A305901 -id:A305901

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A305900

Filter sequence for a(primes > 3) = constant sequences.

+10
26

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

For all i, j:

a(i) = a(j) => A305801(i) = A305801(j) => A305800(i) = A305800(j).

a(i) = a(j) => A007949(i) = A007949(j).

a(i) = a(j) => A305893(i) = A305893(j).

LINKS

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

FORMULA

For n <= 5, a(n) = n, for >= 5, a(n) = 5 when n is a prime, and a(n) = 3+n-A000720(n) when n is a composite.

PROG

(PARI) A305900(n) = if(n<=5, n, if(isprime(n), 5, 3+n-primepi(n)));

CROSSREFS

Cf. A305800, A305801, A305893.

Cf. also A305901, A305902, A305903 (this filter applied to various permutations of N).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 14 2018

STATUS

approved

A305973

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

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

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

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; };

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];

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 15 2018

STATUS

approved

A305810

Filter sequence for a(Sophie Germain primes > 3) = constant sequences.

+10
5

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 5, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 5, 22, 23, 24, 25, 26, 5, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 5, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 5, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 5, 78, 79, 80, 81, 82, 5, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 5

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Filer sequence for all such sequences S, for which S(A005384(k)) = constant for all k >= 3.

Restricted growth sequence transform of the ordered pair [A305900(n), A305901(1+n)].

For all i, j:

a(i) = a(j) => A305900(i) = A305900(j),

a(i) = a(j) => A305901(1+i) = A305901(1+j),

a(i) = a(j) => A305978(i) = A305978(j),

a(i) = a(j) => A305985(i) = A305985(j).

LINKS

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

FORMULA

If n < 5, a(n) = n; for n >= 5, a(n) = 5 if A156660(n) == 1 [when n is in A005384[3..] = 5, 11, 23, 29, 41, 53, 83, 89, 113, ...], otherwise a(n) = 3+n-A156874(n).

PROG

(PARI)

up_to = 100000;

A156660(n) = (isprime(n)&&isprime(2*n+1)); \\ From A156660

partialsums(f, up_to) = { my(v = vector(up_to), s=0); for(i=1, up_to, s += f(i); v[i] = s); (v); }

v156874 = partialsums(A156660, up_to);

A156874(n) = v156874[n];

A305810(n) = if(n<5, n, if(A156660(n), 5, 3+n-A156874(n)));

CROSSREFS

Cf. A005384, A156660, A156874, A305900, A305901, A305978, A305985.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 16 2018

STATUS

approved

A319706

Filter sequence which for primes p records the prime signature of 2p+1, and for all other numbers assigns a unique number.

+10
4

1, 2, 2, 3, 2, 4, 5, 6, 7, 8, 2, 9, 10, 11, 12, 13, 5, 14, 5, 15, 16, 17, 2, 18, 19, 20, 21, 22, 2, 23, 24, 25, 26, 27, 28, 29, 24, 30, 31, 32, 2, 33, 5, 34, 35, 36, 5, 37, 38, 39, 40, 41, 2, 42, 43, 44, 45, 46, 5, 47, 5, 48, 49, 50, 51, 52, 53, 54, 55, 56, 5, 57, 24, 58, 59, 60, 61, 62, 5, 63, 64, 65, 2, 66, 67, 68, 69, 70, 2, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 5

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Restricted growth sequence transform of function f defined as f(n) = A046523(2n+1) when n is a prime, otherwise -n.

For all i, j:

A305810(i) = A305810(j) => a(i) = a(j),

and

a(i) = a(j) => A305800(i) = A305800(j),

a(i) = a(j) => A305978(i) = A305978(j),

a(i) = a(j) => A305985(i) = A305985(j).

LINKS

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

PROG

(PARI)

up_to = 100000;

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

A319706aux(n) = if(isprime(n), A046523(n+n+1), -n);

v319706 = rgs_transform(vector(up_to, n, A319706aux(n)));

A319706(n) = v319706[n];

CROSSREFS

Cf- A046523, A305800, A305810, A305900, A305901, A305978, A305985.

Cf. A005384 (positions of 2's), A234095 (positions of 5's).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Sep 26 2018

STATUS

approved

A305902

Restricted growth sequence transform of A305900(A048673(n)).

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

For all i, j:

a(i) = a(j) => A278224(i) = A278224(j).

a(i) = a(j) => A286583(i) = A286583(j).

a(i) = a(j) => A292251(i) = A292251(j).

LINKS

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

PROG

(PARI)

up_to = 65537;

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961

A048673(n) = (A003961(n)+1)/2;

A305900(n) = if(n<=5, n, if(isprime(n), 5, 3+n-primepi(n)));

v305902 = rgs_transform(vector(up_to, n, A305900(A048673(n))));

A305902(n) = v305902[n];

CROSSREFS

Cf. A048673, A305900.

Cf. also A305901.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 14 2018

STATUS

approved

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