A101296 -id:A101296 - cp's OEIS frontend

A050334 Number of ordered factorizations of n into numbers with an odd number of prime divisors (prime factors counted with multiplicity).

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 8, 1, 2, 2, 4, 1, 7, 1, 5, 2, 2, 2, 10, 1, 2, 2, 8, 1, 7, 1, 4, 4, 2, 1, 15, 1, 4, 2, 4, 1, 8, 2, 8, 2, 2, 1, 18, 1, 2, 4, 8, 2, 7, 1, 4, 2, 7, 1, 23, 1, 2, 4, 4, 2, 7, 1, 15, 3, 2, 1, 18, 2, 2, 2, 8, 1, 18, 2, 4, 2, 2, 2, 28, 1, 4, 4

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).

			From _R. J. Mathar_, May 25 2017: (Start)
a(p) = 1: factorizations p.
a(p^2) = 1: factorizations p*p.
a(p^3) = 2: factorizations p^3, p*p*p.
a(p^4) = 3: factorizations p^3*p, p*p^3, p*p*p*p.
a(p^5) = 5: factorizations p^5, p^3*p*p, p*p^3*p, p*p*p^3, p*p*p*p*p.
a(p*q) = 2: factorizations p*q, q*p. (End)

R. J. Mathar, Table of n, a(n) for n = 1..10000

Cf. A002033, A026424, A050333.

read(transforms):
A066829m := proc(n)
        if n = 1 or isA026424(n) then
                1;
        else
                0;
        end if;
end proc:
[1,seq(-A066829m(n),n=2..10000)] ;
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 A026424 (essentially A066829).
a(p^k) = A000045(k).
a(A002110(k)) = A006154(k).
a(n) = A050335(A101296(n)). - R. J. Mathar, May 26 2017

A077565 Number of factorizations of n where each factor has a different prime signature.

Original entry on oeis.org

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

Offset: 1

Amarnath Murthy, Nov 11 2002

nonn

In contrast to A001055 this sequence excludes from the count all such factorizations of n that include two such factors, f and g, for which it would hold that A046523(f) = A046523(g), or equally A101296(f) = A101296(g). - Antti Karttunen, Nov 24 2017

			a(24) = 4, 24 = 12*2 = 8*3 = 6*4. The factorizations 2*3*4, 2*2*2*3 etc. are not counted.
From _Antti Karttunen_, Nov 24 2017: (Start)
For n = 30 the solutions are 30, 2*15, 3*10, 5*6, thus a(30) = 4.
For n = 36 the solutions are 36, 2*18, 3*12, thus a(36) = 3.
For n = 60 the solutions are 60, 2*30, 3*20, 4*15, 5*12, thus a(60) = 5.
For n = 72 the solutions are 72, 2*36, 3*24, 4*18, 6*12, 8*9, 3*4*6, thus a(72) = 7.
(End)

Amarnath Murthy, Generalization of partition function. Introducing Smarandache Factor Partition. Smarandache Notions Journal, Vol. 11, 1-2-3,2000.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 6143 terms from Antti Karttunen, computed with the given Scheme-program)
Antti Karttunen, Scheme-program for computing this sequence
Index entries for sequences computed from exponents in factorization of n

Cf. A000009, A001055, A025487, A077563, A077564, A077566, A046523, A101296.

Table[1 + Count[Subsets[Rest@ Divisors@ n, {2, Infinity}], ?(And[Times @@ # == n, UnsameQ @@ Map[Sort[FactorInteger[#][[All, -1]], Greater] &, #]] &)], {n, 105}] (* _Michael De Vlieger, Nov 24 2017 *)

a(n) <= A001055(n). - Antti Karttunen, Nov 24 2017

a(p^e) = A000009(p^e). - David A. Corneth, Nov 24 2017

Corrected and extended by Ray Chandler, Aug 26 2003

Name improved by Antti Karttunen and David A. Corneth, Nov 24 2017

A305978 Filter sequence combining prime signatures of n and 2n+1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 15 2018

nonn

Restricted growth sequence transform of A286258.

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

Cf. A046523, A101296, A286258, A305810, A305973, A305977.

Cf. A005384 (positions of 2's), A234095 (of 5's).

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
Aux305978(n) = [A046523(n),A046523(n+n+1)];
v305978 = rgs_transform(vector(up_to,n,Aux305978(n)));
A305978(n) = v305978[n];

A319994 Let g = A006530(n), the largest prime factor of n. This filter sequence combines (g mod 4), n/g (A052126), and a single bit A319988(n) telling whether the largest prime factor is unitary.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 05 2018

nonn

Restricted growth sequence transform of triple [A010873(A006530(n)), A052126(n), A319988(n)], with a separate value allotted for a(1).
Here among the first 100000 terms, only 2331 have a unique value, compared to 69714 in A320004.
For all i, j:
  a(i) = a(j) => A024362(i) = A024362(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).

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

Cf. A006530, A052126, A319988, A320004.

