A001751 -id:A001751 - cp's OEIS frontend

A007922 Smallest k such that k!! is a multiple of n.

1, 2, 3, 4, 5, 6, 7, 4, 9, 10, 11, 6, 13, 14, 5, 6, 17, 12, 19, 10, 7, 22, 23, 6, 15, 26, 9, 14, 29, 10, 31, 8, 11, 34, 7, 12, 37, 38, 13, 10, 41, 14, 43, 22, 9, 46, 47, 6, 21, 20, 17, 26, 53, 18, 11, 14, 19, 58, 59, 10, 61, 62, 9, 8, 13, 22, 67, 34, 23, 14, 71, 12, 73, 74, 15, 38, 11, 26

Offset: 1

R. Muller

nonn

a(n) <= n, with equality if n is 1, 9, a prime or twice a prime. Are those the only cases of equality? - Robert Israel, Apr 22 2024

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
F. Smarandache, Only Problems, Not Solutions!.
Index entries for sequences related to factorial numbers

Cf. A001751, A006882.

V:= Vector(100): count:= 0:
S:= {$1..100}:
for k from 1 while count < 100 do
  v:= doublefactorial(k);
  SP:= select(t -> v mod t = 0, S);
  count:= count + nops(SP);
  V[convert(SP,list)]:= k;
  S:= S minus SP;
od:
convert(V,list); # Robert Israel, Apr 21 2024

df[n_]:=Module[{k=1},While[!Divisible[k!!,n],k++];k]; Array[df,80] (* Harvey P. Dale, Jun 04 2013 *)

A085118 Primes together with twice the odd primes.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 67, 71, 73, 74, 79, 82, 83, 86, 89, 94, 97, 101, 103, 106, 107, 109, 113, 118, 122, 127, 131, 134, 137, 139, 142, 146, 149, 151, 157, 158, 163, 166, 167, 173, 178

Offset: 1

Leroy Quet, Apr 25 2004

Probably the same sequence as: numbers n such that phi(n)+1 divides n.

Cohen and Segal showed that in case there were other solutions to this problem (which appeared to be posed by Schinzel), then they should have at least 15 distinct prime factors. Moreover, there is a connection with the Lehmer's totient problem which asks whether there is a composite n such that phi(n)|(n-1). If no such composite exists, then p and 2p are the only members for Leroy's sequence. - Francisco Salinas (franciscodesalinas(AT)hotmail.com), Apr 25 2004

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
G. L. Cohen and S. L. Segal, A note concerning those n for which phi(n)+1 divides n, Fibonacci Quarterly, Vol. 27, No. 3 (1989), pp. 285-286.
Eric Weisstein's World of Mathematics, Lehmer's Totient Problem

Cf. A000010, A068422.

Equals A001751\{4}.

With[{nn=40},Take[Sort[Join[Prime[Range[2nn]],2Prime[Range[2,nn]]]],2nn]] (* Harvey P. Dale, Oct 03 2013 *)

from sympy import primepi
def A085118(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-primepi(x)-primepi(x>>1)+(x>=4))
    return bisection(f,n,n) # Chai Wah Wu, Oct 17 2024

More terms from David Wasserman, Jan 27 2005

A105661 a(n)=1 if n is a prime, 2 if n is an even semiprime, otherwise 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 0, 0, 2, 1, 0, 1, 2, 0, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 2, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 1, 2, 0, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 1, 2, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 2, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Giovanni Teofilatto, May 04 2005

			a(3) = 1 because n = 3 is a prime;
a(4) = 2 because n = 4 = 2*2 is an even semiprime;
a(8) = 0 because n = 8 is neither prime nor an even semiprime.
a(15) = 0 because n = 15 = 3*5 is an odd semiprime.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001222, A001751, A105700.

Table[Which[PrimeQ[n],1,EvenQ[n]&&PrimeOmega[n]==2,2,True,0],{n,120}] (* Harvey P. Dale, Jul 05 2021 *)

for(n=1,105,if((n%2==0)&&(bigomega(n)==2),r=2,r=isprime(n));print1(r,","))

(define (A105661 n) (cond ((= 1 (A001222 n)) 1) ((and (even? n) (= 2 (A001222 n))) 2) (else 0))) ;; Antti Karttunen, Jul 26 2017

Corrected and extended by Rick L. Shepherd, May 17 2005

A251561 A permutation of the natural numbers: interchange p and 2p for every prime p.

