A108951 -id:A108951 - cp's OEIS frontend

A307866 K-champion numbers: numbers m such that K(m) > K(j) for all j < m, where K(m) is the Kalmár function (A074206).

0, 1, 4, 6, 8, 12, 24, 36, 48, 72, 96, 120, 144, 192, 240, 288, 360, 432, 480, 576, 720, 864, 960, 1152, 1440, 1728, 1920, 2160, 2304, 2880, 3456, 4320, 5760, 6912, 8640, 11520, 17280, 23040, 25920, 30240, 34560, 46080, 51840, 60480, 69120, 86400, 103680, 120960

Offset: 1

Amiram Eldar, May 02 2019

nonn

The corresponding record values are 0, 1, 2, 3, 4, 8, 20, 26, 48, 76, 112, 132, 208, ... (see the link for more values).
Deléglise et al. (2008) proved that the number of powerful (A001694) terms in this sequence is finite. They ask whether a(391) = 485432135516160000 (the 112th powerful term) is the largest. - Amiram Eldar, Aug 20 2019
Is abs(omega(a(n)) - omega(a(n+1))) <= 1? (Cf. A001221.) - David A. Corneth, Apr 16 2020

David A. Corneth, Table of n, a(n) for n = 1..884 (terms 1..762 from Amiram Eldar, calculated by Deléglise et al.)
David A. Corneth, Table of n, a(n) for n = 1..1544 if abs(omega(a(n)) - omega(a(n + 1))) <= 1
M. Deléglise, M. O. Hernane, and J.-L. Nicolas, Grandes valeurs et nombres champions de la fonction arithmétique de Kalmár, Journal of Number Theory, Vol. 128, No. 6 (2008), pp. 1676-1716.
M. Deléglise, M. O. Hernane, and J.-L. Nicolas, Tables des 761 premiers champions de la fonction de Kalmar, alternative link.
Amiram Eldar, Table of n, a(n), A074206(a(n)), calculated by Deléglise et al.
T. M. A. Fink, Number of ordered factorizations and recursive divisors, arXiv:2307.16691 [math.NT], 2023.

Cf. A001221, A001694, A002093, A033833, A074206, A163272, A330686 (after primorial deflation).

a[0] = 0; a[1] = 1; a[n_] := a[n] = Total[a /@ Most[Divisors[n]]]; s = {}; am=-1; Do[a1 = a[n]; If[a1>am, am=a1; AppendTo[s, n]], {n, 0, 10000}]; s

For n >= 1, a(1+n) = A108951(A330686(n)). - Antti Karttunen, Dec 31 2019

A342456 A276086 applied to the primorial inflation of Doudna-tree, where A276086(n) is the prime product form of primorial base expansion of n.

Original entry on oeis.org

2, 3, 5, 9, 7, 25, 35, 15, 11, 49, 117649, 625, 717409, 1225, 55, 225, 13, 121, 1771561, 2401, 36226650889, 184877, 1127357, 875, 902613283, 514675673281, 3780549773, 1500625, 83852850675321384784127, 3025, 62004635, 21, 17, 169, 4826809, 14641, 8254129, 143, 2924207, 77, 8223741426987700773289, 59797108943, 546826709

Offset: 0

Antti Karttunen and Michael De Vlieger, Mar 14 2021

nonn

This sequence (which could be viewed as a binary tree, like the underlying A005940 and A329886) is similar to A324289, but unlike its underlying tree A283477 that generates only numbers that are products of distinct primorial numbers (i.e., terms of A129912), here the underlying tree A329886 generates all possible products of primorial numbers, i.e., terms of A025487, but in different order.

Cf. A005940, A025487, A108951, A129912, A276086, A283980, A324886, A342457 [= 2*A246277(a(n))], A342461 [= A001221(a(n))], A342462 [= A001222(a(n))], A342463 [= A342001(a(n))], A342464 [= A051903(a(n))].

Cf. A324289 (a subset of these terms, in different order).

