A322026 -id:A322026 - cp's OEIS frontend

A323884 Sum of A322026 and its Dirichlet inverse.

2, 0, 0, 4, 0, 12, 0, 8, 9, 4, 0, 8, 0, 4, 6, 8, 0, 4, 0, 4, 6, 4, 0, 0, 1, 4, 15, 4, 0, -2, 0, 12, 6, 4, 2, 13, 0, 4, 6, 4, 0, -2, 0, 4, 5, 4, 0, 22, 1, 2, 6, 4, 0, 7, 2, 4, 6, 4, 0, 8, 0, 4, 5, 20, 2, -2, 0, 4, 6, 0, 0, 38, 0, 4, 3, 4, 2, -2, 0, 10, 13, 4, 0, 8, 2, 4, 6, 4, 0, 16, 2, 4, 6, 4, 2, 28, 0, 2, 5, 4, 0, -2, 0, 4, 0

Offset: 1

Antti Karttunen, Feb 08 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A007814, A007949, A322026, A323882, A323883, A323885.

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];
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA323883(n) = v323883[n];
A323884(n) = (A322026(n)+A323883(n));

a(n) = A322026(n) + A323883(n).

A323883 Dirichlet inverse of A322026.

Original entry on oeis.org

1, -2, -3, 0, -1, 7, -1, 2, 2, 2, -1, 0, -1, 2, 3, -1, -1, -6, -1, 0, 3, 2, -1, -11, 0, 2, 3, 0, -1, -7, -1, -1, 3, 2, 1, -1, -1, 2, 3, -2, -1, -7, -1, 0, -2, 2, -1, 7, 0, 0, 3, 0, -1, -9, 1, -2, 3, 2, -1, 0, -1, 2, -2, 3, 1, -7, -1, 0, 3, -2, -1, 20, -1, 2, 0, 0, 1, -7, -1, 1, -6, 2, -1, 0, 1, 2, 3, -2, -1, 6, 1, 0, 3, 2, 1, 8

Offset: 1

Antti Karttunen, Feb 08 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A007814, A007949, A322026, A323881, A323884.

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)]));
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA323883(n) = v323883[n];

A065331 Largest 3-smooth divisor of n.

Original entry on oeis.org

1, 2, 3, 4, 1, 6, 1, 8, 9, 2, 1, 12, 1, 2, 3, 16, 1, 18, 1, 4, 3, 2, 1, 24, 1, 2, 27, 4, 1, 6, 1, 32, 3, 2, 1, 36, 1, 2, 3, 8, 1, 6, 1, 4, 9, 2, 1, 48, 1, 2, 3, 4, 1, 54, 1, 8, 3, 2, 1, 12, 1, 2, 9, 64, 1, 6, 1, 4, 3, 2, 1, 72, 1, 2, 3, 4, 1, 6, 1, 16, 81, 2, 1, 12, 1, 2, 3, 8, 1, 18, 1, 4, 3, 2, 1, 96

Offset: 1

Reinhard Zumkeller, Oct 29 2001

Bennett, Filaseta, & Trifonov show that if n > 8 then a(n^2 + n) < n^0.715. - Charles R Greathouse IV, May 21 2014

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
M. A. Bennett, M. Filaseta, and O. Trifonov, On the factorization of consecutive integers, J. Reine Angew. Math. 629 (2009), pp. 171-200.
Jon Maiga, Computer-generated formulas for A065331, Sequence Machine.

Cf. A003586, A065330, A006519, A038500, A007814, A007949, A007310, A047229, A071521, A322026.
Related to A053165 via A225546.
Cf. A126760 (ordinal transform of this sequence, from its term a(1) = 1 onward).

a065331 = f 2 1 where
   f p y x | r == 0    = f p (y * p) x'
           | otherwise = if p == 2 then f 3 y x else y
           where (x', r) = divMod x p
-- Reinhard Zumkeller, Nov 19 2015

[Gcd(n,6^n): n in [1..100]]; // Vincenzo Librandi, Feb 09 2016

A065331 := proc(n) n/A065330(n) ; end: # R. J. Mathar, Jun 24 2009
seq(2^padic:-ordp(n,2)*3^padic:-ordp(n,3), n=1..100); # Robert Israel, Feb 08 2016

Table[GCD[n, 6^n], {n, 100}] (* Vincenzo Librandi, Feb 09 2016 *)
a[n_] := Times @@ ({2, 3}^IntegerExponent[n, {2, 3}]); Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)

