A072078 -id:A072078 - cp's OEIS frontend

A065333 Characteristic function of 3-smooth numbers, i.e., numbers of the form 2^i*3^j (i, j >= 0).

1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Oct 29 2001

Dirichlet inverse of b(n) where b(n) = 0 except for: b(1) = b(6) = -b(2) = -b(3) = 1. - Alexander Adam, Dec 26 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
A. Pakapongpun and T. Ward, Functorial Orbit counting, JIS 12 (2009) 09.2.4, example 9.
Index entries for characteristic functions

Characteristic function of A003586.

Cf. A000265, A007814, A007949, A038502, A065330, A065332, A071521 (partial sums), A072078 (inverse Möbius transform).

a065333 = fromEnum . (== 1) . a038502 . a000265
-- Reinhard Zumkeller, Jan 08 2013, Apr 12 2012

a[n_] := Boole[ 2^IntegerExponent[n, 2] * 3^IntegerExponent[n, 3] == n]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, May 16 2013, after Charles R Greathouse IV *)

a(n)=sumdiv(n,d,moebius(6*d)) \\ Benoit Cloitre, Oct 18 2009

a(n)=3^valuation(n,3)<Charles R Greathouse IV, Aug 21 2011

from sympy import multiplicity
def A065333(n): return int(3**(multiplicity(3,m:=n>>(~n&n-1).bit_length()))==m) # Chai Wah Wu, Dec 20 2024

a(n) = if n = A003586(k) for some k then 1 else 0.
a(n) = signum(A065332(n)), where signum = A057427.
a(n) = if A065330(n) = 1 then 1 else 0 = 1 - signum(A065330(n) - 1).
a(n) = Product_{p prime and p|n} 0^floor(p/4). - Reinhard Zumkeller, Nov 19 2004
Multiplicative with a(2^e) = a(3^e) = 1, a(p^e) = 0 for prime p > 3. Dirichlet g.f. 1/(1-2^-s)/(1-3^-s). - Franklin T. Adams-Watters, Sep 01 2006
a(n) = 0^(A038502(A000265(n)) - 1). - Reinhard Zumkeller, Sep 28 2008
a(n) = Sum_{d|n} mu(6*d). - Benoit Cloitre, Oct 18 2009

A060753 Denominator of 1246...(prime(n-1)-1) / (2357...prime(n-1)).

Original entry on oeis.org

1, 2, 3, 15, 35, 77, 1001, 17017, 323323, 676039, 2800733, 86822723, 3212440751, 131710070791, 5663533044013, 11573306655157, 47183480978717, 95993978542907, 5855632691117327, 392327390304860909

Offset: 1

Frank Ellermann, Apr 23 2001

Equivalently, numerator of Product_{k=1..n-1} prime(k)/(prime(k)-1) (cf. A038110). - N. J. A. Sloane, Apr 17 2015
a(n)/A038110(n) is the supremum of the abundancy index sigma(k)/k = A000203(k)/k of the prime(n-1)-smooth numbers, for n>1 (Laatsch, 1986). - Amiram Eldar, Oct 26 2021
From Amiram Eldar, Jul 10 2022: (Start)
a(n)/A038110(n) is the sum of the reciprocals of the prime(n-1)-smooth numbers, for n>1.
a(n)/A038110(n) is the asymptotic mean of the number of prime(n-1)-smooth divisors of the positive integers, for n>1 (cf. A001511, A072078, A355583). (End)

			A038110(50)/ a(50) = 0.1020..., exp(-gamma)/log(229) = 0.1033...
1*2*4/(2*3*5) = 4/15 has denominator a(4) = 15. - _Jonathan Sondow_, Jan 31 2014

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 429.

Michael De Vlieger, Table of n, a(n) for n = 1..423
Frank Ellermann, Illustration for A002110, A005867, A038110, A060753.
Richard Laatsch, Measuring the abundancy of integers, Mathematics Magazine, Vol. 59, No. 2 (1986), pp. 84-92.
Jonathan Sondow and Eric Weisstein, Euler Product, MathWorld.

