A319693 Filter sequence combining sopfr(d) from all proper divisors d of n, where sopfr(d) is A001414(d) = sum of primes dividing d with repetition.

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, 76, 2, 77, 2, 78, 79, 80, 2, 73, 2, 81, 82, 83, 2, 84, 85

Offset

1,2

Comments

Restricted growth sequence transform of A319692.

For all i, j: a(i) = a(j) => A305611(i) = A305611(j).

Links

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

Example

The proper divisors of  96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, while
the proper divisors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54.
It happens that sopfr(8) = sopfr(9), sopfr(16) = sopfr(18), sopfr(24) = sopfr(27), sopfr(32) = sopfr(36) and sopfr(48) = sopfr(54), and the rest of proper divisors (1, 2, 3, 4, 6, 12) are shared by both numbers, from which follows that by taking product of sopfr over proper divisors gives an identical result for both, thus a(96) = a(108). Here sopfr = A001414.

Prog

(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; };
A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
A319692(n) = { my(m=1); fordiv(n, d, if(d<n, m *= prime(1+A001414(d)))); (m); };
v319693 = rgs_transform(vector(up_to, n, A319692(n)));
A319693(n) = v319693[n];

Crossrefs

Cf. A001414, A319692.

Cf. also A319353.

Differs from A305800, A296073 and A317943 for the first time at n=108, as here a(108) = 73.

Sequence in context: A300243 A300241 A320014 * A296073 A317943 A305800

Adjacent sequences: A319690 A319691 A319692 * A319694 A319695 A319696

Keyword

nonn

Author

Antti Karttunen, Oct 02 2018

Status

approved