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

a(n)=gcd(n,6^n) \\ Not very efficient, but simple. Stanislav Sykora, Feb 08 2016

a(n)=gcd(6^logint(n,2),n) \\ 'optimized' version of Sykora's script; Charles R Greathouse IV, Jul 23 2024

a(n) = n / A065330(n).
a(n) = A006519(n) * A038500(n).
a(n) = (2^A007814 (n)) * (3^A007949(n)).
Multiplicative with a(2^e)=2^e, a(3^e)=3^e, a(p^e)=1, p>3. - Vladeta Jovovic, Nov 05 2001
Dirichlet g.f.: zeta(s)*(1-2^(-s))*(1-3^(-s))/ ( (1-2^(1-s))*(1-3^(1-s)) ). - R. J. Mathar, Jul 04 2011
a(n) = gcd(n,6^n). - Stanislav Sykora, Feb 08 2016
a(A225546(n)) = A225546(A053165(n)). - Peter Munn, Jan 17 2020
Sum_{k=1..n} a(k) ~ n*(log(n)^2 + (2*gamma + 3*log(2) + 2*log(3) - 2)*log(n) + (2 + log(2)^2/6 + 3*log(2)*(log(3) - 1) - 2*log(3) + log(3)^2/6 + gamma*(3*log(2) + 2*log(3) - 2) - 2*sg1)) / (6*log(2)*log(3)), where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Sep 19 2020
a(n) = A003586(A322026(n)), A322026(n) = A071521(a(n)). - Antti Karttunen, Sep 08 2024

A126760 a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n+5) = 2n + 2.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 5, 3, 2, 1, 6, 1, 7, 2, 3, 4, 8, 1, 9, 5, 1, 3, 10, 2, 11, 1, 4, 6, 12, 1, 13, 7, 5, 2, 14, 3, 15, 4, 2, 8, 16, 1, 17, 9, 6, 5, 18, 1, 19, 3, 7, 10, 20, 2, 21, 11, 3, 1, 22, 4, 23, 6, 8, 12, 24, 1, 25, 13, 9, 7, 26, 5, 27, 2, 1, 14, 28, 3, 29, 15, 10, 4, 30, 2

Offset: 0

N. J. A. Sloane, Feb 19 2007

nonn

For further information see A126759, which provided the original motivation for this sequence.
From Antti Karttunen, Jan 28 2015: (Start)
The odd bisection of the sequence gives A253887, and the even bisection gives the sequence itself.
A254048 gives the sequence obtained when this sequence is restricted to A007494 (numbers congruent to 0 or 2 mod 3).
For all odd numbers k present in square array A135765, a(k) = the column index of k in that array. (End)
A322026 and this sequence (without the initial zero) are ordinal transforms of each other. - Antti Karttunen, Feb 09 2019
Also ordinal transform of A065331 (after the initial 0). - Antti Karttunen, Sep 08 2024

Antti Karttunen, Table of n, a(n) for n = 0..19683

One less than A126759.
Cf. A003586, A003602, A007310, A064989, A249746, A253887, A254048, A273669, A322317, A323881 (Dirichlet inverse), A323882.
Cf. A347233 (Möbius transform) and also A349390, A349393, A349395 for other Dirichlet convolutions.
Ordinal transform of A065331 and of A322026 (after the initial 0).
Related arrays: A135765, A254102.

f[n_] := Block[{a}, a[0] = 0; a[1] = a[2] = a[3] = 1; a[x_] := Which[EvenQ@ x, a[x/2], Mod[x, 3] == 0, a[x/3], Mod[x, 6] == 1, 2 (x - 1)/6 + 1, Mod[x, 6] == 5, 2 (x - 5)/6 + 2]; Table[a@ i, {i, 0, n}]] (* Michael De Vlieger, Feb 03 2015 *)

A126760(n)={n&&n\=3^valuation(n,3)<M. F. Hasler, Jan 19 2016

a(n) = A126759(n)-1. [The original definition.]
From Antti Karttunen, Jan 28 2015: (Start)
a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n+5) = 2n + 2.
Or with the last clause represented in another way:
a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n-1) = 2n.
Other identities. For all n >= 1:
a(n) = A253887(A003602(n)).
a(6n-3) = a(4n-2) = a(2n-1) = A253887(n).
(End)
a(n) = A249746(A003602(A064989(n))). - Antti Karttunen, Feb 04 2015
a(n) = A323882(4*n). - Antti Karttunen, Apr 18 2022

Name replaced with an independent recurrence and the old description moved to the Formula section - Antti Karttunen, Jan 28 2015

A071521 Number of 3-smooth numbers <= n.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Jun 02 2002

A 3-smooth number is a number of the form 2^x * 3^y where x >= 0 and y >= 0.