Original entry on oeis.org

1, 4, 6, 2, 10, 3, 14, 8, 9, 5, 22, 12, 26, 7, 15, 16, 34, 18, 38, 20, 21, 11, 46, 24, 25, 13, 27, 28, 58, 30, 62, 32, 33, 17, 35, 36, 74, 19, 39, 40, 82, 42, 86, 44, 45, 23, 94, 48, 49, 50, 51, 52, 106, 54, 55, 56, 57, 29, 118, 60, 122, 31, 63, 64, 65, 66

Offset: 1

N. J. A. Sloane, Dec 26 2014

nonn

a(A001751(n)) != A001751(n). - Reinhard Zumkeller, Dec 27 2014

Michael De Vlieger, Table of n, a(n) for n = 1..10000
A. B. Frizell, Certain non-enumerable sets of infinite permutations. Bull. Amer. Math. Soc. 21 (1915), no. 10, 495-499.
Index entries for sequences that are permutations of the natural numbers

Cf. A064614, A253046.

Cf. A020639, A010051, A001751.

a251561 1 = 1
a251561 n | q == 1                    = 2 * p
          | p == 2 && a010051' q == 1 = q
          | otherwise                 = n
          where q = div n p; p = a020639 n
-- Reinhard Zumkeller, Dec 27 2014

a251561[n_] := Block[{f}, f[x_] := Which[PrimeQ[x], 2 x, PrimeQ[x/2], x/2, True, x]; Array[f, n]]; a251561[66] (* Michael De Vlieger, Dec 26 2014 *)

from sympy import isprime
def A251561(n):
    if n == 2:
        return 4
    q,r = divmod(n,2)
    if r :
        if isprime(n):
            return 2*n
        return n
    if isprime(q):
        return q
    return n # Chai Wah Wu, Dec 26 2014

A290541 Record gaps between numbers that are either prime or twice a prime.

Original entry on oeis.org

1, 3, 6, 10, 11, 12, 13, 15, 19, 21, 25, 28, 35, 40, 48, 57, 60, 64, 70, 75, 83, 90, 95, 117, 120, 144, 148, 150, 153, 167, 168, 196, 205, 212, 214, 221, 234, 244, 254, 255

Offset: 1

Bobby Jacobs, Aug 05 2017

Records in A290496.

The gap of 57 between 31397 and 31454 is due to the record prime gap between 31397 and 31469 being almost exactly twice the record prime gap between 15683 and 15727.

			a(3) = 6 because the next number that is a prime or twice a prime after 47 is 53, and that is a record gap of size 6.

Cf. A001751, A290488, A290489, A290496.

With[{nn = 10^7}, Union@ FoldList[Max, Differences@ #] &@ Union@ Flatten@ {#, 2 TakeWhile[#, # < Prime[nn]/2 &]} &@ Prime@ Range@ nn] (* Michael De Vlieger, Aug 06 2017 *)

a(n) = A290489(n) - A290488(n).

a(36)-a(40) from Giovanni Resta, Aug 06 2017

A238268 The number of unordered ways that n can be written as the sum of two numbers of the form p or 2p, where p is prime.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 3, 2, 3, 3, 3, 3, 4, 4, 3, 3, 4, 4, 3, 3, 5, 4, 4, 4, 5, 4, 4, 3, 4, 6, 4, 3, 7, 4, 3, 5, 6, 5, 5, 5, 6, 7, 4, 4, 9, 5, 5, 7, 6, 5, 5, 4, 5, 7, 4, 3, 10, 4, 4, 8, 8, 7, 7, 5, 6, 8, 5, 4, 10, 5, 5, 9, 8, 7, 8, 5, 7, 9, 5, 4, 13, 8, 6, 8, 8, 7

Offset: 4

Lei Zhou, Feb 21 2014

nonn

p and 2p are terms of A001751.
Sequence defined for n >= 4.
It is conjectured that all terms of this sequence are greater than zero.

			n=4, 4=2+2, one case found. So a(4)=1;
...
n=24, 24 = 2+2*11 = 5+19 = 7+17 = 2*5+2*7 = 11+13, 5 cases found.  So a(24)=5;
...
n=33, 33 = 2+31 = 2*2+29 = 7+2*13 = 2*5+23 = 11+2*11 = 2*7+19, 6 cases found.  So a(33)=6.

Lei Zhou, Table of n, a(n) for n = 4..10000

Cf. A001751, A002375, A103151.

