A051037 -id:A051037 - cp's OEIS frontend

A355583 a(n) is the number of the 5-smooth divisors of n.

1, 2, 2, 3, 2, 4, 1, 4, 3, 4, 1, 6, 1, 2, 4, 5, 1, 6, 1, 6, 2, 2, 1, 8, 3, 2, 4, 3, 1, 8, 1, 6, 2, 2, 2, 9, 1, 2, 2, 8, 1, 4, 1, 3, 6, 2, 1, 10, 1, 6, 2, 3, 1, 8, 2, 4, 2, 2, 1, 12, 1, 2, 3, 7, 2, 4, 1, 3, 2, 4, 1, 12, 1, 2, 6, 3, 1, 4, 1, 10, 5, 2, 1, 6, 2, 2

Offset: 1

Amiram Eldar, Jul 08 2022

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

Cf. A000005, A007814, A007949, A051037, A072078, A112765, A355582, A355584.

a[n_] := Times @@ (1 + IntegerExponent[n, {2, 3, 5}]); Array[a, 100]

a(n) = (valuation(n, 2) + 1) * (valuation(n, 3) + 1) * (valuation(n, 5) + 1);

from sympy import multiplicity as v
def a(n): return (v(2, n)+1)*(v(3, n)+1)*(v(5, n)+1)
print([a(n) for n in range(1, 87)]) # Michael S. Branicky, Jul 08 2022

Multiplicative with a(p^e) = e+1 if p <= 5 and 1 otherwise.
a(n) = (A007814(n) + 1)*(A007949(n) + 1)*(A112765(n) + 1).
a(n) = A000005(A355582(n)).
a(n) <= A000005(n), with equality if and only if n is in A051037.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 15/4.
Dirichlet g.f.: zeta(s)/((1-1/2^s)*(1-1/3^s)*(1-1/5^s)). - Amiram Eldar, Dec 25 2022

A108347 Numbers of the form (3^i)(5^j)(7^k), with i, j, k >= 0.

Original entry on oeis.org

1, 3, 5, 7, 9, 15, 21, 25, 27, 35, 45, 49, 63, 75, 81, 105, 125, 135, 147, 175, 189, 225, 243, 245, 315, 343, 375, 405, 441, 525, 567, 625, 675, 729, 735, 875, 945, 1029, 1125, 1215, 1225, 1323, 1575, 1701, 1715, 1875, 2025, 2187, 2205, 2401, 2625, 2835, 3087

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jul 01 2005

nonn

The Heinz numbers of the partitions into parts 2,3, and 4 (including the number 1, the Heinz number of the empty partition). 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 [2,3,3,4] the Heinz number is 3*5*5*7 = 525; it is in the sequence. - Emeric Deutsch , May 21 2015

Numbers m | 105^e with integer e >= 0. - Michael De Vlieger, Aug 22 2019

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A003586, A003591-A003595, A051037, A108319, A108513, A215366.

The odd terms of A002473.

[n: n in [1..4000] | PrimeDivisors(n) subset [3,5,7]]; // Bruno Berselli, Sep 24 2012

with(numtheory): S := {}: for j to 3100 do if `subset`(factorset(j), {3, 5, 7}) then S := `union`(S, {j}) else end if end do: S; # Emeric Deutsch, May 21 2015
# alternative
isA108347 := proc(n)
      if n = 1 then
        true;
    else
        return (numtheory[factorset](n) minus {3, 5, 7} = {} );
    end if;
end proc:
A108347 := proc(n)
     option remember;
     if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA108347(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A108347(n),n=1..80); # R. J. Mathar, Jun 06 2024

With[{n = 3087}, Sort@ Flatten@ Table[3^i * 5^j * 7^k, {i, 0, Log[3, n]}, {j, 0, Log[5, n/2^i]}, {k, 0, Log[7, n/(3^i*5^j)]}]] (* Michael De Vlieger, Aug 22 2019 *)

from sympy import integer_log
def A108347(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n+x
        for i in range(integer_log(x,7)[0]+1):
            for j in range(integer_log(m:=x//7**i,5)[0]+1):
                c -= integer_log(m//5**j,3)[0]+1
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

Sum_{n>=1} 1/a(n) = (3*5*7)/((3-1)*(5-1)*(7-1)) = 35/16. - Amiram Eldar, Sep 22 2020

a(n) ~ exp((6*log(3)*log(5)*log(7)*n)^(1/3)) / sqrt(105). - Vaclav Kotesovec, Sep 23 2020

A003965 Fully multiplicative with a(prime(k)) = Fibonacci(k+2).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 8, 9, 10, 13, 12, 21, 16, 15, 16, 34, 18, 55, 20, 24, 26, 89, 24, 25, 42, 27, 32, 144, 30, 233, 32, 39, 68, 40, 36, 377, 110, 63, 40, 610, 48, 987, 52, 45, 178, 1597, 48, 64, 50, 102, 84, 2584, 54, 65, 64, 165, 288, 4181, 60, 6765, 466, 72, 64, 105, 78, 10946

Offset: 1

Marc LeBrun, N. J. A. Sloane

Numbers k such that a(k) = k are exactly 5-smooth numbers (A051037). - Ivan Neretin, Aug 30 2015

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000045, A000720, A051037, A337669.

Table[Times @@ (Fibonacci[PrimePi[#[[1]]] + 2]^#[[2]] & /@ FactorInteger[n]), {n, 67}] (* Ivan Neretin, Aug 30 2015 *)

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = fibonacci(primepi(f[k, 1])+2)); factorback(f); \\ Michel Marcus, Jan 14 2021

If n = Product p(k)^e(k) then a(n) = Product Fibonacci(k+2)^e(k).
Multiplicative with a(p^e) = A000045(A000720(p)+2)^e. - David W. Wilson, Aug 01 2001
Sum_{n>=1} 1/a(n) = 1 / A337669 = 5.269005... . - Amiram Eldar, Dec 24 2022

A051916 The Greek sequence: 2^a * 3^b * 5^c where a = 0,1,2,3,..., b,c in {0,1}, excluding the terms 1,2; that is: (a,b,c) != (0,0,0), (1,0,0).

Original entry on oeis.org

3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, 80, 96, 120, 128, 160, 192, 240, 256, 320, 384, 480, 512, 640, 768, 960, 1024, 1280, 1536, 1920, 2048, 2560, 3072, 3840, 4096, 5120, 6144, 7680, 8192, 10240, 12288, 15360, 16384, 20480

Offset: 1

Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Dec 17 1999

From Reinhard Zumkeller, Mar 19 2010: (Start)
Union of A007283, A020707, A020714, and A110286.
Intersection of A051037 and A003401 apart from terms 1 and 2. (End)

George E. Martin, Geometric Constructions, New York: Springer, 1997, p. 140.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2).

Cf. A003401, A007283, A020707, A020714, A051037, A110286.

CoefficientList[Series[x(3x^7+2x^6+2x^5+2x^4+6x^3+5x^2+4x+3)/(1-2x^4),{x,0,60}],x] (* Harvey P. Dale, Dec 23 2012 *)

Vec(x*(3*x^7+2*x^6+2*x^5+2*x^4+6*x^3+5*x^2+4*x+3)/(1-2*x^4)+O(x^99)) \\ Charles R Greathouse IV, Oct 12 2012

def A051916(n): return n+2 if n<5 else (15,1,5,3)[m:=n&3]<<(n>>2)+(-2,2,0,1)[m] # Chai Wah Wu, Apr 02 2025

G.f.: x*(3*x^7 + 2*x^6 + 2*x^5 + 2*x^4 + 6*x^3 + 5*x^2 + 4*x + 3)/(1 - 2*x^4).
a(n+4) = 2*a(n) for n > 8. - Reinhard Zumkeller, Mar 19 2010
Sum_{n>=1} 1/a(n) = 17/10. - Amiram Eldar, Jan 18 2023

More terms from James Sellers, Dec 18 1999

Offset corrected by Reinhard Zumkeller, Mar 10 2010

A108319 Numbers of the form (2^i)(3^j)(7^k), with i, j, k >= 0.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 27, 28, 32, 36, 42, 48, 49, 54, 56, 63, 64, 72, 81, 84, 96, 98, 108, 112, 126, 128, 144, 147, 162, 168, 189, 192, 196, 216, 224, 243, 252, 256, 288, 294, 324, 336, 343, 378, 384, 392, 432, 441, 448, 486, 504, 512, 567

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Jun 30 2005

Numbers m | 42^e with integer e >= 0. - Michael De Vlieger, Aug 22 2019

Sum_{n>=1} 1/a(n) = (2*3*7)/((2-1)*(3-1)*(7-1)) = 7/2. - Amiram Eldar, Sep 24 2020

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

Cf. A051037, A003586, A003591, A003594.