Bruce C. Berndt and Robert A. Rankin, "Ramanujan : letters and commentary", History of Mathematics Volume 9, AMS-LMS, p. 23, p. 35.
G. H. Hardy, Ramanujan: Twelve lectures on subjects suggested by his life and work, AMS Chelsea Pub., 1999, pages 67-82.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Thierry Bousch, La Tour de Stockmeyer, Séminaire Lotharingien de Combinatoire 77 (2017), Article B77d.
M. Haussman and H. N. Shapiro, On Ramanujan right triangle conjecture, Comm. Pure Appl. Math. 42 (1989), 885-889.
A. M. Ostrowski, Bemerkungen zur Theorie der Diophantischen Approximationen, Abh. Math. Sem. Univ. Hamburg 1 (1922), 77-98; 250-251.
Raphael Schumacher, The Formulas for the Distribution of the 3-Smooth, 5-Smooth, 7-Smooth and all other Smooth Numbers, arXiv preprint arXiv:1608.06928 [math.NT], 2016.

Cf. A003586.

Cf. A000079, A000244, A022330, A022331, A065331, A322026.

a071521 n = length $ takeWhile (<= n) a003586_list
-- Reinhard Zumkeller, Aug 14 2011

N:= 10000: # to get a(1) to a(N)
V:= Vector(N):
for y from 0 to floor(log[3](N)) do
  for x from 0 to ilog2(N/3^y) do
    V[2^x*3^y]:= 1
od od:
convert(map(round,Statistics:-CumulativeSum(V)),list); # Robert Israel, Dec 16 2014

a[n_] := Sum[ MoebiusMu[6k]*Floor[n/k], {k, 1, n}]; Table[a[n], {n, 1, 75}] (* Jean-François Alcover, Oct 11 2011, after Benoit Cloitre *)
f[n_] := Sum[Floor@Log[3, n/2^i] + 1, {i, 0, Log[2, n]}]; Array[f, 75] (* faster, or *)
f[n_] := Sum[Floor@Log[2, n/3^i] + 1, {i, 0, Log[3, n]}]; Array[f, 75] (* Robert G. Wilson v, Aug 18 2012 *)
Accumulate[Table[If[Max[FactorInteger[n][[All,1]]]<4,1,0],{n,80}]] (* Harvey P. Dale, Jan 11 2017 *)

for(n=1,100,print1(sum(k=1,n,if(sum(i=3,n,if(k%prime(i),0,1)),0,1)),","))

a(n)=sum(k=1,n,moebius(2*3*k)*floor(n/k)) \\ Benoit Cloitre, Jun 14 2007

a(n)=my(t=1/3); sum(k=0,logint(n,3), t*=3; logint(n\t,2)+1) \\ Charles R Greathouse IV, Jan 08 2018

from sympy import integer_log
def A071521(n): return sum((n//3**i).bit_length() for i in range(integer_log(n,3)[0]+1)) # Chai Wah Wu, Sep 15 2024

a(n) = Card{ k | A003586(k) <= n }. Asymptotically: let a=1/(2*log(2)*log(3)), b=sqrt(6), then from Ramanujan a(n) ~ a*log(2*n)*log(3*n) or equivalently a(n) ~ a*log(b*n)^2.
A022331(n) = a(A000079(n)); A022330(n) = a(A000244(n)). - Reinhard Zumkeller, May 09 2006
a(n) = Sum_{k=1..n} mu(6k)*floor(n/k). - Benoit Cloitre, Jun 14 2007
a(n) = Sum_{k=1..n} (floor(6^k/k)-floor((6^k-1)/k)). - Anthony Browne, May 19 2016
From Ridouane Oudra, Jul 17 2020: (Start)
a(n) = Sum_{i=0..floor(log_2(n))} (floor(log_3(n/2^i)) + 1).
a(n) = Sum_{i=0..floor(log_3(n))} (floor(log_2(n/3^i)) + 1). (End)
A322026(n) = a(A065331(n)). - Antti Karttunen, Sep 08 2024

A072078 Number of 3-smooth divisors of n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jun 13 2002

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A000005, A003586, A007814, A007949, A072079, A122841, A322026.

