A319714 -id:A319714 - cp's OEIS frontend

A319704 Filter sequence which records for primes their residue modulo 4, and for all other numbers assigns a unique number.

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, 59, 60, 3, 61, 62, 63, 3, 64, 65, 66, 67, 68, 5, 69, 70, 71, 72, 73, 74, 75, 5, 76, 77, 78, 5, 79, 3

Offset: 1

Antti Karttunen, Sep 26 2018

nonn

Restricted growth sequence transform of function f defined as f(n) = A010873(n) when n is a prime, otherwise -n.
For all i, j:
  a(i) = a(j) => A010873(i) = A010873(j),
  a(i) = a(j) => A305801(i) = A305801(j),
  a(i) = a(j) => A319714(i) = A319714(j).

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

Cf. A010873, A305801, A319714.
Cf. A002145 (positions of 3's), A002144 (positions of 5's).
Cf. also A319350, A319705, A319706.

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; };
A319704aux(n) = if(isprime(n),-(n%4),n);
v319704 = rgs_transform(vector(up_to,n,A319704aux(n)));
A319704(n) = v319704[n];

A319984 Fully multiplicative with a(p^e) = prime(p mod 4)^e.

Original entry on oeis.org

1, 3, 5, 9, 2, 15, 5, 27, 25, 6, 5, 45, 2, 15, 10, 81, 2, 75, 5, 18, 25, 15, 5, 135, 4, 6, 125, 45, 2, 30, 5, 243, 25, 6, 10, 225, 2, 15, 10, 54, 2, 75, 5, 45, 50, 15, 5, 405, 25, 12, 10, 18, 2, 375, 10, 135, 25, 6, 5, 90, 2, 15, 125, 729, 4, 75, 5, 18, 25, 30, 5, 675, 2, 6, 20, 45, 25, 30, 5, 162, 625, 6, 5, 225, 4, 15, 10, 135, 2, 150, 10, 45, 25, 15, 10

Offset: 1

Antti Karttunen, Oct 06 2018

For all i, j:

A319714(i) = A319714(j) => a(i) = a(j) => A065338(i) = A065338(j).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. also A065338, A319714, A319985, A319986, A319987, A320114, A320115.

A319984(n) = { my(f=factor(n)); prod(i=1, #f~, (prime(f[i, 1]%4))^f[i, 2]); };

For all n, A003963(a(n)) = A065338(n).

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

Antti Karttunen, Oct 04 2018

nonn

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).

			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.

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

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.

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];

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.

Original entry on oeis.org

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

Antti Karttunen, Oct 04 2018

nonn

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).

			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).

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

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.

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];

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A319704 Filter sequence which records for primes their residue modulo 4, and for all other numbers assigns a unique number.

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A319984 Fully multiplicative with a(p^e) = prime(p mod 4)^e.

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

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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.

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