With[{n = 567}, Sort@ Flatten@ Table[2^i * 3^j * 7^k, {i, 0, Log2@ n}, {j, 0, Log[3, n/2^i]}, {k, 0, Log[7, n/(2^i*3^j)]}]] (* Michael De Vlieger, Aug 22 2019 *)

list(lim)=my(v=List(), s, t); for(i=0, logint(lim\=1, 7), t=7^i; for(j=0, logint(lim\t, 3), s=t*3^j; while(s<=lim, listput(v, s); s<<=1))); Set(v) \\ Charles R Greathouse IV, Nov 20 2024

a(n) ~ exp((6*log(2)*log(3)*log(7)*n)^(1/3)) / sqrt(42). - Vaclav Kotesovec, Sep 23 2020

A112759 Total number of prime factors of n-th 5-smooth number.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Sep 18 2005

nonn

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

Cf. A001222, A051037, A069352, A112754, A112758, A112760, A112761, A112762.

PrimeOmega @ Select[Range[3000], Last @ Map[First, FactorInteger[#]] <= 5 &] (* Amiram Eldar, Feb 07 2020 *)

a(n) = A001222(A051037(n));

a(n) = A112760(n) + A112761(n) + A112762(n).

A125624 Array read by antidiagonals: n-th row contains the positive integers with their largest prime factor equal to the n-th prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 7, 10, 9, 16, 11, 14, 15, 12, 32, 13, 22, 21, 20, 18, 64, 17, 26, 33, 28, 25, 24, 128, 19, 34, 39, 44, 35, 30, 27, 256, 23, 38, 51, 52, 55, 42, 40, 36, 512, 29, 46, 57, 68, 65, 66, 49, 45, 48, 1024, 31, 58, 69, 76, 85, 78, 77, 56, 50, 54, 2048, 37, 62, 87, 92

Offset: 1

Leroy Quet, Jan 27 2007

This sequence is a permutation of the integers >= 2.
Since the table has been entered by rising instead of falling antidiagonals, the sequence represents the transpose, with columns instead of rows: cf. the "table" link, section "infinite square array". - M. F. Hasler, Oct 22 2019
Start with table headed by primes in the first row, then list beneath each prime(k) the ordered prime(k)-smooth numbers. Read the table by falling antidiagonals to get the terms of this sequence. - David James Sycamore, Jun 23 2024

			Array begins: (rows here appear as columns in the "table" display of the sequence)
   2,  4,  8, 16, 32, 64, 128, 256, 512, ... (A000079)
   3,  6,  9, 12, 18, 24,  27,  36,  48, ... (A065119)
   5, 10, 15, 20, 25, 30,  40,  45,  50, ... (A080193)
   7, 14, 21, 28, 35, 42,  49,  56,  63, ... (A080194)
  11, 22, 33, 44, 55, 66,  77,  88,  99, ... (A080195)
  13, 26, 39, 52, 65, 78,  91, 104, 117, ... (A080196)
The 3rd row, for example, contains the positive integers where the 3rd prime, 5, is the largest prime divisor. That is, each integer in this row is divisible by 5 and may be divisible by 2 or 3 as well, but none of the integers in this row are divisible by primes larger than 5. (So for example, 35 = 5*7 is excluded from the 3rd row.)

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A083140, A006530, A000040 (1st col), A033286 (main diag), A077320.

Cf. A000079, A003586, A051037, A002473, A051038, A080197, A080681.

lpf[n_] := FactorInteger[n][[ -1, 1]];f[n_, m_] := f[n, m] = Block[{k},k = If[m == 1, Prime[n], f[n, m - 1] + 1];While[lpf[k] != Prime[n], k++ ];k];Table[f[ d - m + 1, m], {d, 12}, {m, d}] // Flatten (* Ray Chandler, Feb 09 2007 *)

T=List(); r=c=1; for(n=1,99, #TT[r][1], ); print1(T[r][c]","); r-- && c++ || r=c+c=1) \\ M. F. Hasler, Oct 22 2019

Extended by Ray Chandler, Feb 09 2007

A369374 Powerful numbers k that have a primorial kernel and more than 1 distinct prime factor.

Original entry on oeis.org

36, 72, 108, 144, 216, 288, 324, 432, 576, 648, 864, 900, 972, 1152, 1296, 1728, 1800, 1944, 2304, 2592, 2700, 2916, 3456, 3600, 3888, 4500, 4608, 5184, 5400, 5832, 6912, 7200, 7776, 8100, 8748, 9000, 9216, 10368, 10800, 11664, 13500, 13824, 14400, 15552, 16200

Offset: 1

Michael De Vlieger, Jan 22 2024

nonn

Numbers k such that Omega(k) > omega(k) > 1 with all prime power factors p^m for m > 1, such that squarefree kernel rad(k) is in A002110, where Omega = A001222, omega = A001221, and rad(k) = A007947(k).
Union of the product of the squares of primorials P(n)^2, n > 1, and the set of prime(n)-smooth numbers.
Superset of A364930.
Proper subset of A367268, which in turn is a proper subset of A126706.

