A319716 Filter sequence combining the largest proper divisor of n (A032742) with modulo 6 residue of the smallest prime factor, A010875(A020639(n)).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 5, 11, 7, 12, 13, 14, 5, 15, 7, 16, 17, 18, 5, 19, 20, 21, 22, 23, 5, 24, 7, 25, 26, 27, 28, 29, 7, 30, 31, 32, 5, 33, 7, 34, 35, 36, 5, 37, 38, 39, 40, 41, 5, 42, 43, 44, 45, 46, 5, 47, 7, 48, 49, 50, 51, 52, 7, 53, 54, 55, 5, 56, 7, 57, 58, 59, 60, 61, 7, 62, 63, 64, 5, 65, 66, 67, 68, 69, 5, 70, 71, 72, 73, 74, 75, 76, 7, 77, 78, 79, 5, 80, 7, 81, 82, 83, 5, 84, 7, 85, 86, 87, 5, 88, 89, 90, 91, 92, 93, 94, 43

Offset

1,2

Comments

Restricted growth sequence transform of A286475, or equally, of A286476.

In each a(n) there is enough information to determine the modulo 6 residues of all the prime factors of n (when counted with multiplicity), thus sequences like A319690 and A319691 (which is the characteristic function of A004611) are essentially functions of this sequence. However, to determine that for all divisors of n, more information is needed. See A319717.

For all i, j:

A319707(i) = A319707(j) => A319717(i) = A319717(j) => a(i) = a(j),

a(i) = a(j) => A319690(i) = A319690(i) => A319691(i) = A319691(j).

Links

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

Example

For n = 55 = 5*11 and 121 = 11*11, 55 = 121 = 1 mod 6 and 11 is their common largest proper divisor, thus they are allotted the same number by the restricted growth sequence transform, that is a(55) = a(121) = 43 (which is the number allotted). Note that such nontrivial equivalence classes may only contain numbers that are 5-rough, A007310, with no prime factors 2 or 3.

Prog

(PARI)
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; };
A032742(n) = if(1==n, n, n/vecmin(factor(n)[, 1]));
A286476(n) = if(1==n, n, (6*A032742(n) + (n % 6)));
v319716 = rgs_transform(vector(up_to, n, A286476(n)));
A319716(n) = v319716[n];

Crossrefs

Cf. A010875, A032742, A286475, A286476, A319707, A319717.

Cf. A007528 (positions of 5's), A002476 (positions of 7's).

Cf. also A319714.

Differs from A319707 and A319717 for the first time at n=121.

Sequence in context: A373593 A320117 A319996 * A319707 A319717 A292266

Adjacent sequences: A319713 A319714 A319715 * A319717 A319718 A319719

Keyword

nonn

Author

Antti Karttunen, Oct 04 2018

Status

approved