A101296 -id:A101296 - cp's OEIS frontend

A291761 Restricted growth sequence transform of ((-1)^n)*A046523(n); filter combining the parity and the prime signature of n.

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

Offset: 1

Antti Karttunen, Sep 11 2017

nonn

Equally, restricted growth sequence transform of sequence b defined as b(1) = 1, and for n > 1, b(n) = A046523(n) + A000035(n), which starts as 1, 2, 3, 4, 3, 6, 3, 8, 5, 6, 3, 12, 3, 6, 7, 16, 3, 12, 3, 12, ...

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

Cf. A291767, A291768 (bisections), A147516.
Cf. A046523, A101296, A286161, A286251, A286367, A291762 (related or similar filtering sequences).
Cf. A065091 (positions of 3's), A100484 (of 4 and 5's), A001248 (of 4 and 7's), A046388 (of 9's), A030078 (of 6 and 12's).
Cf. A098108 (one of the matching sequences).

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
write_to_bfile(1,rgs_transform(vector(65537,n,((-1)^n)*A046523(n))),"b291761.txt");
\\ Or alternatively:
f(n) = if(1==n,n,A046523(n)+(n%2));
write_to_bfile(1,rgs_transform(vector(16385,n,f(n))),"b291761.txt");

A300827 Lexicographically earliest sequence such that a(i) = a(j) => A324193(i) = A324193(j) for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 13 2018

nonn

Apart from primes, the sequence contains duplicate values at points p*q and p^3, where p*q are the product of two successive primes, with p < q (sequences A006094, A030078). Question: are there any other cases where a(x) = a(y), with x < y ?
The reason why this is not equal to A297169: Even though A297112 contains only powers of two after the initial zero, as A297112(n) = 2^A033265(A156552(d)) for n > 1, and A297168(n) is computed as Sum_{d|n, dA297112(d), still a single 1-bit in binary expansion of A297168(n) might be formed as a sum of several terms of A297112(d), i.e., could be born of carries.
From Antti Karttunen, Feb 28 2019: (Start)
  A297168(n) = Sum_{d|n, dA297112(d) will not produce any carries (in base-2) if and only if n is a power of prime. Only in that case the number of summands (A000005(n)-1) is equal to the number of prime factors counted with multiplicity, A001222(n) = A000120(A156552(n)). (A notable subset of such numbers is A324201, numbers that are mapped to even perfect numbers by A156552). Precisely because there are so few points with duplicate values (apart from primes), this sequence is not particularly good for filtering other sequences, because the number of false positives is high. Any of the related sequences like A324203, A324196, A324197 or A324181 might work better in that respect. In any case, the following implications hold (see formula section of A324193 for the latter): (End)
For all i, j:
  a(i) = a(j) => A297168(i) = A297168(j). (The same holds for A297169).
  a(i) = a(j) => A324181(i) = A324181(j) => A324120(i) = A324120(j).

			For n = 15, with proper divisors 3 and 5, we have f(n) = prime(1+A297167(3)) * prime(1+A297167(5)) = prime(2)*prime(3) = 3*5 = 15.
For n = 27, with proper divisors 3 and 9, we have f(n) = prime(1+A297167(3)) * prime(1+A297167(9)) = prime(2)*prime(3) = 3*5 = 15.
Because f(15) = f(27), the restricted growth sequence transform allots the same number (in this case 9) for both, so a(15) = a(27) = 9.

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

Cf. A000120, A000961, A006094, A030078, A046660, A061395, A101296, A156552, A297112, A297167, A297168, A297169, A324193, A324120, A324201.

Cf. also A324181, A323914, A324196, A324197, A324203.

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1]))); \\ After M. F. Hasler's code for A006530.
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) -1));
Aux300827(n) = { my(m=1); if(n<=2, n-1, fordiv(n,d,if((d>1)&(dA297167(d)))); (m)); };
write_to_bfile(1,rgs_transform(vector(up_to,n,Aux300827(n))),"b300827.txt");

Restricted growth sequence transform of sequence f, defined as f(1) = 0, f(2) = 1, and for n > 2, f(n) = Product_{d|n, 1A297167(d)).
a(p) = 2 for all primes p.
a(A006094(n)) = a(A030078(n)), for all n >= 1.

Name changed by Antti Karttunen, Feb 21 2019

A050330 Number of factorizations of n into numbers with an odd number of prime factors.

Original entry on oeis.org

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

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).

R. J. Mathar, Table of n, a(n) for n = 1..2519

Cf. A001055, A026424, A050331. a(p^k)=A000009. a(A002110)=A003724.

Dirichlet g.f.: Product_{n in A026424} (1/(1-1/n^s)).

a(n) = A050331(A101296(n)). - R. J. Mathar, May 26 2017

A050338 Number of ways of factoring n with 2 levels of parentheses.

