A065036 -id:A065036 - cp's OEIS frontend

A089233 Number of coprime pairs of divisors > 1 of n.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 6, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 6, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 11, 0, 1, 2, 0, 1, 6, 0, 2, 1, 6, 0, 6, 0, 1, 2, 2, 1, 6, 0, 4, 0, 1, 0, 11, 1, 1, 1, 3, 0, 11, 1, 2, 1, 1, 1, 5, 0, 2, 2, 4, 0, 6, 0, 3, 6

Offset: 1

Reinhard Zumkeller, Dec 11 2003

nonn

Also the number of divisors of n^2 which do not divide n and which are less than n. See link for proof. - Andrew Weimholt, Dec 06 2009
a(A000961(n)) = 0; a(A006881(n)) = 1; a(A054753(n)) = 2; a(A065036(n)) = 3. - Robert G. Wilson v, Dec 16 2009
First occurrence of k beginning with 0: 1, 6, 12, 24, 36, 96, 30, 384, 144, 216, 288, 60, 432, 24576, 1152, 864, 120, 393216, 1728, 1572864, 180, 240, 18432, 25165824, 5184, 210, 480, 13824, 10368, 360, 15552, 960, 20736, 55296, 1179648, 31104, 900, ..., . Except for 1, each is divisible by 6. Also the first occurrence of k must occur at or before 6*2^(n-1). - Robert G. Wilson v, Dec 16 2009
a(3*2^n) = n; if x = 2^n, then a(x) = a(2*x); and if x is not a power of two, then a(x) = y then a(2*x) > y. - Robert G. Wilson v, Dec 16 2009
a(n) = 0 iff n is a prime power. - Franklin T. Adams-Watters, Aug 20 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Andrew Weimholt, Proof of an alternative characterization

a089233 n = sum $ [a063524 $ gcd u v | let ds = tail $ a027750_row n,
                                       u <- ds, v <- dropWhile (<= u) ds]
-- Reinhard Zumkeller, Sep 04 2013

[(NumberOfDivisors(n^2)-1)/2 - NumberOfDivisors(n)+1: n in [1..100]]; // Vincenzo Librandi, Dec 23 2018

a[n_] := (DivisorSigma[0, n^2] - 1)/2 - DivisorSigma[0, n] + 1; Array[a, 104] (* Robert G. Wilson v, Dec 16 2009 *)

a(n) = (numdiv(n^2)-1)/2 - numdiv(n) + 1; \\ Michel Marcus, Feb 17 2016

a(n) = #{(x,y): 1 < x < y, x|n, y|n and gcd(x, y) = 1}.
a(n) = A063647(n) - A000005(n) + 1.
a(n) = A018892(n) - A000005(n). - Franklin T. Adams-Watters, Aug 20 2013

A095904 Triangular array of natural numbers (greater than 1) arranged by prime signature.

Original entry on oeis.org

2, 3, 4, 5, 9, 6, 7, 25, 10, 8, 11, 49, 14, 27, 12, 13, 121, 15, 125, 18, 16, 17, 169, 21, 343, 20, 81, 24, 19, 289, 22, 1331, 28, 625, 40, 30, 23, 361, 26, 2197, 44, 2401, 54, 42, 32, 29, 529, 33, 4913, 45, 14641, 56, 66, 243, 36, 31, 841, 34, 6859, 50, 28561, 88, 70

Offset: 0

Alford Arnold, Jul 10 2004

The unit, 1, has the empty prime signature { } (thus not in triangle).
Downwards diagonals:
* Rightmost diagonal: smallest numbers of a given prime signature in increasing order (A025487). This defines the order of signatures used.
  This special ordering of prime signatures (by increasing smallest numbers of a given prime signature, A181087) is unrelated to any of the 8 variants of graded lexicographic or colexicographic orderings (based on the exponents only) since it depends on the magnitudes of the prime numbers. It is not even graded by Omega(n).
* Second rightmost diagonal: second smallest numbers of a given prime signature (A077560). (They are not increasing anymore.)
Upwards diagonals:
* Leftmost diagonal: primes.                           {1}     (A000040)
* 2nd leftmost diagonal: squares of primes.            {2}     (A001248)
* 3rd leftmost diagonal: squarefree biprimes.          {1,1}   (A006881)
* 4th leftmost diagonal: cubes of primes.              {3}     (A030078)
* 5th leftmost diagonal: signature (Achilles numbers)  {1,2}   (A054753)
* 6th leftmost diagonal: fourth powers of primes.      {4}     (A030514)
* 7th leftmost diagonal: signature (Achilles numbers)  {1,3}   (A065036)
* 8th leftmost diagonal: squarefree triprimes.         {1,1,1} (A007304)
  The Achilles numbers are nonsquarefree while not perfect powers.
Prime signatures are often expressed in increasing order of exponents. The decreasing order of exponents (as on the Wiki page, see links) has the advantage of listing the exponents in the same order (with the canonical factorization convention) as the smallest number of a given prime signature.

