then becomes a power of two in the next iteration). If this conjecture is false then sequence A285332 cannot be a permutation of natural numbers. On the other hand, if the conjecture is true, then A285332 must be a permutation of natural numbers, because all primes and powers of 2 occur in definite positions in that tree. This conjecture also implies the conjectures made in A019565 and A285320 that essentially claim that there are neither finite nor infinite cycles in A019565.
If there are any 2-cycles in this sequence, then both terms of the cycle should be present in A286611 and the larger one should be present in A286612.
(End)
Binary rank of the distinct prime indices of n, where the binary rank of an integer partition y is given by Sum_i 2^(y_i-1). For all prime indices (with multiplicity) we have A048675. - Gus Wiseman, May 25 2024

			a(38) = 129 because 38 = 2*19 = prime(1)*prime(8) and 129 = 2^0 + 2^7 (in binary 10000001).
a(140) = 13, binary 1101 because 140 is divisible by the first, third and fourth primes and 2^(1-1) + 2^(3-1) + 2^(4-1) = 13.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000 [First 1000 terms from _T. D. Noe_]
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization

For partial sums see A288566.
Cf. A000040, A000120, A001221, A005117, A008479, A019565, A055396, A285320, A285321, A285329, A285330, A285332.
Sequences with related definitions: A007947, A008472, A027748, A048675, A248663, A276379 (same sequence shown in base 2), A288569, A289271, A297404.
Cf. A286608 (numbers n for which a(n) < n), A286609 (n for which a(n) > n), and also A286611, A286612.
A003986, A003961, A059896 are used to express relationship between terms of this sequence.
Related to A267116 via A225546.
Positions of particular values are: A000079\{1} (1), A000244\{1} (2), A033845 (3), A000351\{1} (4), A033846 (5), A033849 (6), A143207 (7), A000420\{1} (8), A033847 (9), A033850 (10), A033851 (12), A147576 (14), A147571 (15), A001020\{1} (16), A033848 (17).
A048675 gives binary rank of prime indices.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Binary indices (listed A048793):
- length A000120, complement A023416
- min A001511, opposite A000012
- sum A029931, product A096111
- max A029837 or A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
- opposite A371572, sum A230877
Cf. A000720, A005940, A018819, A023506, A071814, A225620, A277319, A277905, A304818, A372689, A372890.

a087207 = sum . map ((2 ^) . (subtract 1) . a049084) . a027748_row
-- Reinhard Zumkeller, Jul 16 2013

a[n_] := Total[ 2^(PrimePi /@ FactorInteger[n][[All, 1]] - 1)]; a[1] = 0; Table[a[n], {n, 1, 69}] (* Jean-François Alcover, Dec 12 2011 *)

a(n) = {if (n==1, 0, my(f=factor(n), v = []); forprime(p=2, vecmax(f[,1]), v = concat(v, vecsearch(f[,1], p)!=0);); fromdigits(Vecrev(v), 2));} \\ Michel Marcus, Jun 05 2017

A087207(n)=vecsum(apply(p->1<M. F. Hasler, Jun 23 2017

from sympy import factorint, primepi
def a(n):
    return sum(2**primepi(i - 1) for i in factorint(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 06 2017

(definec (A087207 n) (if (= 1 n) 0 (+ (A000079 (+ -1 (A055396 n))) (A087207 (A028234 n))))) ;; This uses memoization-macro definec
(define (A087207 n) (A048675 (A007947 n))) ;; Needs code from A007947 and A048675. - Antti Karttunen, Jun 19 2017

Additive with a(p^e) = 2^(i-1) where p is the i-th prime. - Vladeta Jovovic, Oct 29 2003
a(n) gives the m such that A019565(m) = A007947(n). - Naohiro Nomoto, Oct 30 2003
A000120(a(n)) = A001221(n); a(n) = Sum(2^(A049084(p)-1): p prime-factor of n). - Reinhard Zumkeller, Nov 30 2003
G.f.: Sum_{k>=1} 2^(k-1)*x^prime(k)/(1-x^prime(k)). - Franklin T. Adams-Watters, Sep 01 2009
From Antti Karttunen, Apr 17 2017, Jun 19 2017 & Dec 06 2018: (Start)
a(n) = A048675(A007947(n)).
a(1) = 0; for n > 1, a(n) = 2^(A055396(n)-1) + a(A028234(n)).
A000035(a(n)) = 1 - A000035(n). [a(n) and n are of opposite parity.]
A248663(n) <= a(n) <= A048675(n). [XOR-, OR- and +-variants.]
a(A293214(n)) = A218403(n).
a(A293442(n)) = A267116(n).
A069010(a(n)) = A287170(n).
A007088(a(n)) = A276379(n).
A038374(a(n)) = A300820(n) for n >= 1.
(End)
From Peter Munn, Jan 08 2020: (Start)
a(A059896(n,k)) = a(n) OR a(k) = A003986(a(n), a(k)).
a(A003961(n)) = 2*a(n).
a(n^2) = a(n).
a(n) = A267116(A225546(n)).
a(A225546(n)) = A267116(n).
(End)

