A101296 -id:A101296 - cp's OEIS frontend

A373250 Lexicographically earliest infinite sequence such that a(i) = a(j) => A181819(i) = A181819(j) and i mod A181819(i) = j mod A181819(j), for all i, j >= 1, where A181819 is the prime shadow of n.

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

Offset: 1

Antti Karttunen, May 30 2024

nonn

Restricted growth sequence transform of the ordered pair [A181819(n), A373247(n)].
For all i, j:
  A373251(i) = A373251(j) => a(i) = a(j),
  a(i) = a(j) => A101296(i) = A101296(j),
  a(i) = a(j) => A373246(i) = A373246(j),
  a(i) = a(j) => A373249(i) = A373249(j),
  a(i) = a(j) => A353566(i) = A353566(j).

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

Cf. A101296, A181819, A353566, A373246, A373247, A373249, A373251.

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; };
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
Aux373250(n) = [A181819(n), n%A181819(n)];
v373250 = rgs_transform(vector(up_to, n, Aux373250(n)));
A373250(n) = v373250[n];

A378601 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A083206(i) = A083206(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, 9, 2, 7, 6, 6, 2, 10, 3, 6, 5, 7, 2, 11, 2, 12, 6, 6, 6, 13, 2, 6, 6, 14, 2, 15, 2, 9, 9, 6, 2, 16, 3, 9, 6, 9, 2, 14, 6, 17, 6, 6, 2, 18, 2, 6, 9, 19, 6, 15, 2, 9, 6, 20, 2, 21, 2, 6, 9, 9, 6, 15, 2, 22, 8, 6, 2, 23, 6, 6, 6, 17, 2, 24, 6, 9, 6, 6, 6, 25, 2, 9, 9, 13, 2, 15, 2, 17, 26

Offset: 1

Antti Karttunen, Dec 01 2024

nonn

Restricted growth sequence transform of the ordered pair [A046523(n), A083206(n)].

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

Cf. A046523, A083206, A101296.

Cf. also A378602, A378602, A378603, A378604.

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]); };
v083206 = readvec("b083206_to.txt"); \\ Precomputed with A083206(n) = { my(p=1); fordiv(n, d, p *= ('x^d + 'x^-d)); (polcoeff(p, 0)/2); };
A083206(n) = v083206[n];
Aux378601(n) = [A046523(n), A083206(n)];
v378601 = rgs_transform(vector(up_to, n, Aux378601(n)));
A378601(n) = v378601[n];

A378602 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A119347(i) = A119347(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, 9, 2, 10, 6, 6, 2, 11, 3, 6, 5, 12, 2, 13, 2, 14, 6, 6, 6, 15, 2, 6, 6, 16, 2, 17, 2, 18, 19, 6, 2, 20, 3, 18, 6, 18, 2, 21, 6, 21, 6, 6, 2, 22, 2, 6, 23, 24, 6, 25, 2, 18, 6, 26, 2, 27, 2, 6, 18, 18, 6, 28, 2, 29, 8, 6, 2, 30, 6, 6, 6, 31, 2, 32, 6, 18, 6, 6, 6, 33, 2, 18, 23, 34, 2, 28, 2, 35, 36

Offset: 1

Antti Karttunen, Dec 01 2024

nonn

Restricted growth sequence transform of the ordered pair [A046523(n), A119347(n)].

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

Cf. A046523, A119347, A101296.

Cf. also A378601, A378603.

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]); };
A119347(n) = { my(c=[0]); fordiv(n,d, c = Set(concat(c,vector(#c,i,c[i]+d)))); (#c)-1; };
Aux378602(n) = [A046523(n), A119347(n)];
v378602 = rgs_transform(vector(up_to, n, Aux378602(n)));
A378602(n) = v378602[n];

A050332 Number of factorizations of n into distinct numbers with an odd number of prime factors.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 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. A026424, A045778, A050333. a(p^k)=A000700. a(A002110)=A003724.

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

a(n) = A050333(A101296(n)). - R. J. Mathar, May 26 2017

A050347 Number of ways to factor n into distinct factors with 2 levels of parentheses.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 4, 1, 4, 1, 10, 1, 4, 4, 7, 1, 10, 1, 10, 4, 4, 1, 26, 1, 4, 4, 10, 1, 22, 1, 14, 4, 4, 4, 34, 1, 4, 4, 26, 1, 22, 1, 10, 10, 4, 1, 63, 1, 10, 4, 10, 1, 26, 4, 26, 4, 4, 1, 74, 1, 4, 10, 29, 4, 22, 1, 10, 4, 22, 1, 105, 1, 4, 10, 10, 4, 22, 1, 63, 7, 4, 1, 74, 4, 4, 4, 26

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).

			6 = ((6)) = ((3*2)) = ((3)*(2)) = ((3))*((2)).

R. J. Mathar, Table of n, a(n) for n = 1..2159

Cf. A045778, A050345-A050350. a(p^k)=A050343. a(A002110)=A000307.

Dirichlet g.f.: Product_{n>=2}(1+1/n^s)^A050345(n).

a(n) = A050348(A101296(n)). - R. J. Mathar, May 26 2017

A224401 a(n) is the row number of triangle A085612 in which n appears.

Original entry on oeis.org

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

Offset: 1

Matthew Goers, Apr 05 2013

nonn

Row number in A085612 triangle of prime signatures.

a(n) is not the same for all numbers n with the same prime signature. For such a sequence, see A101296. - Peter Munn, Oct 23 2023

			a(9) = 3, because 9 is in the 3rd row (1st 3 prime^2) of A085612.
a(10) = 5, because 10 is in the 5th row (1st 5 semiprimes) of A085612.
a(11) = 4, because 11 is in the 4th row (4 primes, prime(3)..prime(6)) of A085612.

Rémy Sigrist, Table of n, a(n) for n = 1..10000 (first 150 terms from Matthew Goers)
Rémy Sigrist, PARI program for A224401

Cf. A085612, A101296.

First and last positions of each number: A085834, A085836.

import Data.List (findIndex); import Data.Maybe (fromJust)
a224401 = (+ 1) . fromJust . (`findIndex` a085612_tabf) . elem
-- Reinhard Zumkeller, Jun 05 2013

PARI
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A291762 Restricted growth sequence transform of ((-1)^A000120(n))*A046523(n); filter combining the parity of binary weight with the prime signature of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 11 2017

nonn

Equally, restricted growth sequence transform of sequence b defined as b(1) = 1; b(n) = A046523(n) + A010060(n) for n > 1, which starts as 1, 3, 2, 5, 2, 6, 3, 9, 4, 6, 3, 12, 3, 7, 6, 17, 2, 12, 3, 12, 7, 7, ...

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

Cf. A000120, A010060, A046523.
Cf. A101296, A286163, A291761 (related or similar filtering sequences).
Cf. A027697 (positions of 2's), A027699 (of 3's), A130593 (of 5's and 7's), A230095 (of 9's).
Cf. also A231431, A235001.

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)^hammingweight(n))*A046523(n))),"b291762_upto65537.txt");
\\ Or alternatively:
A010060(n) = (hammingweight(n)%2);
f(n) = if(1==n,n,A046523(n)+A010060(n));
write_to_bfile(1,rgs_transform(vector(16385,n,f(n))),"b291762.txt");

A292587 Compound filter: a(n) = P(A001221(n), A292582(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 5, 1, 3, 3, 7, 1, 5, 1, 5, 3, 3, 1, 8, 2, 3, 4, 5, 1, 6, 1, 11, 3, 3, 3, 23, 1, 3, 3, 8, 1, 6, 1, 5, 5, 3, 1, 12, 2, 5, 3, 5, 1, 8, 3, 8, 3, 3, 1, 9, 1, 3, 5, 22, 3, 6, 1, 5, 3, 6, 1, 38, 1, 3, 5, 5, 3, 6, 1, 12, 7, 3, 1, 9, 3, 3, 3, 8, 1, 9, 3, 5, 3, 3, 3, 17, 1, 5, 5, 23, 1, 6, 1, 8, 6

Offset: 1

Antti Karttunen, Sep 26 2017

This is essentially also a filter constructed from the runlengths of numbers of the form 4k+0 and the runlengths of numbers of the form 4k+2 encountered in trajectories of A005940-tree. See comments in A083399 and A292586.
For all i, j: A291757(i) = A291757(j) => a(i) = a(j), that is, this filter matches to a subset of the sequences matched by filter A291757.
Moreover, for all i, j: a(i) = a(j) <=> A101296(i) = A101296(j), thus the subset is exactly the sequences matched by A101296 (A046523). This follows because the prime signature of n can be recovered from the two components as A046523(n) = A046523(A003557(n)) * A292586(n) and also vice versa as A046523(A003557(n)) = A003557(A046523(n)).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A000027, A001221, A003557, A005940, A046523, A101296, A291757, A292582, A292584, A292586, A292588.

a(n) = (1/2)*(2 + ((A001221(n) + A292582(n))^2) - A001221(n) - 3*A292582(n)).

A319337 Filter sequence combining gcd(n,tau(n)) (= A009191) with the prime signature of n (A046523).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 24 2018

nonn

Restricted growth sequence transform of ordered pair [A009191(n), A046523(n)].

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

Cf. A009191, A046523, A101296, A300230, A300240, A305801, A319338.

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; };
A009191(n) = gcd(n, numdiv(n));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p=0); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
v319337 = rgs_transform(vector(up_to,n,[A009191(n),A046523(n)]));
A319337(n) = v319337[n];

A319357 Filter sequence combining A003415(d) from all proper divisors d of n, where A003415(d) = arithmetic derivative of d.

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

Offset: 1

Antti Karttunen, Oct 02 2018

nonn

Restricted growth sequence transform of A319356.
The only duplicates in range 1..65537 with a(n) > 4 are the following six pairs: a(1445) = a(2783), a(4205) = a(11849), a(5819) = a(8381), a(6727) = a(15523), a(8405) = a(31211) and a(28577) = a(44573). All these have prime signature p^2 * q^1. If all the other duplicates respect the prime signature as well, 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) => A319683(i) = A319683(j),
  a(i) = a(j) => A319686(i) = A319686(j),
  a(i) = a(j) => A101296(i) = A101296(j). [Conjectural, see notes above]

			Proper divisors of 1445 are [1, 5, 17, 85, 289], while the proper divisors of 2783 are [1, 11, 23, 121, 253]. 1 contributes 0 and primes contribute 1, so only the last two matter in each set. We have A003415(85) = 22 = A003415(121) and A003415(289) = 34 = A003415(253), thus the value of arithmetic derivative coincides for all proper divisors, thus a(1445) = a(2783).

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

Cf. A003415, A319356.
Cf. A000041 (positions of 2's), A001248 (positions of 3's), A006881 (positions of 4's),
Cf. also A300245, A300249, A319353, A319693.

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
A319356(n) = { my(m=1); fordiv(n, d, if(dA003415(d)))); (m); };
v319357 = rgs_transform(vector(up_to,n,A319356(n)));
A319357(n) = v319357[n];

cp's OEIS Frontend

A373250 Lexicographically earliest infinite sequence such that a(i) = a(j) => A181819(i) = A181819(j) and i mod A181819(i) = j mod A181819(j), for all i, j >= 1, where A181819 is the prime shadow of n.

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A378601 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A083206(i) = A083206(j), for all i, j >= 1.

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A378602 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A119347(i) = A119347(j), for all i, j >= 1.

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A050332 Number of factorizations of n into distinct numbers with an odd number of prime factors.

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A050347 Number of ways to factor n into distinct factors with 2 levels of parentheses.

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A224401 a(n) is the row number of triangle A085612 in which n appears.

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Haskell

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A291762 Restricted growth sequence transform of ((-1)^A000120(n))*A046523(n); filter combining the parity of binary weight with the prime signature of n.

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A292587 Compound filter: a(n) = P(A001221(n), A292582(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A319337 Filter sequence combining gcd(n,tau(n)) (= A009191) with the prime signature of n (A046523).

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