Block[{a, f, r = MixedRadix[Reverse@ Prime@ Range@ 24]}, f[n_] :=
Times @@ MapIndexed[Prime[First[#2]]^#1 &, Reverse@ IntegerDigits[n, r]]; a[0] = 1; a[1] = 2; a[n_] := a[n] = If[EvenQ@ n, (Times @@ Map[Prime[PrimePi@ #1 + 1]^#2 & @@ # &, FactorInteger[#]] - Boole[# == 1])*2^IntegerExponent[#, 2] &[a[n/2]], 2 a[(n - 1)/2]]; Array[f@ a[#] &, 43, 0]] (* Michael De Vlieger, Mar 17 2021 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)};
A329886(n) = if(n<2,1+n,if(!(n%2),A283980(A329886(n/2)),2*A329886(n\2)));
A342456(n) = A276086(A329886(n));

a(n) = A276086(A329886(n)) = A324886(A005940(1+n)).
For all n >= 0, gcd(a(n), A329886(n)) = 1.
For all n >= 1, A055396(a(n))-1 = A061395(A329886(n)) = A290251(n) = 1+A080791(n).
For all n >= 0, a(2^n) = A000040(2+n).

A342462 Sum of digits when A329886(n) is written in primorial base, where A329886 is the primorial inflation of Doudna-tree.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 6, 4, 6, 4, 2, 4, 1, 2, 6, 4, 10, 6, 6, 4, 8, 12, 10, 8, 22, 4, 8, 2, 1, 2, 6, 4, 6, 2, 6, 2, 18, 10, 8, 6, 18, 12, 16, 4, 26, 16, 24, 8, 20, 14, 4, 6, 26, 16, 14, 8, 30, 6, 8, 4, 1, 2, 6, 4, 14, 12, 12, 8, 18, 12, 24, 4, 8, 12, 14, 4, 24, 20, 28, 20, 26, 16, 16, 12, 32, 26, 24, 14, 28, 16

Offset: 0

Antti Karttunen, Mar 15 2021

nonn

From David A. Corneth's Feb 27 2019 comment in A276150 follows that the only odd terms in this sequence are 1's occurring at 0 and at two's powers.

Subsequences starting at each n = 2^k are slowly converging towards A329886: 1, 2, 6, 4, 30, 12, 36, 8, 210, 60, 180, 24, etc.. Compare also to the behaviors of A324342 and A342463.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to primorial base

Cf. A001222, A005940, A108951, A276150, A329886, A342456, A342457, A342461, A342463, A342464.

Cf. also A324342, A324383, A324387, A324888.

A342462(n) = bigomega(A342456(n)); \\ Other code as in A342456.

A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)};
A329886(n) = if(n<2,1+n,if(!(n%2),A283980(A329886(n/2)),2*A329886(n\2)));
A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); };
A342462(n) = A276150(A329886(n));

a(n) = A001222(A342456(n)) = A001222(A342457(n)).
a(n) = A276150(A329886(n)) = A324888(A005940(1+n)).
a(n) >= A342461(n).
For n >= 0, a(2^n) = 1.

A062977 Difference between largest and smallest positive exponent in prime factorization of n; a(1) = 0 by convention.

Original entry on oeis.org

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

Offset: 1

Henry Bottomley, Jul 24 2001

			a(24) = 2 since 24 = 2^3*3^1 and max(3,1) - min(3,1) = 3 - 1 = 2;
a(25) = 0 since 25 = 5^2 and max(2) - min(2) = 2 - 2 = 0.

Antti Karttunen, Table of n, a(n) for n = 1..100000 (first 4000 terms from Harry J. Smith)
Index entries for sequences computed from exponents in factorization of n.

Cf. A072774 (positions of zeros), A059404 (of nonzeros).

Cf. A001222, A033150, A051903, A051904, A071178, A108951, A325226.

dlsp[n_]:=Module[{xp=FactorInteger[n][[All,2]]},Max[xp]-Min[xp]]; Join[ {0},Array[ dlsp,120]] (* Harvey P. Dale, Jan 28 2021 *)

{ for (n=1, 4000, if (n<2, M=m=0, f=factor(n)~; M=m=f[2, 1]; for (i=2, length(f), M=max(M, f[2, i]); m=min(m, f[2, i]))); write("b062977.txt", n, " ", M - m) ) } \\ Harry J. Smith, Aug 14 2009

