Dimitris Valianatos - cp's OEIS frontend

A339480 Numbers of the form (k^2 - 2) / 2 where k - 1 and k + 1 are both odd composite numbers.

337, 577, 1249, 1567, 2047, 2887, 3697, 4231, 4417, 6727, 6961, 7199, 7441, 7687, 8977, 10081, 10367, 10657, 11857, 12799, 14449, 15487, 16927, 17297, 17671, 20401, 20807, 21217, 21631, 22897, 23327, 23761, 24199, 27847, 29767, 30257, 30751, 32257, 33799, 35377, 37537, 40897

Offset: 1

Dimitris Valianatos, Apr 24 2021

nonn

			For k = 26, k - 1 = 25 and k + 1 = 27 are both odd composite numbers. So (26^2 - 2) / 2 = 337 is a term of the sequence.

Cf. A071904, A129820.

k = 1; forcomposite(c = 1, 287, if(c%2 <> 0, if(c-k == 2, print1((c * (c - 2) - 1) / 2", ")); k = c))

a(n) = (A129820(2*n - 1) * A129820(2*n) - 1) / 2.

A342163 a(n) is the number of numbers greater than 1 and up to prime(n)^2 whose prime factors are all less than or equal to prime(n).

Original entry on oeis.org

2, 6, 15, 29, 60, 87, 137, 176, 247, 360, 422, 568, 689, 776, 923, 1136, 1369, 1494, 1764, 1978, 2128, 2451, 2710, 3074, 3562, 3870, 4077, 4411, 4638, 4995, 6026, 6426, 6987, 7271, 8180, 8493, 9134, 9802, 10319, 11030, 11767, 12139, 13314, 13712, 14329, 14742

Offset: 1

Dimitris Valianatos, Mar 03 2021

nonn

			For n=3, prime(3) = 5. Then the numbers up to 5^2 = 25 that have prime factors <= 5 are 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25. So a(3) = 15.

Robert Israel, Table of n, a(n) for n = 1..4000

Cf. A000040, A001248, A184677.

A[1]:= 2: p:= 2: P:= 1:
f:= proc(n) local x,y;
  x:= n;
  do
    y:= igcd(x,P);
    x:= x/y;
    if x = 1 then return true fi;
    if y = 1 then return false fi
  od;
end proc:
for nn from 2 to 100 do
  q:= p; p:= nextprime(p); P:= P*q;
  A[nn]:= A[nn-1] + p + numboccur(true,map(f, [$q^2+1 .. p^2-1]))
od:
seq(A[i],i=1..100); # Robert Israel, Apr 06 2021

Block[{nn = 46, w}, w = Array[FactorInteger[#][[All, 1]] &, Prime[nn]^2]; Table[-1 + Count[w[[1 ;; p^2]], ?(AllTrue[#, # <= p &] &)], {p, Prime@ Range@ nn}]] (* _Michael De Vlieger, Mar 13 2021 *)

forprime(n = 2, prime(35), i = 0; for(k = 2, n^2, v = factor(k)~[1,]; if(vecmax(v) <= n, i++)); print1(i", "))

a(n) = my(p=prime(n)); sum(k=2, p^2, vecmax(factor(k)[,1]) <= p); \\ Michel Marcus, Mar 03 2021

a(n) = A184677(n) - 1.

Definition clarified by Robert Israel, Apr 06 2021

A339640 a(n) = (A062772(n) + A054270(n)) / 2 - A001248(n).

Original entry on oeis.org

0, 0, 1, 1, -1, 1, -1, 2, 3, 5, -1, 1, 0, 5, 1, 2, -1, 2, -1, 4, -1, -3, 2, 2, -1, 1, 1, 8, -4, 3, 4, 2, -4, 5, 10, -4, -4, -2, -1, 8, -1, -1, 5, -1, 3, -7, 4, 4, 1, 2, 1, 4, 5, 8, 8, 8, -1, 2, -4, -2, 3, 1, -8, -4, 1, -1, -4, 10, -2, 15, 8, 10, 2

Offset: 1

Dimitris Valianatos, Dec 11 2020

sign

Conjecture: The partial sums of this sequence are greater than or equal to zero. This means that the squares of the prime numbers are smaller than the average of the previous and the next prime number most of the time.

			For n = 10 prime(10)^2 = 29^2 = 841. The previous prime of 841 is 839 and the next 853. The average of 839 and 853 is (839 + 853)/2 = 846. So a(10) = 846 - 841 = 5.