			This sequence is the union of the following infinite sets:
P(2)^2 * A003586 = {36, 72, 108, 144, 216, 288, 324, ...}
                 = { m*P(2)^2 : rad(m) | P(2) }.
P(3)^2 * A051037 = {900, 1800, 2700, 3600, 4500, 5400, ...}
                 = { m*P(3)^2 : rad(m) | P(3) }.
P(4)^2 * A002473 = {44100, 88200, 132300, 176400, ...}
                 = { m*P(4)^2 : rad(m) | P(4) }, etc.

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

Cf. A001221, A001222, A001694, A002110, A007947, A055932, A126706, A286708, A364930, A367268, A369417.

With[{nn = 2^14},
  Select[
    Select[Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
      Not@*PrimePowerQ],
    And[EvenQ[#],
      Union@ Differences@ PrimePi[FactorInteger[#][[All, 1]]] == {1}] &] ]

{a(n)} = { m*P(n)^2 : P(n) = Product_{j = 1..n} prime(n), rad(m) | P(n), n > 1 }.
Intersection of A286708 and A055932.
A286708 is the union of A369417 and this sequence.

A018255 Divisors of 30.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 15, 30

Offset: 1

N. J. A. Sloane

For n > 1, These are also numbers m such that k^4 + (k+1)^4 + ... + (k + m - 1)^4 is prime for some k and numbers m such that k^8 + (k+1)^8 + ... + (k + m - 1)^8 is prime for some k. - Derek Orr, Jun 12 2014
These seem to be the numbers m such that tau(n) = n*sigma(n) mod m for all n. See A098108 (mod 2), A126825 (mod 3), and A126832 (mod 5). - Charles R Greathouse IV, Mar 17 2022
The squarefree 5-smooth numbers: intersection of A051037 and A005117. - Amiram Eldar, Sep 26 2023

			From the second comment: 1^3 + 2^3 + 2^3 + 2^3 + 4^3 + 4^3 + 4^3 + 8^3 = (1 + 2 + 2 + 2 + 4 + 4 + 4 + 8)^2 = 729. - _Bruno Berselli_, Dec 28 2014

Boris A. Kordemsky, The Moscow Puzzles: 359 Mathematical Recreations, C. Scribner's Sons (1972), Chapter XIII, Paragraph 349.

Edward Barbeau and Samer Seraj, Sum of Cubes is Square of Sum, arXiv:1306.5257 [math.NT], 2013.
Index entries for sequences related to divisors of numbers.

Cf. A000005, A005117, A051037, A158649, A161715.

Cf. A098108, A126825, A126832.

Divisors(30); // Bruno Berselli, Dec 28 2014

Divisors[30] (* Vladimir Joseph Stephan Orlovsky, Dec 04 2010 *)

PARI
```
divisors(30)
```

a(n) = A161715(n-1). - Reinhard Zumkeller, Jun 21 2009

Sum_{i=1..8} A000005(a(i))^3 = (Sum_{i=1..8} A000005(a(i)))^2, see Kordemsky in References and Barbeau et al. in Links section. - Bruno Berselli, Dec 28 2014

A112762 Exponent of 5 (value of k) in n-th number of the form 2^i3^j5^k.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Sep 18 2005

nonn

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

Cf. A022337, A051037, A112765, A112760, A112761.

IntegerExponent[#, 5] & /@ Select[Range[3000], Last @ Map[First, FactorInteger[#]] <= 5 &] (* Amiram Eldar, Feb 07 2020 *)

a(n) = A112765(A051037(n));

a(n) = A112759(n) - A112760(n) - A112761(n).

cp's OEIS Frontend

A355583 a(n) is the number of the 5-smooth divisors of n.

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A108347 Numbers of the form (3^i)*(5^j)*(7^k), with i, j, k >= 0.

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A003965 Fully multiplicative with a(prime(k)) = Fibonacci(k+2).

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A051916 The Greek sequence: 2^a * 3^b * 5^c where a = 0,1,2,3,..., b,c in {0,1}, excluding the terms 1,2; that is: (a,b,c) != (0,0,0), (1,0,0).

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A108319 Numbers of the form (2^i)*(3^j)*(7^k), with i, j, k >= 0.

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A112759 Total number of prime factors of n-th 5-smooth number.

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A125624 Array read by antidiagonals: n-th row contains the positive integers with their largest prime factor equal to the n-th prime.

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A108347 Numbers of the form (3^i)(5^j)(7^k), with i, j, k >= 0.

A108319 Numbers of the form (2^i)(3^j)(7^k), with i, j, k >= 0.

A112762 Exponent of 5 (value of k) in n-th number of the form 2^i3^j5^k.