A319996 -id:A319996 - cp's OEIS frontend

A319717 Filter sequence combining the largest proper divisor of n (A032742) with modulo 6 residue of the smallest prime factor, A010875(A020639(n)), and a single bit A319710(n) telling whether the smallest prime factor is unitary.

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, 95

Offset: 1

Antti Karttunen, Oct 04 2018

nonn

Restricted growth sequence transform of triple [A010875(A020639(n)), A032742(n), A319710(n)] (with a separate value allotted for a(1)), or equally, of ordered pair [A319716(n), A319710(n)].
In addition to A319716, this filter sequence also records in the value of a(n) also the fact whether the smallest prime factor of n is unitary or not. This information is enough to determine the modulo 6 residues of all the divisors of n, thus sequences like A002324 are essentially functions of this sequence. Moreover, a lot of other information is immediately (and unavoidably) present, for example the exact prime signature of n, including also the relative order of exponents.
Any such filtering sequence can be perceived also in terms of what information it leaves out from a(n) that would be needed to reconstruct whole n from each a(n). If the whole n could be reconstructed from a(n) each time, then sequence a would be injective, and would be useless for filtering, because then it would match with any sequence. In this filter, what is left out is only the exact identity of the smallest prime factor, although its residue class mod 6 is retained. However, when the smallest prime factor is 2 or 3, this can be seen from that residue value, so for any number x in A047229, both A020639(x) and A032742(x) are known, and as x = A020639(x)*A032742(x), it means such numbers must occur in their own singleton equivalence classes.
Likewise, for any n in A283050, even if not divisible by 2 or 3, when we have A319710(n) stored in the triple as 1, this immediately gives away the exact identity of the smallest prime factor, which is equal to A014673(n) = A020639(A032742(n)) in these cases.
Thus there is a substantial subset of N (containing at least the union of A047229 and A283050) which is actually in the "blind sector" of this filter, "where anything goes", as this sequence obtains only unique values in that subdomain.
There is a related filter sequence A319996, which operates by "cleaving n from its high end" (by storing the residue class of the largest prime factor, A006530, instead of the smallest, together with n/A006530(n)), which has its own blind spots, but fortunately, they do not fully coincide with the blind spots of this filter. Naturally, any sequence like A002324 should match both to this sequence and A319996.
For all i, j:
  a(i) = a(j) => A002324(i) = A002324(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),
  a(i) = a(j) => A319716(i) = A319716(j) => A319690(i) = A319690(j).

			For n = 65 = 5*13 and 143 = 11*13, the smallest prime factor is of the form 6k+5,  doesn't occur more than once in the factorization, and the largest proper divisor is the same number (13) in both cases, thus a(65) = a(143) (= 51, a running count value allotted by rgs-transform for this equivalence class).
For n = 1805 (5*19^2), 3971 (11*19^2), 6137 (17*19^2), it's like above, but the largest proper divisor is in all three cases 361 = 19^2, thus a(1805) = a(3971) = a(6137) (= 1405).
Note that such nontrivial equivalence classes may only contain numbers that are 5-rough, A007310, with no prime factors 2 or 3, and also, they may not contain numbers from A283050. See the comments section.

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

Cf. A101296, A319707, A319716, A319996.
Cf. also A320004 (analogous sequence for modulo 4 residues).
Differs from A319707 for the first time at n=143, where a(143) = 51, differs from A319716 for the first time at n=121, where a(121) = 95.

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)));
A319710(n) = ((n>1)&&(factor(n)[1,2]>1));
v319717 = rgs_transform(vector(up_to,n,[A286476(n),A319710(n)]));
A319717(n) = v319717[n];

A320117 Filter sequence for counting the residue classes mod 6 of divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 5, 11, 7, 12, 13, 14, 5, 15, 7, 16, 17, 10, 5, 18, 19, 12, 20, 21, 5, 22, 7, 23, 13, 10, 24, 25, 7, 12, 17, 26, 5, 27, 7, 16, 28, 10, 5, 29, 30, 31, 13, 21, 5, 32, 24, 33, 17, 10, 5, 34, 7, 12, 35, 36, 24, 22, 7, 16, 13, 37, 5, 38, 7, 12, 39, 21, 24, 27, 7, 40, 41, 10, 5, 42, 24, 12, 13, 26, 5, 43, 44, 16, 17, 10, 24, 45, 7, 46, 28, 47, 5, 22, 7, 33

Offset: 1

Antti Karttunen, Oct 06 2018

nonn

Restricted growth sequence transform of A320116.
For all i, j:
  A319717(i) = A319717(j) => a(i) = a(j),
  A319996(i) = A319996(j) => a(i) = a(j),
  A320113(i) = A320113(j) => a(i) = a(j),
  a(i) = a(j) => A002324(i) = A002324(j).