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

Cf. A000040, A001248, A054270, A062772, A123993.

f:= p -> (nextprime(p^2) + prevprime(p^2))/2 - p^2:
map(f, [seq(ithprime(i),i=1..100)]); # Robert Israel, Nov 24 2024

Array[(Total@ NextPrime[#, {-1, 1}])/2 - # &[Prime[#]^2] &, 73] (* Michael De Vlieger, Dec 11 2020 *)

forprime(n = 2, 370, print1((nextprime(n^2) + precprime(n^2)) / 2 - n^2", "))

a(n) = (nextprime(prime(n)^2) + precprime(prime(n)^2)) / 2 - prime(n)^2.

A335266 Numbers k such that b(n) = 2*b(n - 1) + k is prime, where b(0) = 1 and k is the smallest positive number.

Original entry on oeis.org

1, 1, 3, 3, 5, 5, 5, 11, 15, 7, 13, 7, 13, 31, 13, 9, 5, 21, 3, 21, 51, 17, 9, 39, 15, 3, 57, 27, 3, 15, 35, 11, 39, 55, 13, 27, 11, 35, 17, 11, 15, 13, 1, 13, 19, 75, 5, 23, 41, 9, 63, 63, 13, 25, 31, 21, 11, 9, 27, 11, 3, 31, 21, 35, 47, 33

Offset: 0

Dimitris Valianatos, May 29 2020

nonn

Sequence b, with an offset change, corresponds to A110930. - Rémy Sigrist, Jul 18 2020

Cf. A000040, A110930.

b[0] = 1; b[n_] := b[n] = NextPrime[2*b[n - 1]]; Table[b[n] - 2*b[n - 1], {n, 1, 66}] (* Amiram Eldar, Aug 18 2020 *)

b=1; for(n = 1, 66, b = 2 * b; p = nextprime(b + 1); k = p - b; b = p; print1(k", "))

A335139 a(n) = (prime(n + 1) +- k) / 2 where k is the smallest possible odd number such that a(n) is prime and a(n + 1) >= a(n).

Original entry on oeis.org

2, 3, 3, 5, 7, 7, 11, 11, 13, 17, 19, 19, 23, 23, 29, 29, 31, 31, 37, 37, 41, 41, 43, 47, 53, 53, 53, 53, 59, 61, 67, 67, 71, 73, 73, 79, 83, 83, 89, 89, 89, 97, 97, 97, 101, 107, 113, 113, 113, 113, 113, 127, 127, 127, 131, 137, 137, 139, 139, 139, 149

Offset: 1

Dimitris Valianatos, May 24 2020

nonn

The sequence of k's begins {1, 1, -1, -1, 1, -3, 3, -1, -3, 3, 1, -3, 3, -1, ...}. I conjecture that the partial sums of the k's sequence change sign infinitely often and that their absolute value is less than the square root of n.

Cf. A000040.

forprime(n = 3, 300, forstep(j = 1, 999, 2, a = (n + j)/2; b =(n - j)/2; if(isprime(a), print1(a", "); break); if(isprime(b), print1(b", "); break)))

A335135 Number of composite numbers between prime(n)^2 and prime(n + 1)^2 - 1.

Original entry on oeis.org

3, 11, 18, 57, 39, 98, 61, 141, 265, 104, 351, 268, 148, 314, 520, 594, 208, 678, 486, 258, 806, 573, 918, 1325, 703, 366, 753, 390, 788, 3006, 933, 1443, 503, 2581, 542, 1666, 1734, 1192, 1842, 1917, 644, 3364, 691, 1416, 717, 4457, 4729

Offset: 1

Dimitris Valianatos, May 24 2020

			For n = 1, prime(1) = 2 and prime(2) = 3. So the composite numbers between 2^2 = 4 and 3^2 - 1 = 9 - 1 = 8 are 4, 6, and 8, so a(1) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Robert Israel)

Cf. A000040, A002808, A053683.

f:= proc(n) local p,q;
p:= ithprime(n); q:= nextprime(p);
q^2 - p^2 - numtheory:-pi(q^2)+numtheory:-pi(p^2)
end proc:
map(f, [$1..50]); # Robert Israel, Jun 24 2020

Array[#1 - #2 - (PrimePi@ #1 - PrimePi@ #2) & @@ {Prime[# + 1]^2, Prime[#]^2} &, 47] (* Michael De Vlieger, May 24 2020 *)

forprime(n = 2, 220, s = 0; forcomposite(k = n^2, nextprime(n + 1)^2 - 1, s++); print1(s", "))

a(n) = prime(n + 1)^2 - prime(n)^2 - (pi(prime(n + 1)^2) - pi(prime(n)^2)).

a(n) = A053683(n+1) - A053683(n). - Michel Marcus, Aug 27 2022

A334912 a(n) = numerator (2^(4n-1) (2^(4n-2) - 1) (Bernoulli(4n-2) / (4n-2)!) * ((2n-2)! / Euler(2n-2))^2).