A062977(n) = if((1==n),0,n=(factor(n)[, 2]); vecmax(n)-vecmin(n)); \\ Antti Karttunen, Nov 17 2019

a(n) = A051903(n) - A051904(n).
a(A108951(n)) = A325226(n) = A001222(n) - A071178(n). - Antti Karttunen, Nov 17 2019
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A033150 - 1 = 0.705211... . - Amiram Eldar, Jan 05 2024

A212181 Largest odd divisor of tau(n): a(n) = A000265(A000005(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 5, 3, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 5, 5, 1, 1, 3, 1, 1, 1, 1

Offset: 1

Matthew Vandermast, Jun 04 2012

Completely determined by the exponents >=2 in the prime factorization of n (cf. A212172).
Not the same as the number of odd divisors of n (A001227(n)); see example.
Multiplicative because A000005 is multiplicative and A000265 is completely multiplicative. - Andrew Howroyd, Aug 01 2018
a(n) = 1 iff the number of divisors of n is a power of 2 (A036537). - Bernard Schott, Nov 04 2022

			48 has a total of 10 divisors (1, 2, 3, 4, 6, 8, 12, 16, 24 and 48). Since the largest odd divisor of 10 is 5, a(48) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A000079, A000265, A036537, A108951, A212172, A295664, A331286 (applied to primorial inflation of n).

Table[Block[{nd=DivisorSigma[0, n]}, nd/2^IntegerExponent[nd, 2]], {n, 100}] (* Indranil Ghosh, Jul 19 2017, after PARI code *)

a(n) = my(nd = numdiv(n)); nd/2^valuation(nd, 2); \\ Michel Marcus, Jul 19 2017

from sympy import divisor_count, divisors
def a(n): return [i for i in divisors(divisor_count(n)) if i%2][-1]
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 19 2017

a(n) = A000265(A000005(n)).
From Antti Karttunen, Jan 14 2020: (Start)
a(n) = A000005(n) / A000079(A295664(n)).
a(A108951(n)) = A331286(n).
(End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p odd prime} ((1 - 1/p)*(1 + Sum_{k>=1} a(k+1)/p^k)) = 2.076325817863586... . - Amiram Eldar, Oct 15 2022

A328521 Smallest highly composite number that has n prime factors counted with multiplicity.

Original entry on oeis.org

1, 2, 4, 12, 24, 48, 240, 720, 5040, 10080, 20160, 221760, 665280, 8648640, 17297280, 294053760, 2205403200, 27935107200, 293318625600, 1927522396800, 8995104518400, 26985313555200, 782574093100800, 24259796886124800, 48519593772249600, 1795224969573235200, 8976124847866176000, 368021118762513216000

Offset: 0

David A. Corneth, Jan 04 2020

nonn

a(n-1) differs from A133411(n) for n in A354880.

Question: Is this sequence strictly growing? If sequence A330748 is monotonic, so is this also, and vice versa. Note that the primorial deflation sequence, A330743, is not monotonic. - Antti Karttunen, Jan 14 2020

Amiram Eldar, Table of n, a(n) for n = 0..394

Cf. A001222 (bigomega), A002182 (highly composite numbers), A108951, A112778 (bigomega of HCN's), A330743 (primorial deflation), A330748 (indices in A002182).
Cf. also A133411.
Cf. A354880.

(* First load the function f at A025487, then: *)
Block[{s = Union@ Flatten@ f@ 17, t}, t = DivisorSigma[0, s]; s = Map[s[[FirstPosition[t, #][[1]] ]] &, Union@ FoldList[Max, t]]; t = PrimeOmega[s]; Array[s[[FirstPosition[t, #][[1]] ]] &, Max@ t + 1, 0]] (* Michael De Vlieger, Jan 12 2020 *)

a(n)=for(k=1,oo,bigomega(A2182[k])==n&&return(A2182[k])) \\ Global variable A2182 must hold a vector of values of A002182. - M. F. Hasler, Jan 08 2020

a(n) = A002182(A330748(n)) = A002182(min{k: A112778(k)=n}). - M. F. Hasler, Jan 08 2020

a(n) = A108951(A330743(n)), where A330743(n) is the first term k of A329902 for which A056239(k) = n. - Antti Karttunen, Jan 13 2020

A328769 The second primorial based variant of arithmetic derivative: a(p) = A034386(p) for p prime, a(uv) = a(u)v + u*a(v), with a(0) = a(1) = 0.