			343 is in the 4th left- and 4th rightmost diagonal, because it is the 4th value with the 4th prime signature {3}.
First 8 rows of triangular array (Cf. table link for this sequence):
                                   2
                              3         4
                         5         9         6
                    7        25        10         8
               11       49        14        27        12
          13      121        15       125        18        16
     17       169       21       343        20        81        24
19       289       22       1331       28       625        40        30

OEIS Wiki, Prime signature.
OEIS Wiki, Orderings.

Cf. A083140, A064839, A181087.

Extended by Ray Chandler, Jul 31 2004
Corrected (minor) by Daniel Forgues, Jan 21 2011
Example, comments by Daniel Forgues, Jan 21 2011
Edited by Alois P. Heinz, Jan 23 2011
Edited by Daniel Forgues, Jan 23 2011

A179668 Products of the 8th power of a prime and a distinct prime (p^8*q).

Original entry on oeis.org

768, 1280, 1792, 2816, 3328, 4352, 4864, 5888, 7424, 7936, 9472, 10496, 11008, 12032, 13122, 13568, 15104, 15616, 17152, 18176, 18688, 20224, 21248, 22784, 24832, 25856, 26368, 27392, 27904, 28928, 32512, 32805, 33536, 35072, 35584, 38144, 38656, 40192

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 23 2010

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures

Cf. A006881, A007304, A065036, A085986, A085987, A092759, A178739, A179642, A179643, A179644, A179645, A179646, A179664, A179665, A179666, A179667.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,8}; Select[Range[40000], f]
With[{nn=40},Take[Union[#[[1]]^8 #[[2]]&/@Flatten[Permutations/@Subsets[ Prime[Range[nn]],{2}],1]],nn]] (* Harvey P. Dale, Jan 20 2016 *)

list(lim)=my(v=List(),t);forprime(p=2,(lim\2)^(1/8),t=p^8;forprime(q=2,lim\t,if(p==q,next);listput(v,t*q)));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from sympy import primepi, primerange, integer_nthroot
def A179668(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//p**8) for p in primerange(integer_nthroot(x,8)[0]+1))+primepi(integer_nthroot(x,9)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

A179692 Numbers of the form p^9*q where p and q are distinct primes.

Original entry on oeis.org

1536, 2560, 3584, 5632, 6656, 8704, 9728, 11776, 14848, 15872, 18944, 20992, 22016, 24064, 27136, 30208, 31232, 34304, 36352, 37376, 39366, 40448, 42496, 45568, 49664, 51712, 52736, 54784, 55808, 57856, 65024, 67072, 70144, 71168, 76288, 77312, 80384, 83456

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures

Cf. A006881, A007304, A065036, A085986, A085987, A092759, A178739, A179642, A179643, A179644, A179645, A179646, A179664, A179665, A179666, A179667, A179668, A179669, A179670, A179671, A179672, A179688, A179689, A179690, A179691.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,9}; Select[Range[90000], f]

list(lim)=my(v=List(),t);forprime(p=2, (lim\2)^(1/9), t=p^9;forprime(q=2, lim\t, if(p==q, next);listput(v,t*q))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

from sympy import primepi, integer_nthroot, primerange
def A179692(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//p**9) for p in primerange(integer_nthroot(x,9)[0]+1))+primepi(integer_nthroot(x,10)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

A189990 Numbers with prime factorization p^2*q^6.

Original entry on oeis.org

576, 1600, 2916, 3136, 7744, 10816, 18225, 18496, 23104, 33856, 35721, 53824, 61504, 62500, 87616, 88209, 107584, 118336, 123201, 140625, 141376, 179776, 210681, 222784, 238144, 263169, 287296, 322624, 341056, 385641, 399424, 440896, 470596

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 03 2011

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, List of Prime Signatures
Index to sequences related to prime signature

Subsequence of A137484.
Cf. A179645, A179664.
Cf. A065036, A085548, A085966, A085968.

