A045536 -id:A045536 - cp's OEIS frontend

A244520 a(n) = A080715(n+1) / 2.

1, 3, 5, 11, 15, 21, 29, 35, 39, 41, 51, 65, 95, 105, 155, 165, 179, 191, 221, 231, 239, 281, 329, 371, 419, 431, 485, 519, 611, 641, 659, 809, 905, 935, 989, 1019, 1031, 1049, 1121, 1199, 1229, 1289, 1451, 1469, 1481, 1509, 1541, 1661, 1821, 1931, 2109, 2129, 2141, 2339, 2549, 2795, 2969, 3021, 3039, 3189, 3299, 3329

Offset: 1

Joerg Arndt, Jul 10 2014

nonn

Numbers k such that 2d + k/d is prime for every d|k. Such k must be an odd squarefree number. Primes in the sequence are A045536. - Thomas Ordowski, Nov 16 2017

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A045536, A080715, A088627.

Select[Range[1, 3400, 2], Function[n, AllTrue[Divisors@ n, PrimeQ[2 # + n/#] &]]] (* Michael De Vlieger, Nov 18 2017 *)

is_ok(n)=n=2*n;fordiv(n,d,if(!isprime(d+n/d),return(0)));return(1);
for(n=1,10^4,if(is_ok(n),print1(n,", ")));

A088627(a(n)) = A000005(a(n)) = 2^m. - Thomas Ordowski, Nov 16 2017

A307833 Smallest k > 1 such that A014574(n)*k is adjacent to a prime.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 3, 5, 3, 4, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 5, 2, 2, 4, 3, 3, 2, 2, 2, 2, 2, 3, 2, 4, 3, 2, 2, 2, 3, 6, 3, 2, 2, 2, 3, 4, 2, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 4, 2, 3, 2, 3, 2, 2, 4, 3, 2, 2, 5, 2, 4, 4, 4, 4, 3, 2, 5, 2, 3, 4, 2, 4, 4, 2, 2, 2, 4, 2, 6, 4, 2, 2, 5, 4, 6

Offset: 1

Dmitry Kamenetsky, May 01 2019

nonn

It is perhaps surprising that the values in this sequence are so small. For n < 8000 the largest value of a(n) is 20, which occurs for n = 4928. Also for n < 8000, a(n) is 2 on 2449 occasions.

a(n)=2 if and only if A014574(n)+1 is in A038869 or A014574(n)-1 is in A045536. - Robert Israel, Jul 17 2019

			72*5 = 360, which is adjacent to the prime 359, so a(8) = 5.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A014574, A038869, A045536, A307775, A309120.

P:= {seq(ithprime(i),i=1..10^4)}:
A014574:= sort(convert(map(t -> t+1, P intersect map(`-`,P,2)),list)):
f:= proc(m) local k;
  for k from 2 do
    if isprime(k*m-1) or isprime(k*m+1) then return k fi
  od
end proc:
map(f, A014574); # Robert Israel, Jul 17 2019

primeNearQ[n_] := AnyTrue[{-1, 1} + n, PrimeQ]; twinMidQ[n_] := AllTrue[{-1, 1} + n, PrimeQ]; f[n_] := Module[{k = 2}, While[! primeNearQ[k*n], k++]; k]; f /@ Select[Range[10^4], twinMidQ] (* Amiram Eldar, Jul 05 2019 *)

isok2(n) = isprime(n-1) && isprime(n+1);
k(n) = my(k=2); while (! (isprime(n*k-1) || isprime(n*k+1)), k++); k;
lista(nn) = for (n=1, nn, if (isok2(n), print1(k(n), ", "))); \\ Michel Marcus, May 01 2019

a(n) = A309120(A014574(n)). - Robert Israel, Jul 17 2019

A347934 Primes p of the form k^2 + 1 such that p+2 and 2p+1 are also prime.

Original entry on oeis.org

5, 25601, 193601, 1144901, 1464101, 4326401, 6100901, 12390401, 23522501, 72931601, 127012901, 245862401, 256960901, 351937601, 441840401, 732784901, 802588901, 951722501, 1621672901, 2024100101, 2070250001, 2217468101, 2219352101, 2428518401, 2930056901, 2963713601

Offset: 1

Angad Singh, Sep 20 2021

nonn

All primes (except 5) in the sequence are of the form 100*k^2 + 1.