Original entry on oeis.org

0, 0, 2, 6, 8, 30, 18, 210, 24, 36, 70, 2310, 48, 30030, 434, 120, 64, 510510, 90, 9699690, 160, 672, 4642, 223092870, 120, 300, 60086, 162, 896, 6469693230, 270, 200560490130, 160, 6996, 1021054, 1260, 216, 7420738134810, 19399418, 90168, 360, 304250263527210, 1386, 13082761331670030, 9328, 450, 446185786, 614889782588491410, 288, 2940, 650, 1531632

Offset: 0

Antti Karttunen, Oct 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1001
Index entries for sequences related to primorial numbers

Cf. A002110, A034386, A108951, A328772.

Cf. also A003415, A258851, A328768, A328845, A328846.

A034386(n) = factorback(primes(primepi(n)));
A328769(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A034386(f[i,1])/f[i, 1]));

A002110(n) = prod(i=1,n,prime(i));
A328769(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1]))/f[i, 1]));

a(n) = n * Sum e_j * (p_j)#/p_j for n = Product p_j^e_j with (p_j)# = A034386(p_j).

A276150(a(n)) = A328772(n).

A329620 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A046523(n), A246277(A324886(n))].

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 4, 2, 7, 2, 4, 8, 9, 2, 10, 2, 7, 8, 4, 2, 11, 12, 4, 13, 7, 2, 14, 2, 15, 8, 4, 16, 17, 2, 4, 8, 18, 2, 19, 2, 7, 20, 4, 2, 21, 22, 23, 8, 7, 2, 24, 25, 26, 8, 4, 2, 27, 2, 4, 28, 29, 30, 31, 2, 7, 8, 32, 2, 33, 2, 4, 34, 7, 35, 31, 2, 36, 37, 4, 2, 38, 39, 4, 8, 26, 2, 40, 41, 7, 8, 4, 42, 43, 2, 44, 45, 46, 2, 31, 2, 26, 47

Offset: 1

Antti Karttunen, Nov 18 2019

nonn

Restricted growth sequence transform of the ordered pair [A046523(n), A246277(A324886(n))].
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A101296(i) = A101296(j),
  a(i) = a(j) => A329345(i) = A329345(j),
  a(i) = a(j) => A329618(i) = A329618(j),
  a(i) = a(j) => A329619(i) = A329619(j).

Cf. A046523, A034386, A108951, A246277, A276086, A305800, A324886, A329345, A329618, A329619.

up_to = 8192;
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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324886(n) = A276086(A108951(n));
A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2);
Aux329620(n) = [A046523(n), A246277(A324886(n))];
v329620 = rgs_transform(vector(up_to, n, Aux329620(n)));
A329620(n) = v329620[n];

A337203 Sum of the divisors of the primorial inflation of n.

Original entry on oeis.org

1, 3, 12, 7, 72, 28, 576, 15, 91, 168, 6912, 60, 96768, 1344, 546, 31, 1741824, 195, 34836480, 360, 4368, 16128, 836075520, 124, 2821, 225792, 600, 2880, 25082265600, 1170, 802632499200, 63, 52416, 4064256, 22568, 403, 30500034969600, 81285120, 733824, 744, 1281001468723200, 9360, 56364064623820800, 34560, 3600, 1950842880

Offset: 1

Antti Karttunen, Aug 22 2020

nonn

Row 0 of A337205, and of A337472.
Cf. A000203, A034386, A108951, A337204.
Cf. also A323173.

Array[DivisorSigma[1, Apply[Times, FactorInteger[#] /. {p_, e_} /; e > 0 :> Apply[Times, Prime@ Range@ PrimePi@ p]^e]] &, 46] (* Michael De Vlieger, Aug 27 2020 *)

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) }; \\ From A108951
A337203(n) = sigma(A108951(n));

A337203(n) = if(1==n,n, my(f=factor(n), h = #f~, prevpid=primepi(f[h,1]), e=f[h,2], p, s=1); forstep(i=h-1,0,-1, if(!i,pid=0,pid=primepi(f[i,1])); forstep(j=prevpid,(1+pid),-1, p=prime(j); s *= ((p^(1+e)-1)/(p-1))); if(!pid,return(s)); prevpid = pid; e += f[i,2]); (s));

a(n) = A000203(A108951(n)).

