Lear Young - cp's OEIS frontend

A235644 Number of decompositions of 12*n into the sum of two (not necessarily distinct) twin prime pairs.

0, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2, 0, 2, 1, 3, 3, 1, 2, 1, 3, 2, 2, 2, 3, 1, 3, 1, 2, 3, 3, 6, 2, 3, 1, 2, 4, 3, 4, 4, 1, 3, 2, 3, 5, 2, 7, 1, 3, 2, 2, 5, 2, 5, 2, 3, 2, 2, 3, 5, 3, 4, 1, 0, 3, 1, 6, 2, 3, 3, 1, 5, 2, 5, 3, 3, 4, 1, 4

Offset: 1

Lear Young, Jun 16 2014

nonn

In the 1980's, Liang conjectured that (6n)^2 = p_1 + p_2 + p_3 + p_4, where (p_1, p_2) and (p_3, p_4) are twin prime pairs. See reference for more details.
It seems there are at least 2 solutions for the decompositions when n > 701.
If the two twin prime pairs are required to be distinct, the sequence is A187759.

			a(736) = 2 because 12*736 = 197 + 199 + 4217 + 4219 = 857 + 859 + 3557 + 3559, so there are 2 ways of expressing 12*n as the sum of two twin prime pairs.

Liang Ding Xiang, Problem 93#, Bulletin of Mathematics (Wuhan), 6(1992),41. ISSN 0488-7395.

Cf. A016910, A243941, A187759, A243914, A243956.

v=select(p->isprime(p-2)&&p>5, primes(200))\6; l=List(); for(i=1, #v, if(2*v[i]<100, listput(l, 2*v[i])); for(j=i+1, #v, if((v[i]+v[j])<100, listput(l, v[i]+v[j])))); l1=vecsort(l); k=1; for(i=1, 100, s=sum(j=k, #l1, l1[j]==i); print1(s", "); k+=s) \\ Lear Young, Jun 16 2014

v=select(p->isprime(p-2)&&p>5,primes(110))\6;for(i=1, 99, print1(sum(j=1,#v,vecsearch(v,i-v[j])>0 && i-v[j]>=v[j])", "))   \\ change i-v[j]>=v[j] to i-v[j]>v[j] is A187759.  Lear Young, Jun 16 2014

A243956 Positive numbers n without a decomposition into a sum n = i+j such that 6i-1, 6i+1, 6j-1, 6j+1 are twin primes.

Original entry on oeis.org

1, 16, 67, 86, 131, 151, 186, 191, 211, 226, 541, 701

Offset: 1

Lear Young, Jun 15 2014

nonn

Conjecture: any integer n > 701 has a decomposition into a sum n = i+j such that 6i-1, 6i+1, 6j-1, 6j+1 are twin primes.

Cf. A002822, A187759.

b:= n-> isprime(6*n-1) and isprime(6*n+1):
a:= proc(n) option remember; local i, k, ok;
      for k from 1 +`if`(n=1, 0, a(n-1)) do ok:= true;
        for i to iquo(k, 2) while ok
          do ok:= not(b(i) and b(k-i)) od;
        if ok then return k fi
      od
    end:
seq(a(n), n=1..12);  # Alois P. Heinz, Jun 20 2014

l=List();a=select(p->isprime(p-2)&&p>5, primes(2000))\6;
for(i=1,#a-1,listput(l,2*a[i]);for(j=i+1,#a,listput(l,(a[i]+a[j]))));
print(setminus(Set(vector(l[#l]/4, i, i)), Set(l)))

A243914 Even numbers which are twice the sum of a twin prime pair, but cannot be expressed as the sum of 2 distinct twin prime pairs.

Original entry on oeis.org

16, 24, 792, 1392

Offset: 1

Lear Young, Jun 14 2014

nonn

Subsequence of A111046 (twice A054735).

It seems that this sequence is probably finite (there are no further terms below 10^7).

			a(1) = 16 = 2*(3+5).
16 is in the sequence since it is twice the sum of twin primes 3 and 5, but cannot be expressed as the sum of 2 distinct twin pairs.
36 is not in the sequence because although it is the sum of twin primes 17 and 19, it can also be expressed as the sum of pairs (5, 7) and (11, 13).