Original entry on oeis.org

2, 16, 7936, 11184128, 209865342976, 2475749026562048, 123460740095103991808, 5779796046952399460368384, 3729407703720529571097509625856, 485491405392529556189699853976076288, 193817991886041515914007312001087567822848, 56920344782482721622150071084079041150980194304

Offset: 1

Dimitris Valianatos, May 16 2020

For every s odd power of odd prime number p applies: pi(p; 4, 3) = pi(p^s; 4, 3) and pi(p; 4, 1) = pi(p^s; 4, 1).
Product_{p = A002144} ((p^(2*n - 1) - 1) / (p^(2*n - 1) + 1)) = (2^(2*n) + 2) * (2*n - 2)! * (Pi^(2*n - 1) / zeta(2*n - 1)) * (zeta(4*n - 2) / Pi^(4*n - 2)) / abs(EulerE(2*n - 2)), n > 1.
Product_{p = A002145} ((p^(2*n - 1) + 1) / (p^(2*n - 1) - 1)) = (2^(2*n) - 2) * (2*n - 2)! * (zeta(2*n - 1) / Pi^(2*n - 1)) / abs(EulerE(2*n - 2)), n > 1.
Product_{p = A065091, m_p = (p mod 4) - 2} ((p^(2*n - 1) + 1) / (p^(2*n - 1) - 1))^m_p) = (2^(4*n - 1) * (2^(4*n - 2) - 1) * (BernoulliB(4*n - 2) / (4*n - 2)!) * ((2*n - 2)! / EulerE(2*n - 2))^2 ) = a(n) / A334835(n).

X. Gourdon and P. Sebah, Some Constants from Number theory

Cf. A000040, A065091, A002144, A002145, A334835 (denominators).

Cf. A000364, A027641/A027642.

Numerator[Table[2^(4*s - 1) * (2^(4*s - 2) - 1) * BernoulliB[4*s - 2] * (2*s - 2)!^2 / (EulerE[2*s - 2]^2 * (4*s - 2)!), {s, 1, 15}]] (* or *) Numerator[Table[(1 - 1/2^(4*s - 2))*Zeta[4*s - 2]/DirichletBeta[2*s - 1]^2, {s, 1, 15}]] (* Vaclav Kotesovec, May 17 2020 *)

E(n) = subst(bernpol(2*n+1), 'x, 1/4)*4^(2*n+1)*(-1)^(n+1)/(2*n+1); \\ see A000364
a(n) = numerator((2^(4*n-1)*(2^(4*n-2)-1)*(bernfrac(4*n-2)/(4*n-2)!)*((2*n-2)!/ E(n-1))^2)); \\ Michel Marcus, May 17 2020

a(n) = numerator (Product_{p = A065091, m_p = (p mod 4) - 2} ((p^(2*n - 1) + 1) / (p^(2*n - 1) - 1))^m_p) = numerator (2^(4*n) - 4) * ((2*n - 2)! / EulerE(2*n - 2))^2 * (zeta(4*n - 2) / Pi^(4*n - 2))).
From Vaclav Kotesovec, May 17 2020: (Start)
a(n) / A334835(n) ~ 1 as n tends to infinity.
a(n) = numerator((1 - 1/2^(4*n-2)) * zeta(4*n-2) / DirichletBeta(2*n-1)^2). (End)

More terms from Michel Marcus, May 17 2020

A334835 a(n) = denominator (2^(4n-1) (2^(4n-2) - 1) (Bernoulli(4n-2) / (4n-2)!) * ((2n-2)! / Euler(2n-2))^2 ).

Original entry on oeis.org

1, 15, 7875, 11174163, 209844223875, 2475721174255329, 123460585419481594375, 5779795241720954566935675, 3729407645972755442722659595875, 485491404557154927712860942825333525, 193817991848984690019014855170410665878125, 56920344781273501874745734859262004352327035925

Offset: 1

Dimitris Valianatos, May 16 2020

See A334912.

X. Gourdon and P. Sebah, Some Constants from Number theory

Cf. A000040, A065091, A334912 (numerators).

Cf. A000364, A027641/A027642.