A337205 Square array A(n,k) read by falling antidiagonals, where row n gives the sum of the divisors of the {primorial inflation of k, from which all primes <= A000040(n) have been discarded}.

Original entry on oeis.org

1, 3, 1, 12, 1, 1, 7, 4, 1, 1, 72, 1, 1, 1, 1, 28, 24, 1, 1, 1, 1, 576, 4, 6, 1, 1, 1, 1, 15, 192, 1, 1, 1, 1, 1, 1, 91, 1, 48, 1, 1, 1, 1, 1, 1, 168, 13, 1, 8, 1, 1, 1, 1, 1, 1, 6912, 24, 1, 1, 1, 1, 1, 1, 1, 1, 1, 60, 2304, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 96768, 4, 576, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1344, 32256, 1, 96, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Antti Karttunen, Aug 22 2020

Array is read by descending antidiagonals with n >= 0 and k >= 1 ranging as: (0, 1), (0, 2), (1, 1), (0, 3), (1, 2), (2, 1), (0, 4), (1, 3), (2, 2), (3, 1), ...

			The top left 15 x 5 corner of the array:
----+------------------------------------------------------------------------
  0 | 1, 3, 12,  7, 72, 28, 576, 15, 91, 168, 6912, 60, 96768, 1344, 546, ...
  1 | 1, 1,  4,  1, 24,  4, 192,  1, 13,  24, 2304,  4, 32256,  192,  78, ...
  2 | 1, 1,  1,  1,  6,  1,  48,  1,  1,   6,  576,  1,  8064,   48,   6, ...
  3 | 1, 1,  1,  1,  1,  1,   8,  1,  1,   1,   96,  1,  1344,    8,   1, ...
  4 | 1, 1,  1,  1,  1,  1,   1,  1,  1,   1,   12,  1,   168,    1,   1, ...
etc.
For example, the row 1 is the sum of the {primorial inflation of k, from which all primes <= prime(1) = 2 have been discarded}, that is, it is the sum of the odd divisors of the primorial inflation of k.

Cf. A337203, A337204 (rows 0 and 1).
Cf. A000203, A034386, A108951.
Cf. also arrays A337470, A337472.

up_to = 105-1;
A337205sq(n,k) = if(1==k,k, my(f=factor(k), h = #f~, prevpid=primepi(f[h,1]), e=f[h,2], p, s=1); forstep(k=h-1,0,-1, if(!k,pid=0,pid=primepi(f[k,1])); forstep(j=prevpid,(1+pid),-1, if(j<=n,return(s));  p=prime(j); s *= ((p^(1+e)-1)/(p-1))); if(pid<=n,return(s)); prevpid = pid; e += f[k,2]); (s));
A337205list(up_to) = { my(v = vector(1+up_to), i=0); for(a=0, oo, for(b=1, a, i++; if(i > #v, return(v)); v[i] = A337205sq(b-1, (a-(b-1))))); (v); };
v337205 = A337205list(up_to);
A337205(n) = v337205[1+n];

A(n,k) is the sum of divisors of A108951(k) from which all primes less than A000040(n) have been removed first.

A(n,k) is a multiple of A(n+1,k).

cp's OEIS Frontend

A307866 K-champion numbers: numbers m such that K(m) > K(j) for all j < m, where K(m) is the Kalmár function (A074206).

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A342456 A276086 applied to the primorial inflation of Doudna-tree, where A276086(n) is the prime product form of primorial base expansion of n.

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A342462 Sum of digits when A329886(n) is written in primorial base, where A329886 is the primorial inflation of Doudna-tree.

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A062977 Difference between largest and smallest positive exponent in prime factorization of n; a(1) = 0 by convention.

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A212181 Largest odd divisor of tau(n): a(n) = A000265(A000005(n)).

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A328521 Smallest highly composite number that has n prime factors counted with multiplicity.

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A328769 The second primorial based variant of arithmetic derivative: a(p) = A034386(p) for p prime, a(u*v) = a(u)*v + u*a(v), with a(0) = a(1) = 0.

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A329620 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A046523(n), A246277(A324886(n))].

A328769 The second primorial based variant of arithmetic derivative: a(p) = A034386(p) for p prime, a(uv) = a(u)v + u*a(v), with a(0) = a(1) = 0.