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,6}; Select[Range[800000],f]

list(lim)=my(v=List(),t);forprime(p=2, (lim\4)^(1/6), t=p^6;forprime(q=2, sqrt(lim\t), if(p==q, next);listput(v,t*q^2))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A189990(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(isqrt(x//p**6)) for p in primerange(integer_nthroot(x,6)[0]+1))+primepi(integer_nthroot(x,8)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Sum_{n>=1} 1/a(n) = P(2)*P(6) - P(8) = A085548 * A085966 - A085968 = 0.003658..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

a(n) = A065036(n)^2. - Chai Wah Wu, Mar 27 2025

A255231 The number of factorizations n = Product_i b_i^e_i, where all bases b_i are distinct, and all exponents e_i are distinct >=1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 7, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 6, 1, 2, 2, 4, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 1, 5, 1, 1, 1, 2, 2, 1, 1, 3

Offset: 1

Saverio Picozzi, Feb 18 2015

nonn

Not multiplicative: a(48) = a(2^4*3) = 5 <> a(2^4)*a(3) = 4*1 = 4. - R. J. Mathar, Nov 05 2016

			From _R. J. Mathar_, Nov 05 2016: (Start)
a(4)=2: 4^1 = 2^2.
a(8)=2: 8^1 = 2^3.
a(9)=2: 9^1 = 3^2.
a(12)=2: 12^1 = 2^2*3^1.
a(16)=4: 16^1 = 4^2 = 2^2*4^1 = 2^4.
a(18)=2: 18^1 = 2*3^2.
a(20)=2: 20^1 = 2^2*5^1.
a(24)=3: 24^1 = 2^2*6^1 = 2^3*3^1.
a(32)=5: 32^1 = 2^1*4^2 = 2^2*8^1 = 2^3*4^1 = 2^5.
a(36)=4: 36^1 = 6^2 = 3^2*4^1 = 2^2*9^1.
a(48)=5: 48^1 = 3^1*4^2 = 2^2*12^1 = 2^3*6^1 = 2^4*3^1.
a(60)=2 : 60^1 = 2^2*15^1.
a(64)=7: 64^1 = 8^2 = 4^3 = 2^2*16^1 = 2^3*8^1 = 2^4*4^1 = 2^6.
a(72)=6 : 72^1 = 3^2*8^1 = 2^1*6^2 = 2^2*18^1 = 2^3*9^1 = 2^3*3^2.
(End)

R. J. Mathar, Table of n, a(n) for n = 1..419
R. J. Mathar, Factorizations of integers into factors with distinct bases and exponents
Index to sequences related to prime signature

Cf. A000688 (b_i not necessarily distinct).

Cf. A001248, A005117, A030078, A030514, A054753, A065036, A085986, A085987, A143610, A178739.

# Count solutions for products if n = dvs_i^exps(i) where i=1..pividx are fixed
Apiv := proc(n,dvs,exps,pividx)
    local dvscnt, expscopy,i,a,expsrt,e ;
    dvscnt := nops(dvs) ;
    a := 0 ;
    if pividx > dvscnt then
        # have exhausted the exponent list: leave of the recursion
        # check that dvs_i^exps(i) is a representation
        if n = mul( op(i,dvs)^op(i,exps),i=1..dvscnt) then
            # construct list of non-0 exponents
            expsrt := [];
            for i from 1 to dvscnt do
                if op(i,exps) > 0 then
                    expsrt := [op(expsrt),op(i,exps)] ;
                end if;
            end do;
            # check that list is duplicate-free
            if nops(expsrt) = nops( convert(expsrt,set)) then
                return 1;
            else
                return 0;
            end if;
        else
            return 0 ;
        end if;
    end if;
    # need a local copy of the list to modify it
    expscopy := [] ;
    for i from 1 to nops(exps) do
        expscopy := [op(expscopy),op(i,exps)] ;
    end do:
    # loop over all exponents assigned to the next base in the list.
    for e from 0 do
        candf := op(pividx,dvs)^e ;
        if modp(n,candf) <> 0 then
            break;
        end if;
        # assign e to the local copy of exponents
        expscopy := subsop(pividx=e,expscopy) ;
        a := a+procname(n,dvs,expscopy,pividx+1) ;
    end do:
    return a;
end proc:
A255231 := proc(n)
    local dvs,dvscnt,exps ;
    if n = 1 then
        return 1;
    end if;
    # candidates for the bases are all divisors except 1
    dvs := convert(numtheory[divisors](n) minus {1},list) ;
    dvscnt := nops(dvs) ;
    # list of exponents starts at all-0 and is
    # increased recursively
    exps := [seq(0,e=1..dvscnt)] ;
    # take any subset of dvs for the bases, i.e. exponents 0 upwards
    Apiv(n,dvs,exps,1) ;
end proc:
seq(A255231(n),n=1..120) ; # R. J. Mathar, Nov 05 2016

