A101296 -id:A101296 - cp's OEIS frontend

A050363 Number of ordered factorizations into prime powers greater than 1.

1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 5, 1, 2, 2, 8, 1, 5, 1, 5, 2, 2, 1, 12, 2, 2, 4, 5, 1, 6, 1, 16, 2, 2, 2, 14, 1, 2, 2, 12, 1, 6, 1, 5, 5, 2, 1, 28, 2, 5, 2, 5, 1, 12, 2, 12, 2, 2, 1, 18, 1, 2, 5, 32, 2, 6, 1, 5, 2, 6, 1, 37, 1, 2, 5, 5, 2, 6, 1, 28, 8, 2, 1, 18, 2, 2, 2, 12, 1, 18, 2, 5, 2, 2, 2, 64

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).
The Dirichlet inverse is in A010055, turning all but the first element in A010055 negative. - R. J. Mathar, Jul 15 2010
Not multiplicative: a(6) =2 <> a(2)*a(3) = 1*1. - R. J. Mathar, May 25 2017

			From _R. J. Mathar_, May 25 2017: (Start)
a(p^2)  = 2: factorizations p^2, p*p.
a(p^3)  = 4: factorizations p^3, p^2*p, p*p^2, p*p*p.
a(p*q)  = 2: factorizations p*q, q*p.
a(p*q^2)= 5: factorizations p*q^2, q^2*p, p*q*q, q*p*q, q*q*p. (End)

R. J. Mathar, Table of n, a(n) for n = 1..5000

Cf. A000961, A050360, A050361, A050362, A050363, A050364, A076277.

read(transforms) ;
[1,seq(-A010055(n),n=2..100)] ;
DIRICHLETi(%) ; # R. J. Mathar, May 25 2017

Dirichlet g.f.: 1/(1-B(s)) where B(s) is D.g.f. of characteristic function of prime powers >1.
a(p^k) = 2^(k-1).
a(A002110(k)) = k!.
a(n) = A050364(A101296(n)). - R. J. Mathar, May 26 2017
G.f. A(x) satisfies: A(x) = x + Sum_{p prime, k>=1} A(x^(p^k)). - Ilya Gutkovskiy, May 11 2019

A179690 Numbers of the form p^2q^2r*s where p, q, r, and s are distinct primes.

Original entry on oeis.org

1260, 1980, 2100, 2340, 2772, 2940, 3060, 3150, 3276, 3300, 3420, 3900, 4140, 4284, 4410, 4788, 4950, 5100, 5148, 5220, 5580, 5700, 5796, 5850, 6468, 6660, 6732, 6900, 7260, 7308, 7350, 7380, 7524, 7644, 7650, 7700, 7740, 7812, 7956, 8460, 8550, 8700

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
Index to sequences related to prime signature

Part of the list A178739 .. A179696 and A030514 .. A030629, A189975 .. A189990 etc., cf. A101296. - M. F. Hasler, Jul 17 2019

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,2,2}; Select[Range[10000], f]

list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtint(lim\60), t1=p^2; forprime(q=2,sqrtint(lim\(6*t1)), if(q==p, next); t2=q^2*t1; forprime(r=2,lim\(2*t2), if(r==p || r==q, next); t3=r*t2; forprime(s=2,lim\t3, if(s==p || s==q || s==r, next); listput(v, t3*s))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

A179691 Numbers p^5q^2r where p, q, r are 3 distinct primes.

Original entry on oeis.org

1440, 2016, 2400, 3168, 3744, 4704, 4860, 4896, 5472, 5600, 6624, 6804, 7840, 8352, 8800, 8928, 10400, 10656, 10692, 11616, 11808, 12150, 12384, 12636, 13536, 13600, 15200, 15264, 16224, 16524, 16992, 17248, 17568, 18400, 18468, 19296, 19360

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
Index to sequences related to prime signature

Part of the list A178739 .. A179696 (and A030514 .. A030629, A189982 .. A189990 etc, cf. A101296). - M. F. Hasler, Jul 17 2019

Subsequence of A175746 (numbers with 36 divisors).

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,2,5}; Select[Range[20000], f]

list(lim)=my(v=List(),t1,t2);forprime(p=2, (lim\12)^(1/5), t1=p^5;forprime(q=2, sqrt(lim\t1), if(p==q, next);t2=t1*q^2;forprime(r=2, lim\t2, if(p==r||q==r, next);listput(v,t2*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A179691(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**5*q**2)) for p in primerange(integer_nthroot(x,5)[0]+1) for q in primerange(isqrt(x//p**5)+1))+sum(primepi(integer_nthroot(x//p**5,3)[0]) for p in primerange(integer_nthroot(x,5)[0]+1))+sum(primepi(isqrt(x//p**6)) for p in primerange(integer_nthroot(x,6)[0]+1))+sum(primepi(x//p**7) for p in primerange(integer_nthroot(x,7)[0]+1))-(primepi(integer_nthroot(x,8)[0])<<1)
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

Name improved by M. F. Hasler, Jul 17 2019

A304751 Filter sequence: Restricted growth sequence transform of function that gives the least natural number with the same prime signature that (0,1)-polynomial encoded in the binary expansion of n has when it is factored over Q.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 08 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A206719, A206074 (gives the positions of 2's), A257000.

Cf. also A046523 (A101296), A278233, A284010, A284011.

up_to = 65537;
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
Aux304751(n) = { my(p=0, f=vecsort((factor(Pol(binary(n)))[, 2]), , 4)); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }
v304751 = rgs_transform(vector(up_to,n,Aux304751(n)));
A304751(n) = v304751[n];

For all i, j: a(i) = a(j) => A206719(i) = A206719(j).

For all i, j: a(i) = a(j) => A257000(i) = A257000(j).

A305973 Filter sequence for the prime signature of 2n-1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 15 2018

nonn

Restricted growth sequence transform of A278223, the least number with the same prime signature as the n-th odd number.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A046523, A101296, A278223, A305901, A305983, A305985.

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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
v305973 = rgs_transform(vector(up_to,n,A046523(n+n-1)));
A305973(n) = v305973[n];

A050340 Number of ways of factoring n with 3 levels of parentheses.

Original entry on oeis.org

1, 1, 1, 5, 1, 5, 1, 15, 5, 5, 1, 25, 1, 5, 5, 55, 1, 25, 1, 25, 5, 5, 1, 105, 5, 5, 15, 25, 1, 35, 1, 170, 5, 5, 5, 145, 1, 5, 5, 105, 1, 35, 1, 25, 25, 5, 1, 425, 5, 25, 5, 25, 1, 105, 5, 105, 5, 5, 1, 205, 1, 5, 25, 571, 5, 35, 1, 25, 5, 35, 1, 660, 1, 5, 25, 25, 5, 35, 1, 425, 55, 5

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

a(n) depends only on the prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1).

			4 = (((4))) = (((2*2))) = (((2)*(2))) = (((2))*((2))) = (((2)))*(((2))).

R. J. Mathar, Table of n, a(n) for n = 1..1295

Cf. A001055, A050336-A050341. a(p^k) = A007714. a(A002110) = A000357.

Dirichlet g.f.: Product{n=2..infinity} (1/(1-1/n^s)^A050338(n)).

a(n) = A050341(A101296(n)). - R. J. Mathar, May 26 2017

A292259 Restricted growth sequence transform of A292258; filter constructed from the prime signatures of the sequence [n, floor(n/2), floor(n/4), ..., 1].

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 4, 6, 7, 8, 9, 10, 8, 8, 8, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 16, 21, 15, 17, 22, 17, 23, 24, 25, 25, 26, 27, 28, 28, 29, 30, 31, 32, 30, 30, 33, 34, 35, 36, 37, 38, 39, 32, 40, 41, 29, 39, 32, 33, 42, 43, 32, 30, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 65, 66, 62, 67, 61, 62

Offset: 1

Antti Karttunen, Sep 22 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65535

Cf. A292258.

Cf. A078349 (one of the matching sequences).

up_to = 65535
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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
v101296 = rgs_transform(vector(up_to, n, A046523(n)));
A101296(n) = v101296[n];
A292258(n) = if(1==n,n,prime(A101296(n)-1) * A292258(n\2));
write_to_bfile(1,rgs_transform(vector(up_to,n,A292258(n))),"b292259.txt");

A305789 Filter-sequence combining prime signature of n (A046523) and similar signature for GF(2)[X]-factorization (A278233).

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 2, 6, 7, 8, 2, 9, 2, 5, 10, 11, 12, 13, 2, 14, 15, 5, 16, 17, 18, 5, 19, 9, 16, 20, 2, 21, 5, 22, 5, 23, 2, 5, 8, 24, 2, 25, 16, 9, 26, 27, 2, 28, 7, 29, 30, 9, 16, 31, 32, 17, 8, 27, 2, 33, 2, 5, 9, 34, 35, 36, 2, 37, 15, 36, 16, 38, 2, 5, 26, 9, 5, 39, 16, 40, 41, 5, 42, 43, 44, 27, 32, 17, 16, 45, 32, 46, 5, 5, 8, 47, 2, 13, 9, 48, 42

Offset: 1

Antti Karttunen, Jun 10 2018

nonn

Restricted growth sequence transform of ordered pair [A046523(n), A278233(n)].

For all i, j: a(i) = a(j) => A305802(i) = A305802(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences operating on GF(2)[X]-polynomials

Cf. A046523, A101296, A278233, A305788, A305802.

Cf. also A305790.

up_to = 65537;
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,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
A278233(n) = { my(p=0, f=vecsort((factor(Pol(binary(n))*Mod(1, 2))[, 2]), , 4)); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
Aux305789(n) = [A046523(n), A278233(n)];
v305789 = rgs_transform(vector(up_to,n,Aux305789(n)));
A305789(n) = v305789[n];

A319717 Filter sequence combining the largest proper divisor of n (A032742) with modulo 6 residue of the smallest prime factor, A010875(A020639(n)), and a single bit A319710(n) telling whether the smallest prime factor is unitary.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 5, 11, 7, 12, 13, 14, 5, 15, 7, 16, 17, 18, 5, 19, 20, 21, 22, 23, 5, 24, 7, 25, 26, 27, 28, 29, 7, 30, 31, 32, 5, 33, 7, 34, 35, 36, 5, 37, 38, 39, 40, 41, 5, 42, 43, 44, 45, 46, 5, 47, 7, 48, 49, 50, 51, 52, 7, 53, 54, 55, 5, 56, 7, 57, 58, 59, 60, 61, 7, 62, 63, 64, 5, 65, 66, 67, 68, 69, 5, 70, 71, 72, 73, 74, 75, 76, 7, 77, 78, 79, 5, 80, 7, 81, 82, 83, 5, 84, 7, 85, 86, 87, 5, 88, 89, 90, 91, 92, 93, 94, 95

