A305818 -id:A305818

Search: a305818 -id:a305818

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A305983

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

+10
5

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

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

OFFSET

1,2

COMMENTS

Restricted growth sequence transform of A305982.

For all i, j: a(i) = a(j) => A305818(i) = A305818(j).

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; };

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

v305983 = rgs_transform(vector(up_to, n, A305982(n)));

A305983(n) = v305983[n];

CROSSREFS

Cf. A305973, A305982, A305985.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 15 2018

STATUS

approved

A086668

Number of divisors d of n such that 2d+1 is a prime.

+10
4

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

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

OFFSET

1,2

COMMENTS

From Antti Karttunen, Jun 15 2018: (Start)

Number of terms of A005097 that divide n.

For all n >= 1, a(n) > A156660(n). Specifically, a(p) = 2 for all p in A005384 (Sophie Germain primes), although 2's occur in other positions as well.

(End)

LINKS

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

FORMULA

From Antti Karttunen, Jun 15 2018: (Start)

a(n) = Sum_{d|n} A101264(d).

a(n) = A305818(n) + A101264(n).

(End)

EXAMPLE

10 has divisors 1,2,5 and 10 of which 2.1+1, 2.2+1 and 2.5+1 are prime, so a(10)=3

MATHEMATICA

Table[Count[Divisors[n], _?(PrimeQ[2#+1]&)], {n, 100}] (* Harvey P. Dale, Apr 29 2015 *)

PROG

(PARI) for (n=2, 100, s=0; fordiv(i=n, i, s+=isprime(2*i+1)); print1(", "s))

(PARI) A086668(n) = sumdiv(n, d, isprime(d+d+1)); \\ Antti Karttunen, Jun 15 2018

CROSSREFS

One less than A046886.

Cf. A005097, A005384, A101264, A305818.

KEYWORD

nonn

AUTHOR

Jon Perry, Jul 27 2003

EXTENSIONS

Definition modified by Harvey P. Dale, Apr 29 2015

STATUS

approved

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