Intersection of A002496 and A045536.

Select[Range[50000]^2 + 1, AllTrue[{#, # + 2, 2*# + 1}, PrimeQ] &] (* Amiram Eldar, Sep 20 2021 *)

A106060 Primes p such that 1p + 8 and 8p + 1 are primes.

Original entry on oeis.org

5, 11, 29, 71, 101, 131, 149, 269, 401, 431, 449, 479, 491, 599, 761, 821, 929, 1229, 1289, 1559, 1571, 1601, 1619, 1871, 2081, 2129, 2339, 2531, 2549, 2609, 2741, 3041, 3209, 3299, 3461, 3719, 3761, 4289, 5189, 5861, 6269, 6359, 6569, 6701, 6959, 7211

Offset: 1

Zak Seidov, May 07 2005

nonn

Cf. A045536: Primes p such that 1*p+2 and 2*p+1 are prime; A007693: Numbers n such that n and 6*n+1 are primes.

Cf. A007693, A045536.

[p: p in PrimesUpTo(10000) | IsPrime(p+8) and IsPrime(8*p+1)]; // Vincenzo Librandi, Nov 13 2010

Select[Prime[Range[220]], PrimeQ[8#+1]&&PrimeQ[1#+8]&]

A155189 Square-weak primes.

Original entry on oeis.org

3, 5, 7, 13, 19, 23, 31, 43, 47, 53, 61, 73, 83, 89, 103, 109, 113, 131, 139, 151, 157, 167, 173, 181, 193, 199, 211, 229, 233, 241, 257, 263, 271, 283, 293, 313, 317, 337, 349, 353, 359, 373, 383, 389, 401, 409, 421, 433, 443, 449, 463, 467, 491, 503, 509, 523

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

5^2 = 25 < 29 = (3^2+7^2)/2, ...

Cf. A045536, A005384, A051634, A006562, A051635, A155188

lst={};Do[p0=Prime[n];p1=Prime[n+1];p2=Prime[n+2];If[p1^2<(p0^2+p2^2)/2,AppendTo[lst,p1]],{n,5!}];lst
Select[Partition[Prime[Range[100]],3,1],#[[2]]^2<(#[[1]]^2+#[[3]]^2)/2&][[All,2]] (* Harvey P. Dale, May 01 2021 *)

A306407 Brazilian primes p such that p+2 and 2p+1 are also prime.

Original entry on oeis.org

78914411, 7294932341, 119637719305001, 937391863673981, 16737518900352251, 54773061508358111, 417560366367249821, 1103799812221103741, 1515990022247085221, 2748614000294776541, 2805758307714748481, 16359900662260777211, 19024521721109192201, 126048913814465881331, 138996334987487396981

Offset: 1

Bernard Schott, Apr 05 2019

The initial terms of this sequence are of the form (11111)_b. The successive bases b are 94, 292, 3307, 5533, 11374, ...
The first term which is not of this form has 43 digits: it is 1137259672818014782224246589454763146442851 = 1 + 16054 + ... + 16054^9 + 16054^10 = (11111111111)_16054 with a string of eleven 1's.
Sophie Germain primes and lesser twins which are Brazilian both have the same property: if p = (b^q - 1)/(b - 1) is a term, necessarily q (prime) == 5 (mod 6) and b == 1 (mod 3). The smallest terms for the first pairs (q,b) are (5,94), (11,16054), (17,3247).
Intersection of A306845 and A306849.
Intersection of A045536 and A085104.

			The prime 78914411 is a term, because 78914411 = 1 + 94 + 94^2 + 94^3 + 94^4 is a Brazilian prime, 78914411 + 2 = 78914413 is prime and 2 * 78914411 + 1 = 157828823 is prime. The prime 78914411 is Brazilian, the lesser of a pair of twin primes and also a Sophie Germain prime.