Cf. A054735, A111046.

with(SignalProcessing): # requires at least Maple 17
N:= 10^6; # to check primes up to N
Primes:= select(isprime,{seq(2*i+1,i=1..N)}):
Twins:= Primes intersect map(t-> t-2,Primes):
nT:= nops(Twins);
T:= Array(1..(Twins[nT]+1)/2, datatype=float[8]);
for i from 1 to nT do T[(Twins[i]+1)/2]:= 1 od:
TTwins:= Convolution(T,T);
map(t -> 4*(t+1), select(n -> round(TTwins[n])=1,[$1..(nT+1)/2])); # Robert Israel, Jun 15 2014

isok(isum1, vsum2) = {for (k=1, #vsum2, ksum2 = vsum2[k]; if (ksum2 > one, break;); if (isum1 - ksum2 != ksum2, if (vecsearch(vsum2, isum1 - ksum2), return (0)););); return (1);}
lista() = {v = readvec("b014574.txt"); vsum1 = 4*v; vsum2 = 2*v; maxs2 = vecmax(vsum2); for (i=1, #v, isum1 = vsum1[i]; if (isum1 < maxs2, if (isok(isum1, vsum2), print1(isum1, ", "));););} \\ Michel Marcus, Jun 15 2014

l1=l2=List();a=select(p->isprime(p+2),primes(1000));for(i=1,#a-1,if(i<#a/4,listput(l1,4*a[i]+4));for(j=i+1,#a,listput(l2,2*(a[i]+a[j])+4)));print(setminus(Set(l1),Set(l2))) \\ Lear Young, Jun 15 2014

A242880 Numbers that are both Poulet and Proth.

Original entry on oeis.org

1729, 4033, 8321, 12801, 65281, 130561, 348161, 3225601, 8355841, 8384513, 16773121, 40280065, 104988673, 2147418113, 4294901761, 4294967297, 53282340865, 68719214593, 137439477761, 1099510579201, 1911029760001, 2199021158401, 8796097216513, 281474959933441, 9007199388958721, 576460753377165313, 2305843011361177601, 18446744073709551617

Offset: 1

Lear Young, May 25 2014

nonn

Intersection of A080075 and A001567.

a(1) = 1729 is known as the Hardy-Ramanujan number (see A001235). - Omar E. Pol, Jun 14 2014

Giovanni Resta, Poulet n and Proth n.

Cf. A080075 (Proth numbers), A001567 (Poulet numbers).

a(20)-a(28) from Max Alekseyev, May 28 2014

A242515 Numbers n such that 12n+1, 12n+5, 12n+7, 12n+11 are all composite numbers.

Original entry on oeis.org

44, 70, 72, 74, 105, 111, 112, 132, 137, 140, 147, 154, 163, 170, 182, 193, 202, 207, 209, 235, 245, 248, 252, 258, 262, 273, 285, 312, 315, 317, 322, 329, 331, 336, 345, 347, 349, 359, 369, 372, 377, 384, 392, 397, 403, 404, 422, 427, 437

Offset: 1

Lear Young, May 16 2014

nonn

			a(1) = 44; 44*12+1=529, 44*12+5=533, 44*12+7=535, 44*12+11=539, and 529, 533, 535 and 539 are all composites.

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Subsequence of A153383.

Select[Range[500],AllTrue[12#+{1,5,7,11},CompositeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 13 2018 *)

for(i=0,1000,if(!isprime(12*i+1) && !isprime(12*i+5) && !isprime(12*i+7) && !isprime(12*i+11),print1(i", "))) \\ Lear Young, May 16 2014

A241213 a(n) is built digit-by-digit (see comments for details).

Original entry on oeis.org

1, 2, 3, 4, 5, 10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 24, 25, 30, 31, 32, 33, 34, 35, 40, 41, 42, 43, 44, 45, 100, 101, 102, 103, 104, 105, 110, 111, 112, 113, 114, 115, 120, 121, 122, 123, 124, 125, 130, 131, 132, 133, 134, 135, 140, 141, 142, 143, 144, 145

Offset: 1

Lear Young, Apr 17 2014

a(n) is built digit-by-digit as a_i ... a_3 a_2 a_1.
Note that in this case, the definition of "digit" is a nonnegative integer. If i > 3, the number of digits of a_i may be greater than 1.
Successively, we have:
a_1 = n mod 6;
a_2 = ((n - a_1)/primorial(2)) mod prime(2+1);
a_3 = ((n - a_1 - a_2*primorial(2))/primorial(3)) mod prime(3+1);
...
a_i = ((n - a_1 - a_2*primorial(2)-...-a_(i-1)*primorial(i-1))/primorial(i)) mod prime(i+1).
So that finally, n = a_1 + a_2*primorial(2) + ... + a_i*primorial(i).

			a(2287) = 10611.
10611 is built digit-by-digit as a_4 a_3 a_2 a_1 = 10 6 1 1.
And a_1 + a_2*primorial(2) + a_3*primorial(3) + a_4*primorial(4) = 1 + 1*6 + 6*30 + 10*210 = 2287.
(The definition of "digit" is a nonnegative integer. See comments for how to get a_1, a_2, a_3, a_4.)

Lear Young, Table of n, a(n) for n = 1..100000

Pr = Primes()
c = oeis(2110)[:10]
def bjz(a):
    d = len(str(a)) + 1
    b  = [0] * (d)
    b[0] = a % 6
    s = 0
    for x in range(1, d):
        if x > 1:
            s += c[x] * b[x-1]
        b[x] = ((a - b[0] - s) / c[x+1] ) % Pr.unrank(x+1)
    return int(''.join(map(str, b[::-1])))
[ bjz(x)  for x in range(1, 101)] # Lear Young, Apr 17 2014

A240718 Number of decompositions of 2n into an unordered sum of two primes, one of the two primes less than sqrt(2n-2).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 0, 1, 0, 0, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 2, 2, 1, 3, 3, 1, 1, 2, 2, 2, 2, 2

Offset: 1

Lear Young, Apr 11 2014

nonn

			For n = 7, the a(7) = 1 solution is 2*7 = 3 + 11 = 7 + 7; one of these pairs, 3 + 11, contains a number less than sqrt(2*7 - 2).

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

Cf. A002375.

P:= NULL: A[1]:= 0: nextp:= 2:
for n from 2 to 100 do
 while nextp^2 < 2*n-2 do
   P:= P, nextp;
   nextp:= nextprime(nextp);
 od;
 A[n]:= numboccur(true, map(t -> isprime(2*n-t), [P]))
od:
seq(A[i],i=1..100); # Robert Israel, Apr 30 2019

a(n)=sum(i=2,primepi(floor(sqrt(2*n-2))),isprime(2*n-prime(i))) \\ Lear Young, Apr 11 2014

A239731 Difference between sum of first n primes and prime(prime(n)).

Original entry on oeis.org

-1, 0, -1, 0, -3, 0, -1, 10, 17, 20, 33, 40, 59, 90, 117, 140, 163, 218, 237, 286, 345, 390, 443, 502, 551, 614, 701, 784, 881, 976, 1011, 1112, 1215, 1330, 1417, 1550, 1665, 1780, 1923, 2056, 2203, 2360, 2485, 2660, 2827, 3010, 3141, 3252, 3455, 3670, 3879, 4090, 4307, 4484, 4717, 4932, 5147, 5400, 5631, 5876, 6135, 6362, 6555, 6830, 7125, 7424, 7633, 7922

Offset: 1

Lear Young, Mar 30 2014

sign

			For n = 2 the a(2) = 0 solutions are prime(1) + prime(2) - prime(prime(2)) = 5 - 5 = 0.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A006450, A007504.

A239731:=n->sum(ithprime(i), i=1..n) - ithprime(ithprime(n)); seq(A239731(n), n=1..50); # Wesley Ivan Hurt, Mar 30 2014

Table[Sum[Prime[i], {i, n}] - Prime[Prime[n]], {n, 50}] (* Wesley Ivan Hurt, Mar 30 2014 *)
#[[1]]-#[[2]]&/@Module[{nn=70,prs},prs=Accumulate[Prime[Range[nn]]];Thread[{prs,Prime[Prime[Range[nn]]]}]] (* Harvey P. Dale, Apr 10 2025 *)

for(i = 1, 100, print1(sum(k = 1, i, prime(k)) - prime(prime(i))", ")) \\ Lear Young, Mar 30 2014

[a - b for a, b in zip(oeis(7504)[1:], oeis(6450))] # Lear Young, Mar 30 2014

a(n) = A007504(n + 1) - A006450(n) = A007504(n + 1) - A000040(A000040(n)). - Wesley Ivan Hurt, Mar 30 2014

a(n) ~ (n^2 log n)/2. - Charles R Greathouse IV, Apr 08 2014

A226181 Primes p such that p-1 divided by the period of the binary expansion of 1/p equals 2^x for some nonnegative integer x.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 113, 131, 137, 139, 149, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 227, 233, 239, 257, 263, 269, 271, 281, 293, 311, 313, 317, 337, 347, 349

Offset: 1

Lear Young, May 30 2013

Equivalently, p-1 divided by the period of the decimal expansion of 1/p equals 2^x for some nonnegative integer x. Composite numbers satisfying this condition are given in A243050. - Lear Young, May 30 2013

Let pi_1(x) and pi(x) be the numbers of primes of this sequence and all primes not exceeding x, respectively. Then, for x>=3, p_1(x)/pi(x) >= C_Artin = 0.37395581... Numerical results suggest that it is likely lim pi_1(x)/pi(x) = 2*C_Artin. - Peter J. C. Moses and Vladimir Shevelev, May 29 2014

			(41-1)/20 = 2. 20 is the period of the binary representation of 1/n, the odd part of 2 is 1.

Lear Young, Table of n, a(n) for n = 1..1000

Cf. A007733, A136042.

Select[Prime[Range[2, 100]], # == 2^IntegerExponent[#, 2] &[(# - 1)/MultiplicativeOrder[2, #]] &] (* Peter J. C. Moses, May 28 2014 *)

is(n) = {
  m = valuation(n+1,2);
      k=(n+1)>>m;
      if(k!=1, for(i=0,(n-1)>>1,
        l=valuation(k+n,2);
        k=(k+n)>>l;
        m+=l;if(k==1,break)));
       ((n-1)/m)>>valuation((n-1)/m, 2)==1
       \\ m  equals znorder(Mod(2,n))
    }
forstep(i=3,1e3,2,if(is(i),print1(i, ", ")))
\\ Lear Young May 30 2013

forstep(i=1,1e3,2,j = (i-1)/znorder(Mod(2,i));if(j>>valuation(j, 2)==1,print1(i, ", "))) \\ Lear Young May 31 2013

A226014 Primes p such that A179382((p+1)/2) = (p-1)/(2^x) for some x>0.

Original entry on oeis.org

3, 7, 11, 13, 17, 19, 29, 31, 37, 41, 53, 59, 61, 67, 83, 97, 101, 107, 113, 127, 131, 137, 139, 149, 163, 173, 179, 181, 193, 197, 211, 227, 257, 269, 281, 293, 313, 317, 347, 349, 353, 373, 379, 389, 401, 409, 419, 421, 443, 449, 461, 467, 491, 509, 521, 523, 541, 547, 557, 563, 569, 577, 587, 593, 613, 617, 619, 653, 659, 661, 677, 701, 709, 757, 761, 769, 773, 787, 797, 809, 821, 827, 829, 853, 857, 859, 877, 883, 907, 929, 941, 947, 977

Offset: 1

Lear Young, May 22 2013

It is conjectured that:
Let n be an odd number and the period of 1/n is n-1 or a divisor of n-1. Call c=A179382((n+1)/2) the "cycle length of n". If c divides n-1 or n+1 = 2^x for some x>0, then n is prime. For details see link and Cf. - Lear Young, with contributions from Peter Košinár, Giovanni Resta, Charles R Greathouse IV, May 22 2013
The numbers in the sequence are the values of n in the above conjecture.

			929 : (929-1)/(2^2)=232=A179382((929+1)/2) and znorder(Mod(10,929))=464=(929-1)/2

Hagen von Eitzen, Details of the "cycle length of n"

Cf. A179382, A136042 (both sequences related to the way to get the "cycle length of n").

oddres(n)=n>>valuation(n, 2)
cyc(d)=my(k=1, t=1); while((t=oddres(t+d))>1, k++); k
forstep(n=3, 1e3, [4, 2, 2, 2], x=cyc(n);z=znorder(Mod(10, n));if((x==1 || (n%x==1 && oddres((n-1)/x)==1)) && (n%z==1 || n%z==0), print1(n", ")))
\\ Charles R Greathouse IV, May 22 2013

cp's OEIS Frontend

User: Lear Young

A235644 Number of decompositions of 12*n into the sum of two (not necessarily distinct) twin prime pairs.

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A243956 Positive numbers n without a decomposition into a sum n = i+j such that 6i-1, 6i+1, 6j-1, 6j+1 are twin primes.

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A243914 Even numbers which are twice the sum of a twin prime pair, but cannot be expressed as the sum of 2 distinct twin prime pairs.

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A242880 Numbers that are both Poulet and Proth.

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A242515 Numbers n such that 12n+1, 12n+5, 12n+7, 12n+11 are all composite numbers.

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Mathematica

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A241213 a(n) is built digit-by-digit (see comments for details).

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Sage

A240718 Number of decompositions of 2n into an unordered sum of two primes, one of the two primes less than sqrt(2n-2).

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Maple

PARI

A239731 Difference between sum of first n primes and prime(prime(n)).

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