a(n)=1 for all n in A005117. a(n)=2 for all n in A001248 and for all n in A054753 and for all n in A085987 and for all n in A030078. a(n)=3 for all n in A065036. a(n)=4 for all n in A085986 and for all n in A030514. a(n)=5 for all n in A178739, all n in A179644 and for all n in A050997. a(n)=6 for all n in A143610, all n in A162142 and all n in A178740. a(n)=7 for all n in A030516. a(n)=9 for all n in A189988 and all n in A189987. a(n)=10 for all n in A092759. a(n) = 11 for all n in A179664. a(n)=12 for all n in A179646. - R. J. Mathar, Nov 05 2016, May 20 2017

Values corrected. Incorrect comments removed. - R. J. Mathar, Nov 05 2016

A343443 If n = Product (p_j^k_j) then a(n) = Product (k_j + 2), with a(1) = 1.

Original entry on oeis.org

1, 3, 3, 4, 3, 9, 3, 5, 4, 9, 3, 12, 3, 9, 9, 6, 3, 12, 3, 12, 9, 9, 3, 15, 4, 9, 5, 12, 3, 27, 3, 7, 9, 9, 9, 16, 3, 9, 9, 15, 3, 27, 3, 12, 12, 9, 3, 18, 4, 12, 9, 12, 3, 15, 9, 15, 9, 9, 3, 36, 3, 9, 12, 8, 9, 27, 3, 12, 9, 27, 3, 20, 3, 9, 12, 12, 9, 27, 3, 18

Offset: 1

Ilya Gutkovskiy, Apr 15 2021

Inverse Moebius transform of A056671.

a(n) depends only on the prime signature of n (see formulas). - Bernard Schott, May 03 2021

Cf. A000005, A001221, A005361, A007425, A025847, A034444, A056671, A064549, A074816, A107758, A107759, A348018, A363194.

a[1] = 1; a[n_] := Times @@ ((#[[2]] + 2) & /@ FactorInteger[n]); Table[a[n], {n, 80}]
a[n_] := Sum[If[GCD[d, n/d] == 1, DivisorSigma[0, d], 0], {d, Divisors[n]}]; Table[a[n], {n, 80}]

a(n) = sumdiv(n, d, if(gcd(d, n/d)==1, numdiv(d))) \\ Andrew Howroyd, Apr 15 2021

for(n=1, 100, print1(direuler(p=2, n, (1 + X - X^2)/(1-X)^2)[n], ", ")) \\ Vaclav Kotesovec, Feb 11 2023

from math import prod
from sympy import factorint
def A343443(n): return prod(e+2 for e in factorint(n).values()) # Chai Wah Wu, Feb 21 2025

a(n) = 2^omega(n) * tau_3(n) / tau(n), where omega = A001221, tau = A000005 and tau_3 = A007425.
a(n) = Sum_{d|n, gcd(d, n/d) = 1} tau(d).
From Bernard Schott, May 03 2021: (Start)
a(p^k) = k+2 for p prime, or signature [k].
a(A006881(n)) =  9 for signature [1, 1].
a(A054753(n)) = 12 for signature [2, 1].
a(A065036(n)) = 15 for signature [3, 1].
a(A085986(n)) = 16 for signature [2, 2].
a(A178739(n)) = 18 for signature [4, 1].
a(A143610(n)) = 20 for signature [3, 2].
a(A007304(n)) = 27 for signature [1, 1, 1]. (End)
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 1/p^s - 1/p^(2*s)). - Vaclav Kotesovec, Feb 11 2023
From Amiram Eldar, Sep 01 2023: (Start)
a(n) = A000005(A064549(n)).
a(n) = A363194(A348018(n)). (End)

A374459 Nonsquarefree exponentially odd numbers.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 88, 96, 104, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 216, 224, 232, 243, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 352, 375, 376, 378, 384, 408, 416, 424, 440, 456, 459, 472, 480, 486, 488, 512, 513, 520, 536

Offset: 1

Amiram Eldar, Jul 09 2024

First differs from A301517 at n = 1213. A301517(1213) = 12500 = 2^2 * 5^5 is not an exponentially odd number.
Numbers whose exponents in their prime factorization are all odd and at least one of them is larger than 1.
All the squarefree numbers (A005117) are exponentially odd. Therefore, the sequence of exponentially odd numbers (A268335) is a disjoint union of the squarefree numbers and this sequence.
The asymptotic density of this sequence is A065463 - A059956 = 0.096515099145... .