Cf. A001359 (lesser of twin primes), A005384 (Sophie Germain primes).
Cf. A045536 (intersection of A001359 and A005384).
Cf. A085104 (Brazilian primes).
Cf. A306845 (Sophie Germain Brazilian primes), A306849 (lesser of twin primes which is Brazilian).

brazilp(N)=forprime(K=5, #binary(N+1)-1, for(n=4, sqrtnint(N-1, K-1), if((K%6==5)&&(n%3==1),if(isprime((n^K-1)/(n-1))&&isprime((n^K-1)/(n-1)+2)&&isprime(2*(n^K-1)/(n-1)+1), print1((n^K-1)/(n-1), ", "))))) \\ Davis Smith, Apr 06 2019

A353702 Composite k such that tau(k') = (tau(k))', where tau(k) is the number of divisors of k (A000005) and k' is the arithmetic derivative of k (A003415).

Original entry on oeis.org

12, 15, 21, 26, 27, 33, 38, 57, 62, 69, 74, 85, 88, 93, 106, 108, 129, 133, 134, 145, 166, 177, 178, 205, 213, 217, 218, 226, 237, 248, 249, 253, 254, 262, 265, 278, 309, 314, 328, 362, 375, 376, 393, 398, 417, 422, 424, 445, 459, 466, 469, 488, 489, 493, 502

Offset: 1

Marius A. Burtea, May 07 2022

nonn

Since for any prime number p, p' = 1 and (tau(p))' = 2' = 1 = tau(1) = tau (p'), the sequence requires only composite numbers that satisfy the given relation.
For p in A092109 the number m = 3*p is a term. Indeed, (tau(m))' = (tau(3*p))' = 4' = 4 and tau(m') = tau((3*p)') = tau(p + 3) = 4, so m is a term.
If p is in A045536 then p, p + 2 and 2*p + 1 are prime numbers and m = 3*(2*p + 1) is a term. Indeed, tau(m') = tau((3*(2*p + 1))') = tau(2*p + 4) = tau(2*(p+2)) = 4 and (tau(m))' = (tau((3*(2*p + 1)))' = 4' = 4, so m is a term.
If k is in A174100 then the numbers 2*k + 1 and 6*k + 1 are prime numbers and the numbers m = 2*(6*k + 1) is a term. Indeed, (tau(m))' = (tau(2*(6*k + 1)) )' = 4' = 4 and tau(m') = tau(2*(6*k + 1))') = tau(6*k + 3) = tau(2*(2*k + 1)) = 4, so m is a term.

			12' = 16, (tau(12)) = 6' = 5 and tau(12') = tau(16) = tau(2^4) = 5, so 12 is a term.
15' = 8, (tau(15))’ = 4' = 4 and tau(15') = tau(8) = tau(2^3) = 4, so 15 is a term.

Cf. A000005, A003415, A045536, A092109, A174100, A260961.

f:=func; [p:p in [3..550]|not IsPrime(p) and  #Divisors(Floor(f(p))) eq Floor(f(#Divisors(p)))];

isA353702 := proc(n)
    if not isprime(n) and numtheory[tau](A003415(n)) = A003415( numtheory[tau](n) ) then
        true ;
    else
        false;
    end if;
end proc:
for n from 2 to 1000 do
    if isA353702(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, May 05 2023

d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[500], CompositeQ[#] && DivisorSigma[0, d[#]] == d[DivisorSigma[0, #]] &] (* Amiram Eldar, May 07 2022 *)

ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(k) = (k>1) && !isprime(k) && numdiv(ad(k)) == ad(numdiv(k)); \\ Michel Marcus, May 08 2022

cp's OEIS Frontend

A244520 a(n) = A080715(n+1) / 2.

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A307833 Smallest k > 1 such that A014574(n)*k is adjacent to a prime.

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Maple

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A347934 Primes p of the form k^2 + 1 such that p+2 and 2p+1 are also prime.

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Programs

Mathematica

A106060 Primes p such that 1*p + 8 and 8*p + 1 are primes.

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Magma

Mathematica

A155189 Square-weak primes.

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Mathematica

A306407 Brazilian primes p such that p+2 and 2p+1 are also prime.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A353702 Composite k such that tau(k') = (tau(k))', where tau(k) is the number of divisors of k (A000005) and k' is the arithmetic derivative of k (A003415).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

A106060 Primes p such that 1p + 8 and 8p + 1 are primes.