More terms from Don Reble, Ray Chandler and Naohiro Nomoto, Oct 28 2003

Name clarified by Antti Karttunen, Jun 18 2017

A285328 a(n) = 1 if n is squarefree (A005117), otherwise a(n) = Max {m < n | same prime factors as n, ignoring multiplicity}.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 6, 1, 1, 1, 8, 1, 12, 1, 10, 1, 1, 1, 18, 5, 1, 9, 14, 1, 1, 1, 16, 1, 1, 1, 24, 1, 1, 1, 20, 1, 1, 1, 22, 15, 1, 1, 36, 7, 40, 1, 26, 1, 48, 1, 28, 1, 1, 1, 30, 1, 1, 21, 32, 1, 1, 1, 34, 1, 1, 1, 54, 1, 1, 45, 38, 1, 1, 1, 50, 27, 1, 1, 42, 1, 1, 1, 44, 1, 60, 1, 46, 1, 1, 1, 72, 1, 56, 33, 80, 1, 1, 1, 52, 1, 1, 1, 96

Offset: 1

Antti Karttunen, Apr 17 2017

nonn

			From _Michael De Vlieger_, Dec 31 2018: (Start)
a(1) = 1 since 1 is squarefree.
a(2) = 1 since 2 is squarefree.
a(4) = 2 since 4 is not squarefree and 2 is the largest number less than 4 that has all the distinct prime divisors that 4 has.
a(6) = 1 since 6 is squarefree.
a(12) = 6 since 12 is not squarefree and 6 is the largest number less than 12 that has all the distinct prime divisors that 12 has. (6 is also the squarefree root of 12).
a(16) = 8 since 16 is not squarefree and 8 is the largest number less than 16 that has all the distinct prime divisors that 16 has.
a(18) = 12 since 18 is not squarefree and 12 is the largest number less than 18 that has all the distinct prime divisors that 18 has.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10000

A left inverse of A065642.
Cf. A005117, A007947, A008479, A008683, A284571, A285111, A285331, A285329.
Cf. also A079277.

Table[With[{r = DivisorSum[n, EulerPhi[#] Abs@ MoebiusMu[#] &]}, If[MoebiusMu@ n != 0, 1, SelectFirst[Range[n - 2, 2, -1], DivisorSum[#, EulerPhi[#] Abs@ MoebiusMu[#] &] == r &]]], {n, 108}] (* Michael De Vlieger, Dec 31 2018 *)

A007947(n) = factorback(factorint(n)[, 1]); \\ From Andrew Lelechenko, May 09 2014
A285328(n) = { my(r=A007947(n)); if(core(n)==n,1,n = n-r; while(A007947(n) <> r, n = n-r); n); }; \\ After Python-code below - Antti Karttunen, Apr 17 2017
A285328(n) = { my(r); if((n > 1 && !bitand(n,(n-1))),(n/2), r=A007947(n); if(r==n,1,n = n-r; while(A007947(n) <> r, n = n-r); n)); }; \\ Version optimized for powers of 2.

from operator import mul
from sympy import primefactors
from sympy.ntheory.factor_ import core
def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))
def a(n):
    if core(n) == n: return 1
    r = a007947(n)
    k = n - r
    while k>0:
        if a007947(k) == r: return k
        else: k -= r
print([a(n) for n in range(1, 121)]) # Indranil Ghosh and Antti Karttunen, Apr 17 2017

(definec (A285328 n) (if (not (zero? (A008683 n))) 1 (let ((k (A007947 n))) (let loop ((n (- n k))) (if (= (A007947 n) k) n (loop (- n k)))))))

If A008683(n) <> 0, a(n) = 1, otherwise a(n) = the largest number k < n for which A007947(k) = A007947(n).
Other identities. For all n >= 1:
a(A065642(n)) = n.