			8 = 2^3 is a term since 3 is odd and larger than 1.

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

Intersection of A013929 (or A046099) and A268335.
Subsequence of A301517.
Subsequences: A062838 \ {1}, A065036, A102838, A113850, A113852, A179671, A190011, A335988 \ {1}.
Cf. A005117, A374460.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, AllTrue[e, OddQ] && ! AllTrue[e, # == 1 &]]; Select[Range[1000], q]

is(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!(e[i] %2), return(0))); for(i = 1, #e, if(e[i] >1, return(1))); 0;}

a(n) = A268335(A374460(n)).

Sum_{n>=1} 1/a(n)^s = zeta(2*s) * (Product_{p prime} (1 + 1/p^s - 1/p^(2*s))) - zeta(s)/zeta(2*s) for s > 1.

A179689 Numbers with prime signature {7,2}, i.e., of form p^7*q^2 with p and q distinct primes.

Original entry on oeis.org

1152, 3200, 6272, 8748, 15488, 21632, 36992, 46208, 54675, 67712, 107163, 107648, 123008, 175232, 215168, 236672, 264627, 282752, 312500, 359552, 369603, 445568, 476288, 574592, 632043, 645248, 682112, 703125, 789507, 798848, 881792, 1013888

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, Numbers with same prime signature.
Will Nicholes, Prime Signatures

Cf. A006881, A007304, A065036, A085986, A085987, A092759, A178739, A179642, A179643, A179644, A179645, A179646, A179664, A179665, A179666, A179667, A179668, A179669, A179670, A179671, A179672, A179688.

Cf. A085548, A085967, A085969.

a:= proc(n) option remember; local k;
      for k from 1+ `if` (n=1, 1, a(n-1))
        while sort (map (x-> x[2], ifactors(k)[2]), `>`)<>[7, 2]
      do od; k
    end:
seq (a(n), n=1..32);  # Alois P. Heinz, Jan 23 2011

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,7}; Select[Range[10^6], f]

list(lim)=my(v=List(),t);forprime(p=2, (lim\4)^(1/7), t=p^7;forprime(q=2, sqrt(lim\t), if(p==q, next);listput(v,t*q^2))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A179689(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(isqrt(x//p**7)) for p in primerange(integer_nthroot(x,7)[0]+1))+primepi(integer_nthroot(x,9)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Sum_{n>=1} 1/a(n) = P(2)*P(7) - P(9) = A085548 * A085967 - A085969 = 0.001741..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

Title edited by Daniel Forgues, Jan 22 2011

A179696 Numbers with prime signature {7,1,1}, i.e., of form p^7qr with p, q and r distinct primes.

Original entry on oeis.org

1920, 2688, 4224, 4480, 4992, 6528, 7040, 7296, 8320, 8832, 9856, 10880, 11136, 11648, 11904, 12160, 14208, 14720, 15232, 15744, 16512, 17024, 18048, 18304, 18560, 19840, 20352, 20608, 21870, 22656, 23424, 23680, 23936, 25728, 25984, 26240, 26752, 27264

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes List of Prime Signatures
OEIS Wiki, Numbers with same prime signature.

Cf. A006881, A007304, A065036, A085986, A085987, A092759, A178739, A179642, A179643, A179644, A179645, A179646, A179664, A179665, A179666, A179667, A179668, A179669, A179670, A179671, A179672, A179688, A179689, A179690, A179691, A179692, A179693, A179694, A179695.

a:= proc(n) option remember; local k;
      for k from 1+ `if` (n=1, 1, a(n-1))
        while sort (map (x-> x[2], ifactors(k)[2]), `>`)<>[7, 1, 1]
      do od; k
    end:
seq (a(n), n=1..40); # Alois P. Heinz, Jan 23 2011

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,7}; Select[Range[30000], f]

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\6)^(1/7), t1=p^7;forprime(q=2, lim\t1, if(p==q, next);t2=t1*q;forprime(r=q+1, lim\t2, if(p==r,next);listput(v,t2*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from math import isqrt
from sympy import primerange, primepi, integer_nthroot
def A179696(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x+sum((t:=primepi(s:=isqrt(y:=x//r**7)))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(integer_nthroot(x,7)[0]+1))+sum(primepi(x//p**8) for p in primerange(integer_nthroot(x,8)[0]+1))-primepi(integer_nthroot(x,9)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

Title edited by Daniel Forgues, Jan 22 2011

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