Original entry on oeis.org

1, 1, 1, 4, 1, 4, 1, 10, 4, 4, 1, 16, 1, 4, 4, 30, 1, 16, 1, 16, 4, 4, 1, 54, 4, 4, 10, 16, 1, 22, 1, 75, 4, 4, 4, 74, 1, 4, 4, 54, 1, 22, 1, 16, 16, 4, 1, 176, 4, 16, 4, 16, 1, 54, 4, 54, 4, 4, 1, 102, 1, 4, 16, 206, 4, 22, 1, 16, 4, 22, 1, 267, 1, 4, 16, 16, 4, 22, 1, 176, 30, 4, 1, 102

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).

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

R. J. Mathar, Table of n, a(n) for n = 1..1679

Cf. A001055, A050336-A050341. a(p^k)=A007713. a(A002110)=A000307.

Dirichlet g.f.: Product_{n>=2}(1/(1-1/n^s)^A050336(n)).

a(n) = A050339(A101296(n)). - R. J. Mathar, May 26 2017

A286610 Restricted growth sequence computed for Euler totient function phi, A000010.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 11 2017

nonn

			Construction: we start with a(1)=1 for phi(1)=1 (where phi = A000010), and then after, for all n > 1, whenever the value of phi(n) has not been encountered before, we set a(n) to the least natural number k not already in sequence among a(1) .. a(n-1), otherwise [whenever phi(n) = phi(m), for some m < n], we set a(n) = a(m), i.e., to the same value that was assigned to a(m).
For n=2, phi(2) = 1, which value was already encountered as phi(1), thus we set also a(2) = 1.
For n=3, phi(3) = 2, which has not been encountered before, thus we allot for a(3) the least so far unused number, which is 2, thus a(3) = 2.
For n=4, phi(4) = 2, which was already encountered as at n=3 for the first time, thus we set a(4) = a(3) = 2.
For n=5, phi(5) = 4, which has not been encountered before, thus we allot for a(5) the least so far unused number, which is now 3, thus a(5) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000010, A210719 (positions of records, and also the first occurrence of each n).

Cf. also A101296, A286603, A286605, A286619, A286621, A286622, A286626, A286378 for similarly constructed sequences.

With[{nn = 99}, Function[s, Table[Position[Keys@ s, k_ /; MemberQ[k, n]][[1, 1]], {n, nn}]]@ Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Array[EulerPhi, nn]] (* Michael De Vlieger, May 12 2017, Version 10 *)

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A000010(n) = eulerphi(n);
write_to_bfile(1,rgs_transform(vector(10000,n,A000010(n))),"b286610.txt");

A305788 Restricted growth sequence transform of A278233, filter-sequence for GF(2)[X]-factorization.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 10 2018

nonn

This is GF(2)[X] analog of A101296.

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

Cf. A278233.
Cf. A014580 (positions of 2's).
Cf. also A101296, A304529, A304751, A305789.

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; };
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]); };
v305788 = rgs_transform(vector(up_to,n,A278233(n)));
A305788(n) = v305788[n];

A189982 Numbers with prime signature (2,1,1,1), i.e., factorization pqr*s^2 with distinct primes p, q, r, s.

Original entry on oeis.org

420, 630, 660, 780, 924, 990, 1020, 1050, 1092, 1140, 1170, 1380, 1386, 1428, 1470, 1530, 1540, 1596, 1638, 1650, 1710, 1716, 1740, 1820, 1860, 1932, 1950, 2070, 2142, 2220, 2244, 2380, 2394, 2436, 2460, 2508, 2550, 2574, 2580, 2604, 2610, 2652, 2660, 2790

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 03 2011

nonn

Theorem 4 in Goldston-Graham-Pintz-Yildirim proves that a(n+1) = a(n) + 1 for infinitely many n. - Charles R Greathouse IV, Jul 17 2015, corrected by M. F. Hasler, Jul 17 2019

T. D. Noe, Table of n, a(n) for n = 1..1000
D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yıldırım, Small gaps between almost primes, the parity problem, and some conjectures of Erdos on consecutive integers (2008)
Will Nicholes, List of Prime Signatures
Index to sequences related to prime signature

Part of the list A178739 .. A179696 and A030514 .. A030629, A189975 .. A189990 etc., cf. A101296.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,1,2}; Select[Range[4000],f]

is(n)=vecsort(factor(n)[,2])==[1, 1, 1, 2]~ \\ Charles R Greathouse IV, Jul 17 2015

Definition reworded by M. F. Hasler, Jul 17 2019

A286605 Restricted growth sequence computed for number of divisors, d(n) (A000005).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 11 2017

nonn

For all i, j: A101296(i) = A101296(j) => a(i) = a(j).

For all i, j: a(i) = a(j) <=> A000005(i) = A000005(j).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A007416 (positions of records, and also the first occurrence of each n).

Cf. also A101296, A286603, A286610, A286619, A286621, A286622, A286626, A286378.

With[{nn = 119}, Function[s, Table[Position[Keys@ s, k_ /; MemberQ[k, n]][[1, 1]], {n, nn}]]@ Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Array[DivisorSigma[0, #] &, nn]] (* Michael De Vlieger, May 12 2017, Version 10 *)

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A000005(n) = numdiv(n);
write_to_bfile(1,rgs_transform(vector(10000,n,A000005(n))),"b286605.txt");

A300250 Restricted growth sequence transform of A297174: a filter sequence recording the prime signatures of divisors of n, with divisors ordered by their magnitude.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 07 2018

nonn

This sequence gives a coarser partitioning of natural numbers than A290110, and finer than A101296:
For all i, j:
  A290110(i) = A290110(j) => a(i) = a(j) => A101296(i) = A101296(j).

			Divisors of 462 are 1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462.
Divisors of 858 are 1, 2, 3, 6, 11, 13, 22, 26, 33, 39, 66, 78, 143, 286, 429, 858.
If one takes the smallest prime-signature representative (A046523) of each these, one gets in both cases [1, 2, 2, 6, 2, 2, 6, 6, 6, 6, 30, 30, 6, 30, 30, 210]. E.g. 462 = 2*3*7*11 and 858 = 2*3*11*13, which both have the same prime signature as 210 = 2*3*5*7. And similarly for all the other divisors, from which follows that a(462) = a(858).
On the other hand, for 12 = 2*2*3 the divisors are 1, 2, 3, 2*2, 2*3, 2*2*3, and for 18 = 2*3*3 the divisors are 1, 2, 3, 2*3, 3*3, 2*3*3, and because the prime signatures differ both in the fourth and in the fifth places, a(18) != a(12).

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

Cf. A046523, A101296, A297174, A300716.

Differs from similar A290110 for the first time at n=858.

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; };
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]); };  \\ From A046523
v101296 = rgs_transform(vector(up_to, n, A046523(n)));
A101296(n) = v101296[n];
A297174(n) = { my(s=0,i=-1); fordiv(n, d, if(d>1, i += (A101296(d)-1); s += 2^i)); (s); };
write_to_bfile(1,rgs_transform(vector(up_to,n,A297174(n))),"b300250.txt");

A330683 a(n) is the position of A283980(A025487(n)) in A025487.

Original entry on oeis.org

1, 4, 11, 9, 23, 20, 44, 41, 22, 79, 38, 73, 43, 131, 69, 124, 77, 212, 118, 72, 201, 54, 110, 129, 327, 191, 123, 312, 93, 181, 209, 493, 300, 199, 474, 154, 286, 128, 324, 725, 190, 454, 147, 272, 310, 697, 245, 434, 208, 490, 1044, 299, 671, 114, 232, 416, 469, 1008, 374, 646, 321, 721, 1481, 451, 974, 186, 359

Offset: 1

Antti Karttunen, Dec 26 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12868

Permutation of A329897.

Cf. A025487, A085089, A101296, A181815, A283980, A329898 (positive integers not in this sequence), A329904 (a left inverse), A329906, A330681.

(* First, load the function f at A025487, then: *)
With[{s = Union@ Flatten@ f@ 10}, TakeWhile[#, # != 0 &] &@ Map[If[# > Max@ s, 0, FirstPosition[s, #][[1]] ] &[(Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1])*2^IntegerExponent[#, 2]] &, s]] (* Michael De Vlieger, Jan 11 2020 *)

upto_e = 101;
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)}; \\ From A283980
A330683list(e) = { my(lista = List([1, 2]), i=2, u = 2^e, t, v025487); while(lista[i] != u, if(2*lista[i] <= u, listput(lista,2*lista[i]); t = A283980(lista[i]); if(t <= u, listput(lista,t))); i++); v025487 = vecsort(Vec(lista)); lista = List([]); for(i=1,oo,if(!(t=vecsearch(v025487,A283980(v025487[i]))),return(Vec(lista)), listput(lista,t))); };
v330683 = A330683list(upto_e);
A330683(n) = v330683[n];

a(n) = A085089(A330681(n)) = A101296(A330681(n)) = A101296(A283980(A025487(n))).

For all n >= 1, A329904(a(n)) = n.

cp's OEIS Frontend

A291761 Restricted growth sequence transform of ((-1)^n)*A046523(n); filter combining the parity and the prime signature of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A300827 Lexicographically earliest sequence such that a(i) = a(j) => A324193(i) = A324193(j) for all i, j >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A050330 Number of factorizations of n into numbers with an odd number of prime factors.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A050338 Number of ways of factoring n with 2 levels of parentheses.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A286610 Restricted growth sequence computed for Euler totient function phi, A000010.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A305788 Restricted growth sequence transform of A278233, filter-sequence for GF(2)[X]-factorization.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A189982 Numbers with prime signature (2,1,1,1), i.e., factorization p*q*r*s^2 with distinct primes p, q, r, s.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A286605 Restricted growth sequence computed for number of divisors, d(n) (A000005).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A300250 Restricted growth sequence transform of A297174: a filter sequence recording the prime signatures of divisors of n, with divisors ordered by their magnitude.

A189982 Numbers with prime signature (2,1,1,1), i.e., factorization pqr*s^2 with distinct primes p, q, r, s.