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

Cf. A319717, A319996, A320109, A320113, A320115, A320116.

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; };
A320116(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(1+(d%6)))); (m); };
v320117 = rgs_transform(vector(up_to,n,A320116(n)));
A320117(n) = v320117[n];

A319994 Let g = A006530(n), the largest prime factor of n. This filter sequence combines (g mod 4), n/g (A052126), and a single bit A319988(n) telling whether the largest prime factor is unitary.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 3, 7, 8, 9, 3, 10, 5, 6, 11, 12, 5, 13, 3, 14, 15, 6, 3, 16, 17, 9, 18, 10, 5, 19, 3, 20, 15, 9, 21, 22, 5, 6, 11, 23, 5, 24, 3, 10, 25, 6, 3, 26, 27, 28, 11, 14, 5, 29, 21, 16, 15, 9, 3, 30, 5, 6, 31, 32, 33, 24, 3, 14, 15, 34, 3, 35, 5, 9, 36, 10, 37, 19, 3, 38, 39, 9, 3, 40, 33, 6, 11, 16, 5, 41, 42, 10, 15, 6, 21, 43, 5, 44, 31, 45, 5, 19, 3, 23, 46

Offset: 1

Antti Karttunen, Oct 05 2018

nonn

Restricted growth sequence transform of triple [A010873(A006530(n)), A052126(n), A319988(n)], with a separate value allotted for a(1).
Here among the first 100000 terms, only 2331 have a unique value, compared to 69714 in A320004.
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).

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

Cf. A006530, A052126, A319988, A320004.

Cf. also A319996 (modulo 6 analog).

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; };
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A052126(n) = (n/A006530(n));
A319988(n) = ((n>1)&&(factor(n)[omega(n),2]>1));
A319994aux(n) = if(1==n,0,[A006530(n)%4, A052126(n), A319988(n)]);
v319994 = rgs_transform(vector(up_to,n,A319994aux(n)));
A319994(n) = v319994[n];

A322826 Lexicographically earliest such sequence a that a(i) = a(j) => A052126(i) = A052126(j) for all i, j.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 4, 2, 1, 3, 1, 2, 4, 5, 1, 6, 1, 3, 4, 2, 1, 5, 7, 2, 8, 3, 1, 6, 1, 9, 4, 2, 7, 10, 1, 2, 4, 5, 1, 6, 1, 3, 8, 2, 1, 9, 11, 12, 4, 3, 1, 13, 7, 5, 4, 2, 1, 10, 1, 2, 8, 14, 7, 6, 1, 3, 4, 12, 1, 15, 1, 2, 16, 3, 11, 6, 1, 9, 17, 2, 1, 10, 7, 2, 4, 5, 1, 13, 11, 3, 4, 2, 7, 14, 1, 18, 8, 19, 1, 6, 1, 5, 16

Offset: 1

Antti Karttunen, Dec 27 2018

nonn

Restricted growth sequence transform of A052126, or equally, of A322820.
For all i, j:
  A300226(i) = A300226(j) => a(i) = a(j),
  a(i) = a(j) => A322813(i) = A322813(j),
  a(i) = a(j) => A322819(i) = A322819(j).
For all i, j > 1:
  a(i) = a(j) => A001222(i) = A001222(j).

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

Cf. A001222, A006530, A052126, A305800, A300226, A319994, A319996, A322813, A322819, A322820.

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; };
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A052126(n) = (n/A006530(n));
v322826 = rgs_transform(vector(up_to,n,A052126(n)));
A322826(n) = v322826[n];

cp's OEIS Frontend

A319717 Filter sequence combining the largest proper divisor of n (A032742) with modulo 6 residue of the smallest prime factor, A010875(A020639(n)), and a single bit A319710(n) telling whether the smallest prime factor is unitary.

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A320117 Filter sequence for counting the residue classes mod 6 of divisors of n.

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A319994 Let g = A006530(n), the largest prime factor of n. This filter sequence combines (g mod 4), n/g (A052126), and a single bit A319988(n) telling whether the largest prime factor is unitary.

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A322826 Lexicographically earliest such sequence a that a(i) = a(j) => A052126(i) = A052126(j) for all i, j.

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