Cf. A000203, A002110, A005867, A038110, A038111.
Cf. A236435, A236436.
Cf. A001511, A072078, A355583.

[1] cat [Denominator((&*[NthPrime(k-1)-1:k in [2..n]])/(&*[NthPrime(k-1):k in [2..n]])):n in [2..20]]; // Marius A. Burtea, Sep 19 2019

Table[Denominator@ Product[EulerPhi@ Prime[i]/Prime@ i, {i, n}], {n, 0, 19}] (* Michael De Vlieger, Jan 10 2015 *)
{1}~Join~Denominator@ FoldList[Times, Table[EulerPhi@ Prime[n]/Prime@ n, {n, 19}]] (* Michael De Vlieger, Jul 26 2016 *)
b[0] := 0; b[n_] := b[n - 1] + (1 - b[n - 1]) / Prime[n]
Denominator@ Table[b[n], {n, 0, 20}] (* Fred Daniel Kline, Jun 27 2017 *)
Join[{1},Denominator[With[{nn=20},FoldList[Times,Prime[Range[nn]]-1]/FoldList[ Times,Prime[Range[nn]]]]]] (* Harvey P. Dale, Apr 17 2022 *)

a(n) = A002110(n) / gcd( A005867(n), A002110(n) ).
A038110(n) / a(n) ~ exp( -gamma ) / log( prime(n) ), Mertens's theorem for x = prime(n) = A000040(n).
A038110(n) / a(n) = A005867(n) / A002110(n). - corrected by Simon Tatham, Jul 26 2016
a(n) = A038111(n) / prime(n). - Vladimir Shevelev, Jan 10 2014
a(n) = A038110(n) + A161527(n-1). - Jamie Morken, Jun 19 2019

Definition corrected by Jonathan Sondow, Jan 31 2014

A322026 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(i) = A007814(j) and A007949(i) = A007949(j), for all i, j, where A007814 and A007949 give the 2- and 3-adic valuations of n.

Original entry on oeis.org

1, 2, 3, 4, 1, 5, 1, 6, 7, 2, 1, 8, 1, 2, 3, 9, 1, 10, 1, 4, 3, 2, 1, 11, 1, 2, 12, 4, 1, 5, 1, 13, 3, 2, 1, 14, 1, 2, 3, 6, 1, 5, 1, 4, 7, 2, 1, 15, 1, 2, 3, 4, 1, 16, 1, 6, 3, 2, 1, 8, 1, 2, 7, 17, 1, 5, 1, 4, 3, 2, 1, 18, 1, 2, 3, 4, 1, 5, 1, 9, 19, 2, 1, 8, 1, 2, 3, 6, 1, 10, 1, 4, 3, 2, 1, 20, 1, 2, 7, 4, 1, 5, 1, 6, 3

Offset: 1

Antti Karttunen, Dec 03 2018

nonn

Restricted growth sequence transform of the ordered pair [A007814(n), A007949(n)].
For all i, j:
  A305900(i) = A305900(j) => a(i) = a(j),
  a(i) = a(j) => A122841(i) = A122841(j),
  a(i) = a(j) => A244417(i) = A244417(j),
  a(i) = a(j) => A322316(i) = A322316(j) => A072078(i) = A072078(j).
If and only if a(k) > a(i) for all k > i then k is in A003586, - David A. Corneth, Dec 03 2018
That is, A003586 gives the positions of records (1, 2, 3, 4, 5, ...) in this sequence.
Sequence A126760 (without its initial zero) and this sequence are ordinal transforms of each other.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Jon Maiga, Computer-generated formulas for A322026, Sequence Machine.

Cf. A003586 (positions of records, the first occurrence of n), A007814, A007949, A065331, A071521, A072078, A087465, A122841, A126760 (ordinal transform), A322316, A323883, A323884.

Cf. also A247714 and A255975.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007814(n) = valuation(n,2);
A007949(n) = valuation(n,3);
v322026 = rgs_transform(vector(up_to, n, [A007814(n), A007949(n)]));
A322026(n) = v322026[n];

A065331(n) = (3^valuation(n, 3)<A065331
A071521(n) = { my(t=1/3); sum(k=0, logint(n, 3), t*=3; logint(n\t, 2)+1); }; \\ From A071521.
A322026(n) = A071521(A065331(n)); \\ Antti Karttunen, Sep 08 2024

For s = A003586(n), a(s) = n = a((6k+1)*s) = a((6k-1)*s), where s is the n-th 3-smooth number and k > 0. - David A. Corneth, Dec 03 2018
A065331(n) = A003586(a(n)). - David A. Corneth, Dec 04 2018
From Antti Karttunen, Sep 08 2024: (Start)
a(n) = Sum{k=1..n} [A126760(k)==A126760(n)], where [ ] is the Iverson bracket.
a(n) = A071521(A065331(n)). [Found by Sequence Machine and also by LODA miner]
a(n) = A323884(25*n). [Conjectured by Sequence Machine]
(End)

A072079 Sum of 3-smooth divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 1, 12, 1, 15, 13, 3, 1, 28, 1, 3, 4, 31, 1, 39, 1, 7, 4, 3, 1, 60, 1, 3, 40, 7, 1, 12, 1, 63, 4, 3, 1, 91, 1, 3, 4, 15, 1, 12, 1, 7, 13, 3, 1, 124, 1, 3, 4, 7, 1, 120, 1, 15, 4, 3, 1, 28, 1, 3, 13, 127, 1, 12, 1, 7, 4, 3, 1, 195, 1, 3, 4, 7, 1, 12, 1, 31, 121

Offset: 1

Reinhard Zumkeller, Jun 13 2002

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

Cf. A000203, A003586, A065331, A072078.

Cf. A001620, A082633.

f[p_, e_] := If[p > 3, 1, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)

a(n) = (2^(valuation(n, 2)+1)-1)*(3^(valuation(n, 3)+1)-1)/2; \\ Amiram Eldar, Dec 01 2022

a(n) = (2^(A007814(n)+1)-1)*(3^(A007949(n)+1)-1)/2.
a(n) = A000203(A065331(n)).
Multiplicative with a(2^e) = 2^(e+1)-1, a(3^e) = (3^(e+1)-1)/2, a(p^e) = 1, p>3. Christian G. Bower, May 20 2005
From Amiram Eldar, Dec 01 2022: (Start)
Dirichlet g.f.: zeta(s)*(2^s/(2^s-2))*(3^s/(3^s-3)).
Sum_{k=1..n} a(k) ~ c_1 * (n * log(n)^2 + c_2 * n * log(n) + c_3 * n), where c_1 = 1/(2*log(2)*log(3)) = 0.656598..., c_2 = (2*gamma - 2 + log(6)) = 0.9461907..., and c_3 = (log(6)^2 + log(2)*log(3))/6 - (log(6)-2)*(1-gamma) - 2*gamma_1 = 0.895656..., gamma is Euler's constant (A001620), and gamma_1 is the 1st Stieltjes constant (A082633). (End)

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

Original entry on oeis.org

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

A382488 The number of unitary 3-smooth divisors of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 29 2025

Period 6: repeat [1, 2, 2, 2, 1, 4].
Decimal expansion of 407380/333333.
Continued fraction expansion of 10/(6 + sqrt(66)) (with offset 0).

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

Cf. A005117, A003586, A034444, A065331, A072078, A181982, A382487.

The number of unitary prime(k)-smooth divisors of n: A134451 (k = 1), this sequence (k = 2), A382489 (k = 3).

Table[{1, 2, 2, 2, 1, 4}, {12}] // Flatten

a(n) = [1, 2, 2, 2, 1, 4][(n-1) % 6 + 1];

Multiplicative with a(p^e) = 2 if p <= 3, and 1 otherwise.
a(n) = A034444(A065331(n)).
a(n) = A034444(n) if and only if n is 3-smooth (A003586).
a(n) = A072078(n) if and only if n is squarefree (A005117).
a(n) = abs(A181982(n+9)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2.
G.f.: -(4*x^6 + x^5 + 2*x^4 + 2*x^3 +2*x^2 + x)/(x^6 - 1).
Dirichlet g.f.: (1 + 1/2^s) * (1 + 1/3^s) * zeta(s).

A171126 Numbers k such that A169611(k) = 1.

Original entry on oeis.org

2, 3, 10, 14, 15, 21, 22, 26, 33, 34, 38, 39, 46, 50, 51, 57, 58, 62, 69, 70, 74, 75, 82, 86, 87, 93, 94, 98, 105, 106, 110, 111, 118, 122, 123, 129, 130, 134, 141, 142, 146, 147, 154, 158, 159, 165, 166, 170, 177, 178, 182, 183, 190, 194, 195, 201, 202, 206, 213

Offset: 1

Juri-Stepan Gerasimov, Dec 04 2009

Appears to be numbers k such that tau(36*k) = 2*tau(6*k). - Gary Detlefs, Jan 13 2020
The asymptotic density of this sequence is 5/18. - Amiram Eldar, Jan 16 2022
Numbers that have exactly two 3-smooth divisors, i.e., numbers k such that A072078(k) = 2. - Amiram Eldar, Mar 29 2025

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

Cf. A000005 (tau), A072078, A169611.

f[n_] := Plus @@ Last /@ Select[ FactorInteger@ n, 1 < #[[1]] < 4 &]; Select[ Range@ 213, f@# == 1 &] (* Robert G. Wilson v, Dec 19 2009 *)

isok(n) = valuation(n, 2)+valuation(n, 3) == 1; \\ Michel Marcus, Jan 13 2020

More terms from Robert G. Wilson v, Dec 19 2009

A322316 Lexicographically earliest such sequence a that a(i) = a(j) => A122841(i) = A122841(j) and A244417(i) = A244417(j), for all i, j.

Original entry on oeis.org

1, 2, 2, 3, 1, 4, 1, 5, 3, 2, 1, 6, 1, 2, 2, 7, 1, 6, 1, 3, 2, 2, 1, 8, 1, 2, 5, 3, 1, 4, 1, 9, 2, 2, 1, 10, 1, 2, 2, 5, 1, 4, 1, 3, 3, 2, 1, 11, 1, 2, 2, 3, 1, 8, 1, 5, 2, 2, 1, 6, 1, 2, 3, 12, 1, 4, 1, 3, 2, 2, 1, 13, 1, 2, 2, 3, 1, 4, 1, 7, 7, 2, 1, 6, 1, 2, 2, 5, 1, 6, 1, 3, 2, 2, 1, 14, 1, 2, 3, 3, 1, 4, 1, 5, 2

Offset: 1

Antti Karttunen, Dec 04 2018

nonn

Restricted growth sequence transform of the ordered pair [A122841(n), A244417(n)].
Essentially also the restricted growth sequence transform of the unordered pair {A007814(n), A007949(n)}.
For all i, j: a(i) = a(j) => A072078(i) = A072078(j).

Antti Karttunen, Table of n, a(n) for n = 1..19683
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A007814, A007949, A122841, A244417, A322026, A322317 (ordinal transform).

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007814(n) = valuation(n,2);
A007949(n) = valuation(n,3);
A122841(n) = min(A007814(n), A007949(n));
A244417(n) = max(valuation(n,2), valuation(n,3));
v322316 = rgs_transform(vector(up_to, n, [A122841(n), A244417(n)]));
\\ The following is equivalent:
\\ v322316 = rgs_transform(vector(up_to, n, Set([A007814(n), A007949(n)])));
A322316(n) = v322316[n];

A358230 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(i) = A007814(j), A007949(i) = A007949(j) and A046523(i) = A046523(j), for all i, j, where A007814 and A007949 give the 2-adic and 3-adic valuation, and A046523 gives the prime signature of its argument.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 5, 7, 8, 9, 5, 10, 5, 9, 11, 12, 5, 13, 5, 14, 11, 9, 5, 15, 16, 9, 17, 14, 5, 18, 5, 19, 11, 9, 20, 21, 5, 9, 11, 22, 5, 18, 5, 14, 23, 9, 5, 24, 16, 25, 11, 14, 5, 26, 20, 22, 11, 9, 5, 27, 5, 9, 23, 28, 20, 18, 5, 14, 11, 29, 5, 30, 5, 9, 31, 14, 20, 18, 5, 32, 33, 9, 5, 27, 20, 9, 11, 22, 5, 34, 20, 14, 11, 9, 20, 35, 5, 25, 23, 36, 5, 18, 5, 22, 37

Offset: 1

Antti Karttunen, Dec 01 2022

nonn

Restricted growth sequence transform of the triple [A007814(n), A007949(n), A046523(n)].
For all i, j:
  A305900(i) = A305900(j) => a(i) = a(j),
  a(i) = a(j) => A305891(i) = A305891(j),
  a(i) = a(j) => A305893(i) = A305893(j),
  a(i) = a(j) => A322026(i) = A322026(j) => A072078(i) = A072078(j),
  a(i) = a(j) => A065333(i) = A065333(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A007814, A007949, A046523, A065333, A072078, A305891, A305893, A322026.

Cf. also A358747.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007814(n) = valuation(n,2);
A007949(n) = valuation(n,3);
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
v358230 = rgs_transform(vector(up_to, n, [A007814(n), A007949(n), A046523(n)]));
A358230(n) = v358230[n];

A355709 Numbers k such that k and k+1 have the same number of 3-smooth divisors.

Original entry on oeis.org

2, 14, 21, 33, 38, 44, 50, 57, 69, 74, 80, 86, 93, 99, 105, 110, 116, 122, 129, 135, 141, 146, 158, 165, 171, 177, 182, 194, 201, 213, 218, 230, 237, 249, 254, 260, 266, 273, 285, 290, 296, 302, 309, 315, 321, 326, 332, 338, 345, 357, 362, 374, 381, 387, 393, 398

Offset: 1

Amiram Eldar, Jul 15 2022

nonn

Numbers k such that A072078(k) = A072078(k+1).

This sequence is infinite since it includes all the numbers of the form 3*(2^(2*k+1)-1), with k>=1.

			2 is a term since A072078(2) = A072078(3) = 2.

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

Cf. A072078, A355710 (5-smooth analog).

Similar sequences: A005237, A006049, A332386, A343819, A344312, A344313, A344314, A348345.

s[n_] := Times @@ (1 + IntegerExponent[n, {2, 3}]); Select[Range[400], s[#] == s[#+1] &]

s(n) = (valuation(n, 2) + 1) * (valuation(n, 3) + 1);
s1 = s(1); for(k = 2, 400, s2 = s(k); if(s1 == s2, print1(k-1,", ")); s1 = s2);

cp's OEIS Frontend

A065333 Characteristic function of 3-smooth numbers, i.e., numbers of the form 2^i*3^j (i, j >= 0).

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A060753 Denominator of 1*2*4*6*...*(prime(n-1)-1) / (2*3*5*7*...*prime(n-1)).

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A322026 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(i) = A007814(j) and A007949(i) = A007949(j), for all i, j, where A007814 and A007949 give the 2- and 3-adic valuations of n.

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A072079 Sum of 3-smooth divisors of n.

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

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A382488 The number of unitary 3-smooth divisors of n.

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A171126 Numbers k such that A169611(k) = 1.

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A060753 Denominator of 1246...(prime(n-1)-1) / (2357...prime(n-1)).