Table[ct = 0; Do[If[((PrimeQ[i]) || (PrimeQ[i/2])) && ((PrimeQ[n - i]) || (PrimeQ[(n - i)/2])), ct++], {i, 2, Floor[n/2]}]; ct, {n, 4, 89}]

isp(i) = isprime(i) || (((i % 2) == 0) && isprime(i/2));
a(n) = sum(i=1, n\2, isp(i) && isp(n-i)); \\ Michel Marcus, Mar 07 2014

A298861 Rank of n-th prime when all the primes and twice-primes are jointly ranked.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 25, 27, 28, 30, 31, 32, 34, 36, 38, 40, 41, 42, 44, 45, 46, 49, 50, 52, 53, 56, 57, 58, 60, 62, 63, 65, 66, 67, 68, 70, 71, 74, 77, 79, 80, 81, 82, 83, 84, 86, 88, 89, 90, 92, 94, 95, 96, 99, 100, 101

Offset: 1

Clark Kimberling, Feb 13 2018

			A001751 = ordered sequence of primes and twice-primes: 2,3,4,5,6,7,10,... in which the primes occupy ranks 1,2,4,6,...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000040, A001751, A298862 (complement).

z = 1000; u = Prime[Range[z]]; w = Take[Union[u, 2 u], z];
p[n_] := If[MemberQ[u, w[[n]]], 0, 1];
Take[w, z];  (* A001751 *)
t = Table[p[n], {n, 1, z}];
Flatten[Position[t, 0]];  (* A298861 *)
Flatten[Position[t, 1]];  (* A298862 *)

A298862 Rank of n-th twice-prime when all the primes and twice-primes are jointly ranked.

Original entry on oeis.org

3, 5, 7, 10, 13, 15, 18, 20, 23, 26, 29, 33, 35, 37, 39, 43, 47, 48, 51, 54, 55, 59, 61, 64, 69, 72, 73, 75, 76, 78, 85, 87, 91, 93, 97, 98, 102, 104, 106, 108, 112, 114, 118, 120, 122, 124, 129, 134, 136, 138, 141, 143, 145, 149, 152, 155, 156, 158, 160

Offset: 1

Clark Kimberling, Feb 13 2018

			A001751 = ordered sequence of primes and twice-primes:  2,3,4,5,6,7,10,... in which twice-primes occupy ranks 3,5,7,...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000040, A001751, A298861 (complement).

z = 1000; u = Prime[Range[z]]; w = Take[Union[u, 2 u], z];
p[n_] := If[MemberQ[u, w[[n]]], 0, 1];
Take[w, z];  (* A001751 *)
t = Table[p[n], {n, 1, z}];
Flatten[Position[t, 0]];  (* A298861 *)
Flatten[Position[t, 1]];  (* A298862 *)

A238269 Smallest number that can be written in n ways as the sum of two numbers of the form p or 2p, where p is prime.

Original entry on oeis.org

4, 6, 8, 16, 24, 33, 36, 63, 48, 60, 93, 140, 84, 108, 132, 189, 165, 144, 120, 210, 297, 168, 204, 180, 276, 252, 285, 288, 462, 240, 372, 432, 336, 300, 396, 609, 360, 492, 552, 468, 564, 528, 576, 504, 708, 1089, 648, 480, 420, 540, 768, 672, 600, 816, 792

Offset: 1

Lei Zhou, Feb 21 2014

nonn

			4 is the smallest number that can be written in only one way as the sum of two primes or doubled primes, 4=2+2. So a(1)=4;
6 = 2+2*2 = 3+3, two ways, so a(2)=6;
...
33 = 2+31 = 2*2+29 = 7+2*13 = 2*5+23 = 11+2*11 = 2*7+19, 6 ways.  And all numbers smaller than 33 can only be split in 5 or fewer ways.  So a(6)=33.

Lei Zhou, Table of n, a(n) for n = 1..10000

Cf. A001751, A002375, A103151, A238268.

target = 55; n = 3; a = {}; sc = 0; Do[AppendTo[a, 0], {i, 1, target}]; While[sc < target, n++; ct = 0; Do[If[((PrimeQ[i]) || (PrimeQ[i/2])) && ((PrimeQ[n - i]) || (PrimeQ[(n - i)/2])), ct++], {i, 2, Floor[n/2]}]; If[ct<=target,If[a[[ct]] == 0, a[[ct]] = n; sc++]]];a

cp's OEIS Frontend

A007922 Smallest k such that k!! is a multiple of n.

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A085118 Primes together with twice the odd primes.

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A105661 a(n)=1 if n is a prime, 2 if n is an even semiprime, otherwise 0.

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A251561 A permutation of the natural numbers: interchange p and 2p for every prime p.

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Haskell

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A290541 Record gaps between numbers that are either prime or twice a prime.

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A238268 The number of unordered ways that n can be written as the sum of two numbers of the form p or 2p, where p is prime.

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Mathematica

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A298861 Rank of n-th prime when all the primes and twice-primes are jointly ranked.

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Mathematica

A298862 Rank of n-th twice-prime when all the primes and twice-primes are jointly ranked.

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