A300834 -id:A300834 - cp's OEIS frontend

A300835 Restricted growth sequence transform of A300834, product_{d|n, dA019565(A003714(d)); Filter sequence related to Zeckendorf-representations of proper divisors of n.

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, 7, 35, 36, 37, 2, 38, 39, 40, 41, 42, 2, 43, 2, 44, 45, 46, 47, 48, 2, 49, 50, 51, 2, 52, 2, 53, 54, 55, 56, 57, 2, 58, 59, 60, 2, 61, 41, 62, 63, 64, 2, 65, 66, 67, 68, 69

Offset: 1

Antti Karttunen, Mar 18 2018

nonn

For all i, j: a(i) = a(j) => A001065(i) = A001065(j).

For all i, j: a(i) = a(j) => A300836(i) = A300836(j).

			For cases n=10 and 49, we see that 10 has proper divisors 1, 2 and 5 and these have Zeckendorf-representations (A014417) 1, 10 and 1000, while 49 has proper divisors 1 and 7 and these have Zeckendorf-representations 1 and 1010. When these Zeckendorf-representations are summed (columnwise without carries), result in both cases is 1011, thus a(10) = a(49).

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

Cf. A003714, A014417, A019565, A300834, A300836.

Cf. also A293215, A293217, A293223, A293224, A293232, A300833 for similar filtering sequences.

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])); }
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); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A300834(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A003714(d)))); m; };
write_to_bfile(1,rgs_transform(vector(up_to,n,A300834(n))),"b300835.txt");

A293214 a(n) = Product_{d|n, dA019565(d).

Original entry on oeis.org

1, 2, 2, 6, 2, 36, 2, 30, 12, 60, 2, 2700, 2, 180, 120, 210, 2, 7560, 2, 6300, 360, 252, 2, 661500, 20, 420, 168, 94500, 2, 23814000, 2, 2310, 504, 132, 600, 43659000, 2, 396, 840, 2425500, 2, 187110000, 2, 207900, 352800, 1980, 2, 560290500, 60, 194040, 264, 485100, 2, 115259760, 840, 254677500, 792, 4620, 2, 264737261250000, 2, 13860

Offset: 1

Antti Karttunen, Oct 03 2017

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences related to binary expansion of n

Cf. A001065, A002110, A019565, A048675, A091954, A292257, A293215 (restricted growth sequence transform).

Cf. also A293216, A293221, A293222, A293225, A293231, A293442, A300830, A300831, A300832, A300833, A300834.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A293214(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(d))); m; };

a(n) = Product_{d|n, dA019565(d).
a(n) = A300830(n) * A300831(n) * A300832(n). - Antti Karttunen, Mar 16 2018
Other identities.
For n >= 0, a(2^n) = A002110(n).
For n >= 1:
A048675(a(n)) = A001065(n).
A001222(a(n)) = A292257(n).
A007814(a(n)) = A091954(n).
A087207(a(n)) = A218403(n).
A248663(a(n)) = A227320(n).

A300836 a(n) is the total number of terms (1-digits) in Zeckendorf representation of all proper divisors of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 7, 1, 4, 3, 5, 1, 7, 1, 7, 4, 4, 1, 11, 2, 3, 4, 8, 1, 10, 1, 7, 4, 5, 4, 14, 1, 5, 3, 11, 1, 10, 1, 8, 7, 4, 1, 15, 3, 8, 5, 7, 1, 12, 4, 12, 5, 4, 1, 21, 1, 5, 7, 10, 3, 13, 1, 8, 4, 11, 1, 19, 1, 4, 8, 10, 5, 10, 1, 16, 7, 5, 1, 20, 5, 5, 4, 12, 1, 20, 4, 10, 5, 4, 5, 21, 1, 9, 10, 16, 1, 13, 1, 11, 10

Offset: 1

Antti Karttunen, Mar 18 2018

nonn

			For n=12, its proper divisors are 1, 2, 3, 4 and 6. Zeckendorf-representations (A014417) of these numbers are 1, 10, 100, 101 and 1001. Total number of 1's present is 7, thus a(12) = 7.

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

Cf. A000045, A007895, A014417, A072649, A300834, A300837.

Cf. also A292257, A293435.