Cf. also A319996 (modulo 6 analog).

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; };
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A052126(n) = (n/A006530(n));
A319988(n) = ((n>1)&&(factor(n)[omega(n),2]>1));
A319994aux(n) = if(1==n,0,[A006530(n)%4, A052126(n), A319988(n)]);
v319994 = rgs_transform(vector(up_to,n,A319994aux(n)));
A319994(n) = v319994[n];

A319996 Let g = A006530(n), the largest prime factor of n. This filter sequence combines (g mod 6), n/g (A052126), and a single bit A319988(n) telling whether the largest 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, 10, 5, 18, 19, 12, 20, 21, 5, 22, 7, 23, 13, 10, 24, 25, 7, 12, 17, 26, 5, 27, 7, 16, 28, 10, 5, 29, 30, 31, 13, 21, 5, 32, 33, 34, 17, 10, 5, 35, 7, 12, 36, 37, 24, 22, 7, 16, 13, 38, 5, 39, 7, 12, 40, 21, 41, 27, 7, 42, 43, 10, 5, 44, 33, 12, 13, 26, 5, 45, 46, 16, 17, 10, 24, 47, 7, 48, 28, 49, 5, 22, 7, 34

Offset: 1

Antti Karttunen, Oct 05 2018

nonn

Restricted growth sequence transform of triple [A010875(A006530(n)), A052126(n), A319988(n)], with a separate value allotted for a(1).
Many of the same comments as given in A319717 apply also here, except for this filter, the "blind spot" area (where only unique values are possible for a(n)) is different, and contains at least all numbers in A070003. Because presence of 2 or 3 in the prime factorization of n do not force the value of a(n) unique, this is substantially less lax (i.e., more exact) filter than A319717. Here among the first 100000 terms, only 2393 have a unique value, compared to 74355 in A319717.
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) => A319690(i) = A319690(j).

			For n = 15 (3*5) and n = 33 (3*11), the mod 6 residue of the largest prime factor is 5, also in both cases it is unitary (A319988(n) = 1), and the quotient n/A006530(n) is equal, in this case 3. Thus a(15) and a(33) are alloted the same running count (13 in this case) by rgs-transform.
For n = 2275 (5^2 * 7 * 13), n = 3325 (5^2 * 7 * 19), 5425 (5^2 * 7 * 31) and 6475 (5^2 * 7 * 37), the largest prime factor = 1 (mod 6), and A052126(n) = 175, thus these numbers are allotted the same running count (394 in this case) by rgs-transform.

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

Cf. A006530, A052126, A319988, A319717.
Cf. A007528 (positions of 5's), A002476 (of 7's), A112774 (after its initial term gives the position of 10's in this sequence).
Cf. also A319994 (modulo 4 analog).

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; };
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A052126(n) = (n/A006530(n));
A319988(n) = ((n>1)&&(factor(n)[omega(n),2]>1));
A319996aux(n) = if(1==n,0,[A006530(n)%6, A052126(n), A319988(n)]);
v319996 = rgs_transform(vector(up_to,n,A319996aux(n)));
A319996(n) = v319996[n];

A327931 Lexicographically earliest infinite sequence such that for all i, j, a(i) = a(j) => A327930(i) = A327930(j).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 39, 28, 40, 41, 42, 2, 43, 2, 44, 45, 46, 47, 48, 2, 49, 50, 51, 2, 52, 2, 53, 54, 55, 47, 56, 2, 57, 58, 59, 2, 60, 41, 61, 62, 63, 2, 64, 37, 65, 66, 67, 68, 69, 2, 70, 71, 72, 2, 73, 2, 74, 75

Offset: 1

Antti Karttunen, Sep 30 2019

nonn

Restricted growth sequence transform of A327930, or equally, of the ordered pair [A003415(n), A319356(n)].
It seems that the sequence takes duplicated values only on primes (A000040) and some subset of squarefree semiprimes (A006881). If this holds, then also the last implication given below is valid.
For all i, j:
  a(i) = a(j) => A000005(i) = A000005(j),
  a(i) = a(j) => A319684(i) = A319684(j),
  a(i) = a(j) => A319685(i) = A319685(j),
  a(i) = a(j) => A101296(i) = A101296(j). [Conjectural, see notes above and in A319357]

			Divisors of 39 are [1, 3, 13, 39], while the divisors of 55 are [1, 5, 11, 55]. Taking their arithmetic derivatives (A003415) yields in both cases [0, 1, 1, 16], thus a(39) = a(55) (= 28, as allotted by restricted growth sequence transform).

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

Cf. A000005, A000040 (positions of 2's), A003415, A006881, A101296, A319356, A319357, A319684, A319685, A327930.
Differs from A300249 for the first time at n=105, where a(105)=75, while A300249(105)=56.
Differs from A300235 for the first time at n=153, where a(153)=110, while A300235(153)=106.
Differs from A305895 for the first time at n=3283, where a(3283)=2502, while A305895(3283)=1845.

