A320004 Filter sequence combining the largest proper divisor of n (A032742) with n's residue modulo 4 (A010873), and a single bit (A319710) telling whether the smallest prime factor is unitary.

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

Offset

1,2

Comments

Restricted growth sequence transform of triple [A010873(A020639(n)), A032742(n), A319710(n)], or equally, of ordered pair [A319714(n), A319710(n)].

Here any nontrivial equivalence classes (that is, when we exclude the singleton classes and two infinite classes of A002144 and A002145), like the example shown, may not contain any even numbers, nor any numbers from A283050. See additional comments in A319717 and A319994.

For all i, j:

a(i) = a(j) => A024362(i) = A024362(j),

a(i) = a(j) => A067029(i) = A067029(j),

a(i) = a(j) => A071178(i) = A071178(j),

a(i) = a(j) => A077462(i) = A077462(j) => A101296(i) = A101296(j).

Links

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

Example

For n = 33 (3*11) and n = 77 (7*11), the modulo 4 residue of the smallest prime factor is 3, and the largest proper divisors (A032742) is also equal 11, and the smallest prime factor is unitary. Thus a(33) = a(77) (= 25, a running count value allotted by rgs-transform).

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]));
A286474(n) = if(1==n, n, (4*A032742(n) + (n % 4)));
A319710(n) = ((n>1)&&(factor(n)[1, 2]>1));
v320004 = rgs_transform(vector(up_to, n, [A286474(n), A319710(n)]));
A320004(n) = v320004[n];

Crossrefs

Cf. A319704, A319714, A319994.

Cf. also A319717 (analogous sequence for modulo 6 residues).

Cf. A002145 (positions of 3's), A002144 (positions of 5's).

Differs from A319704 for the first time at n=77, and from A319714 for the first time at n=49.

Sequence in context: A320115 A319994 A319714 * A319704 A331304 A343721

Adjacent sequences: A320001 A320002 A320003 * A320005 A320006 A320007

Keyword

nonn

Author

Antti Karttunen, Oct 04 2018

Status

approved