cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A320010 Filter sequence combined from those proper divisors of n that are multiples of 3; Restricted growth sequence transform of A319990.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Oct 03 2018

Keywords

Comments

For all i, j: a(i) = a(j) => A320003(i) = A320003(j).

Links

Crossrefs

Cf. A319990, A320011, A320012, A320014.

Programs

  • PARI
    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; };
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A319990(n) = { my(m=1); fordiv(n,d,if((dA019565(d))); m; };
v320010 = rgs_transform(vector(up_to,n,A319990(n)));
A320010(n) = v320010[n];

A319991 a(n) = Product_{d|n, dA019565(d)^[1 == d mod 3].

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 10, 2, 2, 2, 10, 2, 60, 2, 10, 2, 2, 2, 210, 60, 2, 2, 10, 2, 140, 2, 300, 2, 42, 2, 110, 2, 2, 60, 10, 2, 132, 140, 210, 2, 60, 2, 1650, 2, 2, 2, 110, 60, 6468, 2, 700, 2, 2, 2, 115500, 132, 2, 2, 210, 2, 4620, 60, 110, 140, 330, 2, 390, 2, 1260, 2, 10, 2, 260, 308, 660, 60, 140, 2, 210210, 2, 2, 2, 115500, 2, 1092, 2
Offset: 1

Views

Author

Antti Karttunen, Oct 03 2018

Keywords

Links

Crossrefs

Cf. A019565, A293214, A293895, A293897, A319990, A319992, A320001, A320011 (rgs-transform).
Cf. also A293221.

Programs

  • PARI
    A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A319991(n) = { my(m=1); fordiv(n,d,if((dA019565(d))); m; };

Formula

a(n) = Product_{d|n, dA019565(d)^[1 == d mod 3].
a(n) = A293214(n) / (A319990(n)*A319992(n)).
For all n >= 1:
A007814(a(n)) = A320001(n).
A048675(a(n)) = A293897(n).
A195017(a(n)) = A293895(n) mod 3.

A319992 a(n) = Product_{d|n, dA019565(d)^[2 == d mod 3].

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 3, 1, 30, 1, 3, 1, 3, 10, 21, 1, 3, 1, 30, 1, 126, 1, 21, 10, 3, 1, 315, 1, 30, 1, 21, 42, 66, 10, 3, 1, 3, 1, 11550, 1, 315, 1, 126, 10, 990, 1, 21, 1, 30, 22, 693, 1, 3, 420, 2205, 1, 2310, 1, 1650, 1, 3, 1, 273, 10, 126, 1, 66, 330, 245700, 1, 21, 1, 3, 10, 585, 42, 693, 1, 11550, 1, 546, 1, 315, 220, 3, 770
Offset: 1

Views

Author

Antti Karttunen, Oct 03 2018

Keywords

Links

Crossrefs

Cf. A019565, A293214, A293896, A293898, A319990, A319991, A320005, A320012 (rgs-transform).
Cf. also A293222.

Programs

  • PARI
    A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A319992(n) = { my(m=1); fordiv(n,d,if((dA019565(d))); m; };

Formula

a(n) = Product_{d|n, dA019565(d)^[2 == d mod 3].
a(n) = A293214(n) / (A319990(n)*A319991(n)).
For all n >= 1:
A007814(a(n)) = A320005(n).
A048675(a(n)) = A293898(n).
A195017(a(n)) = -A293896(n) mod 3.

A320003 Number of proper divisors of n of the form 6*k + 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 3, 0, 0, 2, 0, 0, 1, 0, 0, 4, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 2, 0, 0, 3
Offset: 1

Views

Author

Antti Karttunen, Oct 03 2018

Keywords

Comments

Number of divisors of n that are odd multiples of 3 and less than n.

Examples

			For n = 18, of its five proper divisors [1, 2, 3, 6, 9] only 3 and 9 are odd multiples of three, thus a(18) = 2.
For n = 108, the odd part is 27 for which 27/3 has 3 divisors. As 108 is even, we don't subtract 1 from that 3 to get a(108) = 3. - _David A. Corneth_, Oct 03 2018

Links

Crossrefs

Cf. A319990, A320001, A320005.
Cf. A001620, A016629, A020759 (psi(1/2)).

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 1 &, # < n && Mod[#, 6] == 3 &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)
  • PARI
    A320003(n) = if(!n,n,sumdiv(n, d, (d
  • PARI
    a(n) = if(n%3==0, my(v=valuation(n, 2)); n>>=v; numdiv(n/3)-(!v), 0) \\ David A. Corneth, Oct 03 2018

Formula

a(n) = Sum_{d|n, dA000035(d))*A079978(d).
a(n) = A007814(A319990(n)).
a(4*n) = a(2*n). - David A. Corneth, Oct 03 2018
G.f.: Sum_{k>=1} x^(12*k-6) / (1 - x^(6*k-3)). - Ilya Gutkovskiy, Apr 14 2021
Sum_{k=1..n} a(k) = n*log(n)/6 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,6) - (2 - gamma)/6 = -0.208505..., gamma(3,6) = -(psi(1/2) + log(6))/6 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A320014 Filter sequence combining the binary expansions of proper divisors of n, grouped by their residue classes mod 3.

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, 40, 41, 42, 43, 2, 44, 2, 45, 46, 47, 48, 49, 2, 50, 51, 52, 2, 53, 2, 54, 55, 56, 57, 58, 2, 59, 60, 61, 2, 62, 63, 64, 65, 66, 2, 67, 68, 69, 70, 71, 72, 73, 2, 74, 75
Offset: 1

Views

Author

Antti Karttunen, Oct 03 2018

Keywords

Comments

Restricted growth sequence transform of triple [A319990(n), A319991(n), A319992(n)], or equally, of ordered pair [A320010(n), A320013(n)].
Apart from trivial cases of primes, all other duplicates in range 1 .. 65537 seem to be squarefree semiprimes of the form 3k+1, i.e., both prime factors are either of the form 3k+1 or of the form 3k+2. Question: Is there any reason that more complicated cases would not occur later?
For all i, j: a(i) = a(j) => A293215(i) = A293215(j).
Differs from A319693 first for n = 108. - Georg Fischer, Oct 16 2018

Examples

			The first set of numbers that forms a nontrivial equivalence class is [295, 583, 799, 943] = [5*59, 11*53, 17*47, 23*41]. The prime factors in these are all of the form 3k+2, and when the binary expansions of the factors (like e.g., "101" for 5 and "111011" for 59 or "10111" for 23 and "101001" for 41) are overlaid, the resulting bit vector is always [1, 1, 1, 1, 1, 1^2], with the least significant bit-position containing 2 copies of 1's. Thus we have a(295) = a(583) = a(799) = a(943).

Links

Crossrefs

Cf. A019565, A319990, A319991, A319992, A320010, A320011, A320012, A320013.
Cf. also A293214, A293215, A293226, A300833.
Differs from A305800 for the first time at n=583, where a(583) = 234, while A305800(478).

Programs

  • PARI
    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; };
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A319990(n) = { my(m=1); fordiv(n,d,if((dA019565(d))); m; };
A319991(n) = { my(m=1); fordiv(n,d,if((dA019565(d))); m; };
A319992(n) = { my(m=1); fordiv(n,d,if((dA019565(d))); m; };
v320014 = rgs_transform(vector(up_to,n,[A319990(n),A319991(n),A319992(n)]));
A320014(n) = v320014[n];
Showing 1-5 of 5 results.

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