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; };
A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
v003415 = vector(up_to,n,A003415(n));
A327930(n) = { my(m=1); fordiv(n,d,if((d>1), m *= prime(v003415[d]))); (m); };
v327931 = rgs_transform(vector(up_to, n, A327930(n)));
A327931(n) = v327931[n];

a(p) = 2 for all primes p.

A340681 The closure under squaring of A051144, the nonsquarefree nonsquares.

Original entry on oeis.org

8, 12, 18, 20, 24, 27, 28, 32, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200, 204, 207, 208

Offset: 1

Antti Karttunen and Peter Munn, Feb 07 2021

Numbers not of the form s^(2^e), where s is a squarefree number, and e >= 0.
The categorization provided by this sequence and its complement, A340682, is an alternative extension (to all integers greater than 1) of the 2-way distinction between squarefree and nonsquarefree as it applies to nonsquares.
All positive integers have a unique factorization into powers of nonunit squarefree numbers with distinct exponents that are powers of 2. This sequence lists the numbers where this factorization has more than one term, that is numbers m such that A331591(m) > 1.
Presence in the sequence is determined by prime signature (A101296). The set of represented signatures starts: {{3}, {2,1}, {3,1}, {2,1,1}, {5}, {4,1}, {3,2}, {3,1,1}, {2,2,1}, {2,1,1,1}, {6}, {5,1}, {4,2}, {4,1,1}, {3,3}, {3,2,1}, {3,1,1,1}, {2,2,1,1}, {2,1,1,1,1}, {7}, ...}.
Gives positions of 1's in A340675 after its initial one.

			24 = 6 * 4 = 6^1 * 2^2 = 6^(2^0) * 2^(2^1), which is the factorization into powers of nonunit squarefree numbers with distinct exponents that are powers of 2. As this factorization has 2 terms, 24 is in the sequence.
The equivalent factorization for 100 is 100 = 10^2 = 10^(2^1). As this factorization has only 1 term, 100 is not in the sequence.

Index entries for sequences related to prime signature

Cf. A101296, A331591, A340675.
Cf. A340682 (complement, apart from 1 which is in neither).
Cf. subsequences: A051144, A059404.
Subsequence of A013929.

isA340681(n) = if(!issquare(n), !issquarefree(n), (n>1)&&isA340681(sqrtint(n)));

from math import isqrt
from sympy import mobius, integer_nthroot
def A340681(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 g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def f(x): return int(n+1+sum(g(integer_nthroot(x,1<Chai Wah Wu, Jun 01 2025

A351235 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j), A327858(i) = A327858(j) and A345000(i) = A345000(j) for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 06 2022

Restricted growth sequence transform of the triplet [A046523(n), A327858(n), A345000(n)].
For all i, j >= 1:
  A305800(i) = A305800(j) => a(i) = a(j) => A351085(i) = A351085(j).

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

Cf. A003415, A046523, A276086, A327858, A345000, A351086.

Cf. also A101296, A305800, A329620, A351085, A351236.

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; };
A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A327858(n) = gcd(A003415(n),A276086(n));
A345000(n) = gcd(A003415(n),A003415(A276086(n)));
Aux351235(n) = [A046523(n), A327858(n), A345000(n)];
v351235 = rgs_transform(vector(up_to,n,Aux351235(n)));
A351235(n) = v351235[n];

A352898 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A046523(n), A352892(n)], except f(n) = -n when <= 2.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 08 2022

nonn

For all i, j: A305801(i) = A305801(j) => a(i) = a(j) => A352897(i) = A352897(j).

Cf. A046523, A101296, A305801, A305897, A352892, A352897, A352899.

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; };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A329603(n) = A005940(2+(3*A156552(n)));
A341515(n) = if(n%2, A064989(n), A329603(n));
A348717(n) = { my(f=factor(n)); if(#f~>0, my(pi1=primepi(f[1, 1])); for(k=1, #f~, f[k, 1] = prime(primepi(f[k, 1])-pi1+1))); factorback(f); }; \\ From A348717
A352892(n) = A348717(A341515(n));
Aux352898(n) = if(n<=2,-n,[A046523(n),A352892(n)]);
v352898 = rgs_transform(vector(up_to, n, Aux352898(n)));
A352898(n) = v352898[n];

A378603 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A033630(i) = A033630(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 01 2024

nonn

Restricted growth sequence transform of the ordered pair [A046523(n), A033630(n)].

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

Cf. A046523, A033630, A101296.

Cf. also A378601, A378602, A378604, A378605.

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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
v033630 = readvec("b033630_to.txt"); \\ Precomputed with A033630(n) = if(!n, 1, my(p=1); fordiv(n, d, p *= (1 + 'x^d)); polcoeff(p, n));
A033630(n) = v033630[n];
Aux378603(n) = [A046523(n), A033630(n)];
v378603 = rgs_transform(vector(up_to, n, Aux378603(n)));
A378603(n) = v378603[n];

cp's OEIS Frontend

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