[(Valuation(n,2)+1)*(Valuation(n,3)+1): n in [1..120]]; // Vincenzo Librandi, Mar 24 2015

a[n_] := DivisorSum[n, MoebiusMu[6*#]*DivisorSigma[0, n/#] &]; Array[a, 100] (* or *) a[n_] := ((1+IntegerExponent[n, 2])*(1+IntegerExponent[n, 3])); Array[a, 100] (* Amiram Eldar, Dec 03 2018 from the pari codes *)

a(n)=sumdiv(n,d,moebius(6*d)*numdiv(n/d)) \\ Benoit Cloitre, Jun 21 2007

A072078(n) = ((1+valuation(n,2))*(1+valuation(n,3))); \\ Antti Karttunen, Dec 03 2018

a(n) = A000005(A065331(n)).
a(n) = (A007814(n) + 1)*(A007949(n) + 1).
1/Product_{k>0} (1 - x^k + x^(2*k))^a(k) is g.f. for A000041(). - Vladeta Jovovic, Jun 07 2004
From Christian G. Bower, May 20 2005: (Start)
Multiplicative with a(2^e) = a(3^e) = e+1, a(p^e) = 1, p>3.
Dirichlet g.f.: 1/((1-1/2^s)*(1-1/3^s))^2 * Product{p prime > 3}(1/(1-1/p^s)). [corrected by Vaclav Kotesovec, Nov 20 2021] (End)
a(n) = Sum_{d divides n} mu(6d)*tau(n/d). - Benoit Cloitre, Jun 21 2007
Dirichlet g.f.: zeta(s)/((1-1/2^s)*(1-1/3^s)). - Ralf Stephan, Mar 24 2015; corrected by Vaclav Kotesovec, Nov 20 2021
Sum_{k=1..n} a(k) ~ 3*n. - Vaclav Kotesovec, Nov 20 2021

More terms from Benoit Cloitre, Jun 21 2007

A244417 Exponents of 6 in appearing in the 6-adic value of 1/n, n>=1 (A244416).

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Jul 02 2014

For the definition of 'g-dic value of 1/n' see a comment on A244416. In the Mahler reference, p. 7, the present exponent of 6 is there called f = f(1/n) for g = 6.

			See A244416.

Kurt Mahler, p-adic numbers and their functions, second ed., Cambridge University Press, 1981.

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A122841, A244416, A007814 (g=2), A007949 (g=3), A244415 (g=4), A112765 (g=5), A051903, A065331.

Cf. also A322026, A322316.

a[n_] := Max[IntegerExponent[n, {2, 3}]]; Array[a, 100] (* Amiram Eldar, Aug 19 2024 *)

A244417(n) = max(valuation(n,2), valuation(n,3)); \\ Antti Karttunen, Dec 04 2018

a(n) = 0 if n is congruent 1 or 5 (mod 6). a(n) = max(A007814(n), A007949(n)) if n == 0 (mod 6). a(n) = A007814(n) if n == 2 or 4 (mod 6) and a(n) = A007949(n) if n == 3 (mod 6).
a(n) = max(A007814(n), A007949(n)), in all cases. - Antti Karttunen, Dec 04 2018
From Amiram Eldar, Aug 19 2024: (Start)
a(n) = A051903(A065331(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 13/10. (End)

A379005 Lexicographically earliest infinite sequence such that a(i) = a(j) => v_2(i) = v_2(j), v_3(i) = v_3(j) and v_5(i) = v_5(j), for all i, j, where v_2 (A007814), v_3 (A007949) and v_5 (A112765) give the 2-, 3- and 5-adic valuations of n respectively.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 15 2024

nonn

Restricted growth sequence transform of A355582.
For all i, j:
  A379001(i) = A379001(j) => a(i) = a(j),
  a(i) = a(j) => A322026(i) = A322026(j),
  a(i) = a(j) => A379004(i) = A379004(j).

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

Cf. A007814, A007949, A112765, A355582, A379006 (ordinal transform).

Cf. A322026, A379001, A379004.

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; };
v379005 = rgs_transform(vector(up_to, n, [valuation(n,2), valuation(n,3), valuation(n,5)]));
A379005(n) = v379005[n];

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];

A340680 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(1+i) = A007814(1+j) and A292251(i) = A292251(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 11 2021

nonn

Restricted growth sequence transform of the ordered pair [A007814(1+n), A292251(n)], where the first element is the 2-adic valuation of 1+n (i.e., the number of trailing 1-digits in the base-2 representation of n), and the latter element is the 3-adic valuation of A048673(n).

For all i, j: a(i) = a(j) => A341345(i) = A341345(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A007814, A048673, A292251, A341345.

Cf. also A322026.

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);
A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A048673(n) = (A003961(n)+1)/2;
A292251(n) = valuation(A048673(n),3);
Aux340680(n) = [A007814(1+n), A292251(n)];
v340680 = rgs_transform(vector(up_to, n, Aux340680(n)));
A340680(n) = v340680[n];

a(2n) = 2.

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