A305983 -id:A305983 - cp's OEIS frontend

A305973 Filter sequence for the prime signature of 2n-1.

1, 2, 2, 2, 3, 2, 2, 4, 2, 2, 4, 2, 3, 5, 2, 2, 4, 4, 2, 4, 2, 2, 6, 2, 3, 4, 2, 4, 4, 2, 2, 6, 4, 2, 4, 2, 2, 6, 4, 2, 7, 2, 4, 4, 2, 4, 4, 4, 2, 6, 2, 2, 8, 2, 2, 4, 2, 4, 6, 4, 3, 4, 5, 2, 4, 2, 4, 9, 2, 2, 4, 4, 4, 6, 2, 2, 6, 4, 2, 4, 4, 2, 8, 2, 3, 6, 2, 6, 4, 2, 2, 4, 4, 4, 9, 2, 2, 8, 2, 2, 4, 4, 4, 6, 4, 2, 4, 4, 4, 4, 4, 2, 10, 2, 2, 8, 2, 4, 4, 2

Offset: 1

Antti Karttunen, Jun 15 2018

nonn

Restricted growth sequence transform of A278223, the least number with the same prime signature as the n-th odd number.

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

Cf. A046523, A101296, A278223, A305901, A305983, A305985.

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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
v305973 = rgs_transform(vector(up_to,n,A046523(n+n-1)));
A305973(n) = v305973[n];

A305985 Filter sequence combining from all divisors d > 1 of n, the prime signature of 2d+1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 5, 6, 7, 8, 2, 9, 10, 8, 4, 11, 5, 12, 5, 13, 8, 14, 2, 15, 16, 17, 8, 18, 2, 19, 20, 18, 4, 21, 8, 22, 20, 21, 17, 23, 2, 24, 5, 25, 26, 8, 5, 27, 28, 29, 8, 30, 2, 19, 8, 31, 21, 14, 5, 32, 5, 33, 26, 34, 17, 35, 36, 18, 4, 37, 5, 32, 20, 14, 26, 38, 21, 39, 5, 40, 41, 42, 2, 43, 44, 8, 14, 45, 2, 46, 47, 18, 44, 48, 8, 49, 50, 51, 12, 52, 5

Offset: 1

Antti Karttunen, Jun 15 2018

nonn

Restricted growth sequence transform of A305984.

For all i, j: a(i) = a(j) => A086668(i) = A086668(j).

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

Cf. A305810, A305973, A305983, A305984.

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; };
A305984(n) = { my(m=1); fordiv(n, d, if((d>1), m *= prime(A305973(1+d)-1))); (m); }; \\ Needs also code from A305973.
v305985 = rgs_transform(vector(up_to,n,A305984(n)));
A305985(n) = v305985[n];

A305982 a(n) = Product_{d|n, dA305793(1+d)-1), where A305973(k) records the prime signature of 2k-1.

Original entry on oeis.org

1, 2, 2, 4, 2, 8, 2, 12, 4, 8, 2, 48, 2, 20, 8, 24, 2, 32, 2, 120, 20, 8, 2, 288, 4, 28, 8, 120, 2, 320, 2, 120, 8, 20, 20, 576, 2, 20, 28, 480, 2, 320, 2, 264, 32, 8, 2, 4320, 10, 200, 20, 168, 2, 320, 8, 1200, 20, 8, 2, 11520, 2, 44, 80, 600, 28, 704, 2, 300, 8, 800, 2, 6912, 2, 44, 80, 300, 20, 448, 2, 31200, 40, 8, 2, 72000, 20, 20, 8

Offset: 1

Antti Karttunen, Jun 15 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A305973, A305983 (rgs-transform), A305984.

A305982(n) = { my(m=1); fordiv(n, d, if((dA305973(1+d)-1))); (m); }; \\ Needs also code from A305973.

a(n) = Product_{d|n, dA008578(A305793(1+d)).

A319343 Filter sequence combining the prime signature of phi(d) from all proper divisors d of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 5, 6, 2, 7, 2, 8, 9, 10, 2, 11, 2, 12, 13, 8, 2, 14, 15, 16, 13, 17, 2, 18, 2, 19, 13, 20, 21, 22, 2, 16, 23, 24, 2, 25, 2, 17, 26, 8, 2, 27, 28, 29, 30, 31, 2, 25, 21, 32, 23, 16, 2, 33, 2, 34, 35, 36, 37, 25, 2, 38, 13, 39, 2, 40, 2, 41, 42, 31, 43, 44, 2, 45, 46, 47, 2, 48, 49, 34, 23, 32, 2, 50, 51, 17, 52, 8, 37, 53, 2, 54, 35

Offset: 1

Antti Karttunen, Sep 17 2018

nonn

Restricted growth sequence transform of A319342.

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

Cf. A318881, A319342, A319345.

Cf. also A305983.

up_to = 65537;
A319342(n) = { my(m=1); fordiv(n, d, if((dA318881(d)))); (m); }; \\ Needs also code from A318881
v319343 = rgs_transform(vector(up_to,n,A319342(n)));
A319343(n) = v319343[n];

cp's OEIS Frontend

A305973 Filter sequence for the prime signature of 2n-1.

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A305985 Filter sequence combining from all divisors d > 1 of n, the prime signature of 2d+1.

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A305982 a(n) = Product_{d|n, dA305793(1+d)-1), where A305973(k) records the prime signature of 2k-1.

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A319343 Filter sequence combining the prime signature of phi(d) from all proper divisors d of n.

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PARI