A355432 a(n) = number of k < n such that rad(k) = rad(n) and k does not divide n, where rad(k) = A007947(k).

Original entry on oeis.org

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

Offset: 1

Michael De Vlieger, Feb 22 2023

nonn

a(n) = 0 for prime powers and squarefree numbers.

			a(1) = 18, since 18/6 >= 3. We note that rad(12) = rad(18) = 6, yet 12 does not divide 18.
a(2) = 24, since 24/6 >= 3. rad(18) = rad(24) = 6 and 24 mod 18 = 6.
a(3) = 36, since 36/6 >= 3. rad(24) = rad(36) = 6 and 36 mod 24 = 12.
a(6) = 54, since 54/6 >= 3. m in {12, 24, 36, 48} are such that rad(m) = rad(54) = 6, but none divides 54, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Plot (k, n) at (x, -y), k = 1..n, n = 1..54, showing k in A126706 in dark blue, n in A360768 in dark red, and for n and nondivisor k such that rad(k) = rad(n), we highlight in large black dots. This sequence counts the number of black dots in row n.
Michael De Vlieger, Condensation of the above plot, showing k = 1..n and only n in A360768 and n <= 54.

Cf. A005361, A007947, A008479, A010846, A013929, A020639, A024619, A027750, A126706, A162306, A243822, A272618, A360589, A360768.

rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]]; Table[Which[PrimePowerQ[n], 0, SquareFreeQ[n], 0, True, r = rad[n]; Count[Select[Range[n], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], _?(And[rad[#] == r, Mod[n, #] != 0] &)]], {n, 120}]

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
a(n) = my(rn=rad(n)); sum(k=1, n-1, if (n % k, rad(k)==rn)); \\ Michel Marcus, Feb 23 2023

a(n) > 0 for n in A360768.
a(n) < A243822(n) < A010846(n).
a(n) = A008479(n) - A005361(n). - Amiram Eldar, Oct 25 2024

A067004 Number of numbers <= n with same number of divisors as n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 2, 2, 3, 5, 1, 6, 4, 5, 1, 7, 2, 8, 3, 6, 7, 9, 1, 3, 8, 9, 4, 10, 2, 11, 5, 10, 11, 12, 1, 12, 13, 14, 3, 13, 4, 14, 6, 7, 15, 15, 1, 4, 8, 16, 9, 16, 5, 17, 6, 18, 19, 17, 1, 18, 20, 10, 1, 21, 7, 19, 11, 22, 8, 20, 2, 21, 23, 12, 13, 24, 9, 22, 2, 2, 25, 23, 3, 26, 27

Offset: 1

Henry Bottomley, Dec 21 2001

nonn

			a(10)=3 since 6,8,10 each have four divisors. a(11)=5 since 2,3,5,7,11 each have two divisors.

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A000005, A008479, A058933, A067003. Inverse of A000040, A001248, A030513, A030514, A030515, A030516, A030626, A030627, A030628, A030629, A030630, A030631, A030632, A030633, A030634, A030635, A030636, A030637, A030638 etc.
Cf. also A079788, A138009.
A047983(n) = a(n)-1.

N:= 1000: # to get a(1) to a(N)
R:= Vector(N):
for n from 1 to N do
  v:= numtheory:-tau(n);
  R[v]:= R[v]+1;
  A[n]:= R[v];
od:
seq(A[n],n=1..N); # Robert Israel, May 04 2015

b[_] = 0;
a[n_] := a[n] = With[{t = DivisorSigma[0, n]}, b[t] = b[t]+1];
Array[a, 105] (* Jean-François Alcover, Dec 20 2021 *)

a(n)=my(d=numdiv(n)); sum(k=1,n,numdiv(k)==d) \\ Charles R Greathouse IV, Sep 02 2015

Ordinal transform of A000005. - Franklin T. Adams-Watters, Aug 28 2006

a(A000040(n)^(p-1)) = n if p is prime. - Robert Israel, May 04 2015

cp's OEIS Frontend

A372720 a(n) = A000005(n) - A008479(n).

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A293780 Where records occur in A008479.

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A373737 a(n) is the smallest number k in the sorted sequence S(q) = {k : rad(k) = q}, q = A120944(n), such that tau(k) - A008479(k) is not positive, where rad = A007947 and tau = A000005.

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A375011 a(n) = A373738(n) - A008479(n).

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A051903 Maximum exponent in the prime factorization of n.

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A051904 Minimal exponent in prime factorization of n.

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A087207 A binary representation of the primes that divide a number, shown in decimal.

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