Denominator[Table[2^(4*s - 1) * (2^(4*s - 2) - 1) * BernoulliB[4*s - 2] * (2*s - 2)!^2 / (EulerE[2*s - 2]^2 * (4*s - 2)!), {s, 1, 15}]] (* or *) Denominator[Table[(1 - 1/2^(4*s - 2))*Zeta[4*s - 2]/DirichletBeta[2*s - 1]^2, {s, 1, 15}]] (* Vaclav Kotesovec, May 17 2020 *)

E(n) = subst(bernpol(2*n+1), 'x, 1/4)*4^(2*n+1)*(-1)^(n+1)/(2*n+1); \\ see A000364
a(n) = denominator((2^(4*n-1)*(2^(4*n-2)-1)*(bernfrac(4*n-2)/(4*n-2)!)*((2*n-2)!/ E(n-1))^2)); \\ Michel Marcus, May 17 2020

a(n) = denominator (Product_{p = A065091, m_p = (p mod 4) - 2} ((p^(2*n - 1) + 1) / (p^(2*n - 1) - 1))^m_p) = denominator (2^(4*n) - 4) * ((2*n - 2)! / EulerE(2*n - 2))^2 * (zeta(4*n - 2) / Pi^(4*n - 2)).

a(n) = denominator((1 - 1/2^(4*n-2)) * zeta(4*n-2) / DirichletBeta(2*n-1)^2). - Vaclav Kotesovec, May 17 2020

More terms from Michel Marcus, May 17 2020

A331370 Decimal expansion of Sum_{(p1, p2) is twin prime pair} log(p2 / p1).

Original entry on oeis.org

1, 8, 7, 2, 1, 7, 8, 8

Offset: 1

Dimitris Valianatos, May 03 2020

This constant is the difference between Brun's constant and the A331369 constant.

This sum converges because log(p_(k+1)/p_k) < 1/p_k + 1/p_(k+1) for every k >= 1 and (p_k, p_(k+1)) twin prime pair.

			1.8721788...

Cf. A077800, A065421, A331369, A001359, A006512.

Equals A065421 - A331369.

A331369 Decimal expansion of Sum_{(p1, p2) is twin prime pair} 1/p1 + 1/p2 - log(p2/p1).

Original entry on oeis.org

0, 2, 9, 9, 8, 1, 7, 1, 0, 8, 3, 8, 9, 0, 6, 2, 6, 9, 6, 8, 2, 6

Offset: 0

Dimitris Valianatos, May 03 2020

Let (p_k, p_(k+1)) twin prime pair. Then log(p_(k+1)/p_k) < 1/p_k + 1/p_(k+1).
Lim_{k -> oo} 1/p_k + 1/p_(k+1) - log(p_(k+1)/p_k) = 0.
This constant is analogous to Euler-Mascheroni constant for twin primes.

			0.0299817108389062696826...

Cf. A077800, A065421, A001359, A006512, A077761.

p = 3; st = 0.0; forprime(n = 5, 1e11, if(n - p == 2, st += 1/p + 1/n - log(n/p)); p = n); print(st)

Equals Sum_{k >= 1} 1/A001359(k) + 1/A006512(k) - log(A006512(k)/A001359(k)).

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User: Dimitris Valianatos

A339480 Numbers of the form (k^2 - 2) / 2 where k - 1 and k + 1 are both odd composite numbers.

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A342163 a(n) is the number of numbers greater than 1 and up to prime(n)^2 whose prime factors are all less than or equal to prime(n).

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A339640 a(n) = (A062772(n) + A054270(n)) / 2 - A001248(n).

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A335266 Numbers k such that b(n) = 2*b(n - 1) + k is prime, where b(0) = 1 and k is the smallest positive number.

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A335139 a(n) = (prime(n + 1) +- k) / 2 where k is the smallest possible odd number such that a(n) is prime and a(n + 1) >= a(n).

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A335135 Number of composite numbers between prime(n)^2 and prime(n + 1)^2 - 1.

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A334912 a(n) = numerator (2^(4*n-1) * (2^(4*n-2) - 1) * (Bernoulli(4*n-2) / (4*n-2)!) * ((2*n-2)! / Euler(2*n-2))^2).

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A334835 a(n) = denominator (2^(4*n-1) * (2^(4*n-2) - 1) * (Bernoulli(4*n-2) / (4*n-2)!) * ((2*n-2)! / Euler(2*n-2))^2 ).

A334912 a(n) = numerator (2^(4n-1) (2^(4n-2) - 1) (Bernoulli(4n-2) / (4n-2)!) * ((2n-2)! / Euler(2n-2))^2).

A334835 a(n) = denominator (2^(4n-1) (2^(4n-2) - 1) (Bernoulli(4n-2) / (4n-2)!) * ((2n-2)! / Euler(2n-2))^2 ).