Offset: 1

Antti Karttunen, Oct 04 2018

nonn

Restricted growth sequence transform of triple [A010875(A020639(n)), A032742(n), A319710(n)] (with a separate value allotted for a(1)), or equally, of ordered pair [A319716(n), A319710(n)].
In addition to A319716, this filter sequence also records in the value of a(n) also the fact whether the smallest prime factor of n is unitary or not. This information is enough to determine the modulo 6 residues of all the divisors of n, thus sequences like A002324 are essentially functions of this sequence. Moreover, a lot of other information is immediately (and unavoidably) present, for example the exact prime signature of n, including also the relative order of exponents.
Any such filtering sequence can be perceived also in terms of what information it leaves out from a(n) that would be needed to reconstruct whole n from each a(n). If the whole n could be reconstructed from a(n) each time, then sequence a would be injective, and would be useless for filtering, because then it would match with any sequence. In this filter, what is left out is only the exact identity of the smallest prime factor, although its residue class mod 6 is retained. However, when the smallest prime factor is 2 or 3, this can be seen from that residue value, so for any number x in A047229, both A020639(x) and A032742(x) are known, and as x = A020639(x)*A032742(x), it means such numbers must occur in their own singleton equivalence classes.
Likewise, for any n in A283050, even if not divisible by 2 or 3, when we have A319710(n) stored in the triple as 1, this immediately gives away the exact identity of the smallest prime factor, which is equal to A014673(n) = A020639(A032742(n)) in these cases.
Thus there is a substantial subset of N (containing at least the union of A047229 and A283050) which is actually in the "blind sector" of this filter, "where anything goes", as this sequence obtains only unique values in that subdomain.
There is a related filter sequence A319996, which operates by "cleaving n from its high end" (by storing the residue class of the largest prime factor, A006530, instead of the smallest, together with n/A006530(n)), which has its own blind spots, but fortunately, they do not fully coincide with the blind spots of this filter. Naturally, any sequence like A002324 should match both to this sequence and A319996.
For all i, j:
  a(i) = a(j) => A002324(i) = A002324(j),
  a(i) = a(j) => A067029(i) = A067029(j),
  a(i) = a(j) => A071178(i) = A071178(j),
  a(i) = a(j) => A077462(i) = A077462(j) => A101296(i) = A101296(j),
  a(i) = a(j) => A319716(i) = A319716(j) => A319690(i) = A319690(j).

			For n = 65 = 5*13 and 143 = 11*13, the smallest prime factor is of the form 6k+5,  doesn't occur more than once in the factorization, and the largest proper divisor is the same number (13) in both cases, thus a(65) = a(143) (= 51, a running count value allotted by rgs-transform for this equivalence class).
For n = 1805 (5*19^2), 3971 (11*19^2), 6137 (17*19^2), it's like above, but the largest proper divisor is in all three cases 361 = 19^2, thus a(1805) = a(3971) = a(6137) (= 1405).
Note that such nontrivial equivalence classes may only contain numbers that are 5-rough, A007310, with no prime factors 2 or 3, and also, they may not contain numbers from A283050. See the comments section.

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

Cf. A101296, A319707, A319716, A319996.
Cf. also A320004 (analogous sequence for modulo 4 residues).
Differs from A319707 for the first time at n=143, where a(143) = 51, differs from A319716 for the first time at n=121, where a(121) = 95.

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; };
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A286476(n) = if(1==n,n,(6*A032742(n) + (n % 6)));
A319710(n) = ((n>1)&&(factor(n)[1,2]>1));
v319717 = rgs_transform(vector(up_to,n,[A286476(n),A319710(n)]));
A319717(n) = v319717[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];

cp's OEIS Frontend

A050363 Number of ordered factorizations into prime powers greater than 1.

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A179690 Numbers of the form p^2*q^2*r*s where p, q, r, and s are distinct primes.

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A179691 Numbers p^5*q^2*r where p, q, r are 3 distinct primes.

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A304751 Filter sequence: Restricted growth sequence transform of function that gives the least natural number with the same prime signature that (0,1)-polynomial encoded in the binary expansion of n has when it is factored over Q.

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A305973 Filter sequence for the prime signature of 2n-1.

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PARI

A050340 Number of ways of factoring n with 3 levels of parentheses.

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A292259 Restricted growth sequence transform of A292258; filter constructed from the prime signatures of the sequence [n, floor(n/2), floor(n/4), ..., 1].

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Programs

PARI

A305789 Filter-sequence combining prime signature of n (A046523) and similar signature for GF(2)[X]-factorization (A278233).

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Programs

PARI

A319717 Filter sequence combining the largest proper divisor of n (A032742) with modulo 6 residue of the smallest prime factor, A010875(A020639(n)), and a single bit A319710(n) telling whether the smallest prime factor is unitary.

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A179690 Numbers of the form p^2q^2r*s where p, q, r, and s are distinct primes.

A179691 Numbers p^5q^2r where p, q, r are 3 distinct primes.