A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A007895(n) = { my(s=0); while(n>0, s++; n -= fibonacci(1+A072649(n))); (s); }
A300836(n) = sumdiv(n,d,(dA007895(d));

a(n) = Sum_{d|n, dA007895(d).
a(n) = A300837(n) - A007895(n).
a(n) = A001222(A300834(n)).
For all n >=1, a(n) >= A293435(n).

A318834 a(n) = Product_{d|n, dA019565(phi(d)), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 2, 2, 4, 2, 12, 2, 12, 6, 20, 2, 108, 2, 60, 30, 60, 2, 540, 2, 300, 90, 84, 2, 2700, 10, 140, 90, 2700, 2, 6300, 2, 420, 126, 44, 150, 121500, 2, 132, 210, 10500, 2, 283500, 2, 5292, 3150, 660, 2, 132300, 30, 5500, 66, 14700, 2, 267300, 210, 472500, 198, 1540, 2, 4630500, 2, 4620, 47250, 4620, 350, 873180, 2, 1452, 990

Offset: 1

Antti Karttunen, Sep 04 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A000010, A019565, A318835 (rgs-transform).

Cf. also A293214, A293231, A300834.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A318834(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(eulerphi(d)))); m; };

a(n) = Product_{d|n, dA019565(A000010(d)).

A048675(a(n)) = A051953(n).

A318469 Multiplicative with a(p^e) = A019565(A003714(e)).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 10, 2, 6, 2, 6, 4, 4, 2, 10, 3, 4, 5, 6, 2, 8, 2, 7, 4, 4, 4, 9, 2, 4, 4, 10, 2, 8, 2, 6, 6, 4, 2, 20, 3, 6, 4, 6, 2, 10, 4, 10, 4, 4, 2, 12, 2, 4, 6, 14, 4, 8, 2, 6, 4, 8, 2, 15, 2, 4, 6, 6, 4, 8, 2, 20, 10, 4, 2, 12, 4, 4, 4, 10, 2, 12, 4, 6, 4, 4, 4, 14, 2, 6, 6, 9, 2, 8, 2, 10, 8

Offset: 1

Antti Karttunen, Aug 30 2018

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

Cf. A003714, A019565, A072649, A318464, A318465.

Cf. also A293442, A293443, A300834, A318470.

A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A318469(n) = factorback(apply(e -> A019565(A003714(e)),factor(n)[,2]));

For all n >= 1, A001222(a(n)) = A318464(n).

A319708 a(n) = Product_{d|n, dA276086(d).

Original entry on oeis.org

1, 2, 2, 6, 2, 36, 2, 54, 12, 108, 2, 1620, 2, 60, 216, 810, 2, 5400, 2, 43740, 120, 540, 2, 607500, 36, 300, 360, 40500, 2, 21870000, 2, 182250, 1080, 2700, 360, 151875000, 2, 1500, 600, 246037500, 2, 101250000, 2, 5467500, 972000, 13500, 2, 85429687500, 20, 6075000, 5400, 5062500, 2, 2531250000, 3240, 3417187500, 3000, 67500, 2

Offset: 1

Antti Karttunen, Oct 03 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2310
Index entries for sequences related to primorial base

Cf. A276085, A276086, A319709 (rgs-transform).

Cf. A293214, A293221, A293222, A300834 for similar constructions for other bases.

A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A319708(n) = { my(m=1); fordiv(n, d, if(dA276086(d))); (m); };

a(n) = Product_{d|n, dA276086(d).
For all n >= 1:
A276085(a(n)) = A001065(n).
A001222(a(n)) = A319713(n).

cp's OEIS Frontend

A300835 Restricted growth sequence transform of A300834, product_{d|n, dA019565(A003714(d)); Filter sequence related to Zeckendorf-representations of proper divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A293214 a(n) = Product_{d|n, dA019565(d).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A300836 a(n) is the total number of terms (1-digits) in Zeckendorf representation of all proper divisors of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A318834 a(n) = Product_{d|n, dA019565(phi(d)), where phi is the Euler totient function A000010.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A318469 Multiplicative with a(p^e) = A019565(A003714(e)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A319708 a(n) = Product_{d|n, dA276086(d).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula