A304102 -id:A304102 - cp's OEIS frontend

A304103 Restricted growth sequence transform of A304102, a filter sequence related to the proper divisors of n expressed in Fibonacci number system.

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

Offset: 1

Antti Karttunen, May 13 2018

nonn

For all i, j: a(i) = a(j) => b(i) = b(j), where b can be any of {A000005, A293435, A304095 or A300836} for example.

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

Cf. A293435, A304095, A300836, A304102.
Cf. also A300835, A304105, A305800.
Cf. A305793 (analogous filter for base 2).

\\ Needs also code from A304101.
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])); }
A304102(n) = { my(m=1); fordiv(n,d,if(dA304101(d)-1))); (m); };
write_to_bfile(1,rgs_transform(vector(up_to,n,A304102(n))),"b304103.txt");

A304101 Restricted growth sequence transform of A278222(A048679(n)).

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 4, 3, 2, 4, 4, 3, 5, 2, 4, 4, 4, 6, 3, 6, 5, 2, 4, 4, 4, 6, 4, 7, 6, 3, 6, 6, 5, 8, 2, 4, 4, 4, 6, 4, 7, 6, 4, 7, 7, 6, 9, 3, 6, 6, 6, 10, 5, 9, 8, 2, 4, 4, 4, 6, 4, 7, 6, 4, 7, 7, 6, 9, 4, 7, 7, 7, 11, 6, 11, 9, 3, 6, 6, 6, 10, 6, 11, 10, 5, 9, 9, 8, 12, 2, 4, 4, 4, 6, 4, 7, 6, 4, 7, 7, 6, 9, 4, 7, 7, 7

Offset: 0

Antti Karttunen, May 13 2018

Positions of 2's is given by the positive Fibonacci numbers: 1, 2, 3, 5, 8, 13, 21, ..., that is, A000045(n) from n >= 2 onward.
Positions of 3's is given by Lucas numbers larger than 3: 4, 7, 11, 18, ..., that is, A000032(n) from n >= 3 onward.
Sequence allots a distinct value for each distinct multiset formed from the lengths of 1-runs in the binary representation of A048679(n). Compare to the scatter plot of A286622.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A000032, A000045, A003603, A007895, A035513, A048679, A278222, A304100, A304102, A304103, A304104, A304105.

Cf. also A286622 (compare the scatter-plots).

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; };
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A106151(n) = if(n<=1, n, if(n%2, 1+(2*A106151((n-1)/2)), A106151(n>>valuation(n, 2))<<(valuation(n, 2)-1)));
A048679(n) = if(!n,n,A106151(2*A003714(n)));
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
A278222(n) = A046523(A005940(1+n));
v304101 = rgs_transform(vector(1+up_to, n, A278222(A048679(n-1))));
A304101(n) = v304101[1+n];

A304104 a(n) = Product_{d|n, d>1} prime(A304101(d)-1).

Original entry on oeis.org

1, 2, 2, 6, 2, 20, 3, 12, 10, 20, 3, 420, 2, 30, 20, 60, 11, 300, 11, 420, 12, 30, 5, 4200, 22, 20, 130, 990, 3, 11000, 11, 420, 102, 44, 30, 31500, 5, 242, 20, 10920, 11, 3000, 13, 1170, 1100, 190, 3, 231000, 33, 2420, 506, 420, 19, 66300, 12, 9900, 110, 30, 11, 8085000, 13, 242, 300, 5460, 52, 56100, 19, 660, 130, 19500, 13, 9135000, 11, 290, 4180, 2178, 99

Offset: 1

Antti Karttunen, May 13 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4181

Cf. A304101, A304102, A304105 (restricted growth sequence transform of this sequence).

\\ Needs also code from A304101:
A304104(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(A304101(d)-1))); (m); };

a(n) = Product_{d|n, d>1} A000040(A304101(d)-1).
a(n) = (1/2) * A304102(n) * A000040(A304101(n)-1).
Other identities. For all n >= 1:
A001222(a(n)) = A032741(n).
A001511(a(n)) = A005086(n).
A007949(a(n)) = A304096(n).

A304095 a(n) is the number of the proper divisors of n that are Lucas numbers larger than 3 (4, 7, 11, 18, ...).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 2, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 1, 1

Offset: 1

Antti Karttunen, May 13 2018

nonn

a(n) is the number of the proper divisors d of n that are of the form d = A000045(k-1) + A000045(k+1), for k >= 3.

			The proper divisors of 28 are 1, 2, 4, 7 and 14. Of these 4 and 7 are Lucas numbers (A000032) larger than 3, thus a(28) = 2.

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

Cf. A000045, A000204, A093540, A102460, A304091, A304093, A304096, A304102.

Cf. also A300836.

A102460(n) = { my(u1=1,u2=3,old_u1); if(n<=2,sign(n),while(n>u2,old_u1=u1;u1=u2;u2=old_u1+u2);(u2==n)); };
A304095(n) = sumdiv(n,d,(d>3)*(dA102460(d));

a(n) = Sum_{d|n, d>3, dA102460(d).
a(n) = A007949(A304102(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A093540 - 4/3 = 0.629524... . - Amiram Eldar, Jul 05 2025

A305792 a(n) = Product_{d|n, dA286622(d)-1).

Original entry on oeis.org

1, 2, 2, 4, 2, 12, 2, 8, 6, 20, 2, 72, 2, 28, 30, 16, 2, 180, 2, 200, 42, 44, 2, 432, 10, 44, 30, 392, 2, 11700, 2, 32, 66, 20, 70, 5400, 2, 44, 66, 2000, 2, 29988, 2, 968, 1950, 76, 2, 2592, 14, 1100, 30, 968, 2, 20700, 110, 5488, 66, 76, 2, 4563000, 2, 116, 3570, 64, 110, 21780, 2, 200, 114, 53900, 2, 162000, 2, 68, 4290, 968, 154, 82764, 2, 20000

Offset: 1

Antti Karttunen, Jun 11 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A008578, A278222, A286622, A305793 (rgs-transform), A305794.

Cf. also A293214, A304102.

A305792(n) = { my(m=1); fordiv(n,d,if(dA286622(d)-1))); (m); }; \\ Needs also code from A286622.

a(n) = Product_{d|n, dA008578(A286622(d)).

For all k >= 0, a(2^k) = 2^k.

A305812 a(1) = 0; for n > 1, a(n) = Product_{d|n, 1 < d < n} prime(A305788(d)-1).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 6, 2, 6, 1, 60, 1, 4, 6, 42, 1, 100, 1, 198, 4, 4, 1, 4620, 3, 4, 10, 60, 1, 4620, 1, 546, 4, 26, 6, 56100, 1, 4, 4, 26334, 1, 600, 1, 60, 210, 10, 1, 1381380, 2, 132, 26, 60, 1, 18700, 6, 4620, 4, 10, 1, 66625020, 1, 4, 60, 15834, 6, 1000, 1, 2418, 10, 3300, 1, 334187700, 1, 4, 84, 60, 4, 2200, 1, 14036022, 110, 4, 1

Offset: 1

Antti Karttunen, Jun 11 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences operating on GF(2)[X]-polynomials

Cf. A008578, A278233, A305788, A305813 (rgs-transform), A305814.

Cf. also A305792, A304102.

A305812(n) = if(1==n,0, my(m=1); fordiv(n,d,if((d>1)&&(dA305788(d)-1))); (m)); \\ Needs also code from A305788.

a(1) = 0; for n > 1, a(n) = Product_{d|n, dA008578(A305788(d)).

cp's OEIS Frontend

A304103 Restricted growth sequence transform of A304102, a filter sequence related to the proper divisors of n expressed in Fibonacci number system.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A304101 Restricted growth sequence transform of A278222(A048679(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A304104 a(n) = Product_{d|n, d>1} prime(A304101(d)-1).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A304095 a(n) is the number of the proper divisors of n that are Lucas numbers larger than 3 (4, 7, 11, 18, ...).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A305792 a(n) = Product_{d|n, dA286622(d)-1).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A305812 a(1) = 0; for n > 1, a(n) = Product_{d|n, 1 < d < n} prime(A305788(d)-1).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula