Craig J. Beisel - cp's OEIS frontend

A356826 Numbers k such that 2^k - 29 is prime.

5, 8, 104, 212, 79316, 102272, 225536, 340688

Offset: 1

Craig J. Beisel, Aug 29 2022

A particularly low-density pseudo-Mersenne sequence. I have verified that there are no additional terms for k < 5*10^4. For k = a(5), a(6), a(7), and a(8), 2^k - 29 is a probable prime (see link).
The terms a(5)-a(8) were discovered by Henri Lifchitz (see link). - Elmo R. Oliveira, Nov 29 2023
Empirically: except for 5, all terms are even. - Elmo R. Oliveira, Nov 29 2023

			5 is a term because 2^5 - 29 = 3 is prime.
8 is a term because 2^8 - 29 = 227 is prime.

Henri Lifchitz and Renaud Lifchitz, Search for 2^n-29, PRP Top Records.

Cf. A096502.

Cf. Sequences of numbers k such that 2^k - d is prime: A000043 (d=1), A050414 (d=3), A059608 (d=5), A059609 (d=7), A059610 (d=9), A096817 (d=11), A096818 (d=13), A059612 (d=15), A059611 (d=17), A096819 (d=19), A096820 (d=21), A057220 (d=23), this sequence (d=29).

for(n=2, 1000, if(isprime(2^n-29), print1(n, ", ")))

A357039 Number of integer solutions to x' = 2n, where x' is the arithmetic derivative of x.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 3, 2, 2, 3, 4, 3, 2, 3, 4, 4, 4, 2, 3, 4, 4, 4, 6, 4, 3, 5, 4, 4, 7, 3, 5, 6, 3, 5, 7, 5, 5, 7, 6, 5, 8, 5, 4, 9, 6, 5, 8, 3, 6, 8, 5, 6, 9, 6, 8, 10, 6, 6, 13, 4, 6, 10, 4, 7, 9, 6, 5, 8, 9, 8, 11, 6, 5, 12, 5, 8, 12, 5, 8, 11, 6, 6, 14, 9, 6, 11, 9, 7, 14, 6, 8, 13, 7, 8, 13, 7, 9, 13, 8

Offset: 1

Craig J. Beisel, Sep 09 2022

nonn

Conjecture: All terms are positive with the exception of a(1).

			Since 12'=16, 39'=16 and 55'=16, a(8)=3. We don't need to search any higher than (x'^2)/4=(16^2)/4=64 from Barbeau lower bound (See links).

E. J. Barbeau, Remarks on an arithmetic derivative, Canadian Mathematical Bulletin, Volume 4, Issue 2, May 1961, pp. 117-122.
Wikipedia, Arithmetic_derivative.

Cf. A003415.

Bisection of A099302.

for(n=1, 100, v=2*n; c=0; for(k=2, v^2/4, d=0; m=factor(k); for(i=1, matsize(m)[1], d+=(m[i,2]/m[i,1])*k; if(d>v, break;); ); if(d==v, c=c+1; ); ); print1(c", "); );

from sympy import factorint
def A357039(n): return sum(1 for m in range(1,n**2+1) if sum((m*e//p for p,e in factorint(m).items())) == n<<1) # Chai Wah Wu, Sep 12 2022

a(n) = A099302(2n).

A352587 Even numbers 2m such that A352612(2m) = A103131(2m).

Original entry on oeis.org

2, 4, 6, 10, 16, 18, 20, 28, 60, 84, 228, 240, 280, 366, 420, 468, 484, 604, 684, 942, 990, 1152, 1170, 1196, 1440, 2064, 5292, 5954, 8968, 9176, 13242, 13680, 14160, 15190, 24524, 28764, 29422, 30558, 30646, 34804, 35190, 38164, 44642, 56772, 62790, 93024

Offset: 1

Craig J. Beisel, Mar 21 2022

nonn

Any counterexample to the Goldbach conjecture must have this form.

Conjecture: For all a(n) > 18, a(n) is never equal to 2*q^x where q is prime and x is an integer x > 0. In other words, the product of its totatives is never congruent to -1 (mod 2m).

			For a(1) we have A352612(228) == -(59)(85) (mod 228) == 1 (mod 228) == A103131(228). Therefore A352612(228) == A103131(228) and 228 belongs to the sequence.

Craig J. Beisel, Table of n, a(n) for n = 1..56

Cf. A103131, A141098, A352612.

for(n=1,150000, prod_t=1; prod_p=1; prod_r=1; for(k=3, 2*n-3, if(gcd(k,2*n)==1, prod_t=prod_t*k; ); if(gcd(k,2*n)==1 && isprime(k), prod_p=prod_p*k*(2*n-k); ); if(gcd(k,2*n)==1 && !isprime(k) && !isprime(2*n-k), prod_r=prod_r*k; ); ); if(-prod_t%(2*n)==(-prod_p*prod_r)%(2*n), print1(2*n,","); ); );

A352612 (n-1)prod(-p^2 where 2 <= p <= n-2 is prime and relatively prime to n)prod(k where both k and (n-k) are composite and relatively prime to n) (mod n).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 4, 7, 4, 9, 10, 11, 4, 9, 11, 1, 16, 17, 4, 1, 17, 17, 22, 23, 4, 1, 26, 1, 28, 29, 4, 17, 29, 25, 1, 25, 36, 11, 35, 39, 40, 25, 4, 7, 41, 27, 46, 23, 4, 31, 1, 23, 52, 1, 51, 55, 1, 49, 58, 1, 4, 37, 59, 55, 1, 49, 66, 33, 65, 31, 70, 25, 4

Offset: 1

Craig J. Beisel, Mar 23 2022

nonn

The convention for the empty product here is 1. The second product exists for all numbers greater than 210. See A141098.

Conjecture: For odd n, if a(n) == -1 (mod n) then n must be a prime power.

			For n=6 there are no prime totatives between 2 and 4 and there are also no composite totative pairs which add to 6 so both products do not exist and a(6)=n-1=5.
For n=25 these products exist and are given -44618574^2*12096 == 4 (mod 25). Therefore, a(25)=4.

Cf. A103131, A141098, A352587.

a(n)= {prod_p=1; prod_r=1; for(k=2, n-2, if(gcd(k,n)==1, if(isprime(k), prod_p=prod_p*k*(n-k); ); if(!isprime(k) && !isprime(n-k), prod_r=prod_r*k; );); ); (-prod_p*prod_r)%n; }

A351899 Integers k for which there exist two distinct prime nondivisors p, q < k such that, for all i, j >= 0, p^i*q^j mod k is either 1 or is divisible by p or q.

Original entry on oeis.org

5, 10, 16, 18, 19, 20, 21, 22, 38, 48, 50, 51, 54, 60, 61, 67, 75, 77, 78, 80, 85, 90, 91, 98, 100, 108, 120, 122, 126, 127, 134, 147, 150, 154, 160, 170, 182, 189, 201, 204, 210, 217, 234, 234, 240, 252, 254, 255, 266, 268, 288, 291, 294, 300, 310, 320, 328, 336, 340, 348, 360, 362, 364

Offset: 1

Craig J. Beisel, Feb 24 2022

nonn

Conjecture: The prime nondivisors p and q are elements of the reduced residue system consisting of the totatives of k. Assume a triple (k,p,q) which satisfies the definition. If P and Q are the two subgroups generated by p and q respectively and p < q then P >= Q.

			For k = 20 and p, q = (3,7), p^i*q^j mod k can only take on the values 1, 3, 7, 9 which, other than 1, are all divisible by 3 or 7, so 20 is a term.

Cf. A306746.

for(k=1, 364, test2=0; forprime(p=2, k-1, forprime(q=p+1, k-2, if(gcd(p, k)==1 && gcd(q, k)==1, test=0; for(i=0, eulerphi(k), for(j=0, eulerphi(k), if(p^i*q^j % k >1 && gcd(p^i*q^j % k, p)==1 && gcd(p^i*q^j % k, q)==1, test=1; ); if(test==1, break(2); ); ); ); if(test==0, test2=1; ); ); ); ); if(test2==1, print1(k, ", "); ); );

A332704 Record indices of the ratio A280008(n) / A002375(n) (Goldbach conjecture related).

Original entry on oeis.org

3, 5, 33, 63, 165, 315, 255255

Offset: 1

Craig J. Beisel, Jun 08 2020

The sequence lists indices n for which A280008(n) / A002375(n) is less than all previous indices n > 2.
We do not consider indices n=1 and n=2, for which the sequence A002375(n) (= number of odd primes {p,q} such that 2n=p+q) is zero.
If the Goldbach conjecture is false, then this sequence is finite. It will end with n such that A280008(n) / A002375(n) = -1 since no further terms could achieve less than this value.
If the Goldbach conjecture is true, then this sequence may be finite or infinite. The ratio A280008(n) / A002375(n) has a lower bound greater than -1 and the value of this ratio for record indices approaches the lower bound.
It is known that this sequence has additional terms beyond a(7) = 17#/2 = 255255 =  A070826(7) since A280008(255255) / A002375(255255) = -0.76119 and for A070826(10) = 20#/2 = 3234846615 we have A280008(3234846615) / A002375(3234846615) = -0.78989.

Cf. A002375, A070826, A280008.

lastx=1; record=999; for(n=4, 1000, x=0; forprime(i=3, n, if(isprime(2*n-i), x=x+1; ); ); y=(x-lastx)/lastx; if(y

A332704(k+1) = min{ n>2 | A280008(n)/A002375(n) < A002375(A332704(k))/A280008(A332704(k)) }.

A335360 Numbers m such that the number of unordered Goldbach partitions of 2m is greater than the number of unordered Goldbach partitions of 4m.

Original entry on oeis.org

17, 32, 38, 143, 353

Offset: 1

Craig J. Beisel, Jun 03 2020

Integers m such that A002375(2*m) > A002375(4*m) .

It is conjectured that a(5)=353 is the last term in this sequence.

			For a(1)=17, 2*17=34 has 4 Goldbach partitions and 4*17=68 has 2.

Index entries for sequences related to Goldbach conjecture

Cf. A002375.

for(n=1, 1000, x=0; y=0; forprime(i=2, 2*n, if(i<=n && isprime(2*n-i), x=x+1; ); if(isprime(4*n-i), y=y+1; ); ); if(x>y, print1(n, ", ")))

A335250 Numbers m such that twice the number of unordered Goldbach partitions of 2m equals the number of unordered Goldbach partitions of 4m.

Original entry on oeis.org

1, 4, 9, 15, 21, 30, 40, 46, 69, 70, 79, 81, 82, 106, 114, 199, 229, 256, 361, 391, 469, 586, 754, 760, 766, 826, 892, 1471, 1483, 1525, 1591, 1609, 1624, 1816, 2194, 2206, 2454, 2629, 2869, 3955, 3961, 3964, 6406, 6946, 11749

Offset: 1

Craig J. Beisel, May 28 2020

Integers m such that 2*A002375(2m) = A002375(4m).

It is conjectured that the last term in this sequence is a(45)=11749.

			m=4 is a term because 2m=8 has the partition (3,5) while 4m=16 has the partitions (3,13) and (5,11).

Index entries for sequences related to Goldbach conjecture

Cf. A002375, A335226.

for(n=1, 200000, x=0; y=0; forprime(i=2, 2*n-1, if(i<=n && isprime(2*n-i), x=x+1; ); if(isprime(4*n-i), y=y+1; ); ); if(2*x==y, print1(n, ", ")))

A335226 Numbers m such that twice the number of unordered Goldbach partitions of 2m is less than the number of unordered Goldbach partitions of 4m.

Original entry on oeis.org

6, 16, 19, 28, 34, 49, 61, 64, 76, 91, 94, 124, 133, 154, 163, 166, 184, 208, 214, 244, 250, 259, 271, 277, 286, 301, 316, 334, 346, 355, 364, 403, 430, 439, 451, 481, 496, 511, 556, 619, 649, 679, 706, 709, 724, 799, 802, 859, 874, 979, 982, 994, 1006, 1024, 1069, 1099

Offset: 1

Craig J. Beisel, May 27 2020

nonn

Integers m such that 2*A002375(2m) < A002375(4m).

It is conjectured that the last term in this sequence is a(114)=22564.

			m=6 is a term because 2m=12 has the partition (5,7) while 4m=24 has the partitions (5,19),(7,17) and (11,13).

Index entries for sequences related to Goldbach conjecture

Cf. A002375, A335250, shares a number of terms with A137820.

for(n=1, 100000, x=0; y=0; forprime(i=2, 2*n-1, if(i<=n && isprime(2*n-i), x=x+1;); if(isprime(4*n-i), y=y+1;);); if(2*x

A329979 Prime numbers which can be represented as p^i * q^j - (p + q) where p and q are distinct odd primes and i,j > 0.

Original entry on oeis.org

7, 11, 19, 23, 31, 37, 43, 47, 53, 59, 67, 71, 79, 83, 101, 103, 107, 127, 131, 137, 139, 149, 163, 167, 179, 181, 191, 199, 211, 223, 229, 233, 239, 251, 263, 271, 283, 293, 307, 311, 331, 347, 349, 359, 367, 373, 379, 383, 397, 419, 421, 431, 439, 443, 463, 467, 479, 491, 499

Offset: 1

Craig J. Beisel, Nov 26 2019

Numbers of this form are an attempt to generalize Mersenne numbers (see link).

Robert Israel, Table of n, a(n) for n = 1..10000
Craig J. Beisel, Can it be shown that numbers of a certain form produce primes more often than expected?, Math StackExchange, November 2019.

N:= 1000: # for terms <= N
P:= select(isprime, [seq(i,i=3..(N+3)/2,2)]):
S:= {}:
for ip from 1 to nops(P) do
  p:= P[ip];
  for i from 1 while p^i*3 - (p+3) <= N do
    for iq from 1 to ip-1 do
       q:= P[iq];
       if p^i*q - (p+q) > N then break fi;
       for j from 1 do
         x:= p^i * q^j - (p+q);
         if x > N then break fi;
         if isprime(x) then S:= S union {x} fi;
od od od od:
sort(convert(S,list));  # Robert Israel, Aug 25 2025

z=[];forprime(a=3,1000, forprime(b=a+2,1000, for(i=1,10, for(j=1,10, y=a+b; x=a^i*b^j-y; if(x<500 && isprime(x) && setsearch(z,x)==0,z=setunion(z,[x])) )))); print(z)

Definition clarified by Robert Israel, Aug 25 2025

cp's OEIS Frontend

User: Craig J. Beisel

A356826 Numbers k such that 2^k - 29 is prime.

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A357039 Number of integer solutions to x' = 2n, where x' is the arithmetic derivative of x.

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A352587 Even numbers 2m such that A352612(2m) = A103131(2m).

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A352612 (n-1)*prod(-p^2 where 2 <= p <= n-2 is prime and relatively prime to n)*prod(k where both k and (n-k) are composite and relatively prime to n) (mod n).

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A351899 Integers k for which there exist two distinct prime nondivisors p, q < k such that, for all i, j >= 0, p^i*q^j mod k is either 1 or is divisible by p or q.

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A332704 Record indices of the ratio A280008(n) / A002375(n) (Goldbach conjecture related).

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Formula

A335360 Numbers m such that the number of unordered Goldbach partitions of 2m is greater than the number of unordered Goldbach partitions of 4m.

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A335250 Numbers m such that twice the number of unordered Goldbach partitions of 2m equals the number of unordered Goldbach partitions of 4m.

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A352612 (n-1)prod(-p^2 where 2 <= p <= n-2 is prime and relatively prime to n)prod(k where both k and (n-k) are composite and relatively prime to n) (mod n).