Giuseppe Coppoletta - cp's OEIS frontend

A288908 Primes p whose distance from next prime and from previous prime is greater than log(p).

5, 7, 23, 37, 47, 53, 89, 157, 173, 211, 251, 257, 263, 293, 331, 337, 359, 367, 373, 389, 409, 479, 631, 691, 701, 709, 719, 787, 797, 839, 919, 929, 1163, 1171, 1201, 1249, 1259, 1381, 1399, 1409, 1471, 1511, 1523, 1531, 1637, 1709, 1733, 1801, 1811, 1823

Offset: 1

Giuseppe Coppoletta, Jun 19 2017

nonn

Primes preceded and followed by larger-than-average prime gaps (see link), then included in A082885.

			n = 5 is a term because 3 + log(5) < 5 < 7 - log(5).
n = 11 is not a term because 13 - 11 < log(11) = 2.39...

Eric Weisstein's MathWorld, Prime Number Theorem

Cf. A082885, A151799, A151800, A288907, A330426, A330427, A330428.

f:=func;  [p:p in PrimesInInterval(3,2000)|f(p)]; // Marius A. Burtea, Dec 19 2019

Select[Prime@ Range[2, 300], Min@ Abs[# - NextPrime[#, {-1,1}]] > Log[#] &] (* Giovanni Resta, Jun 19 2017 *)

[n for n in prime_range(3,2000) if next_prime(n)-n>log(n) and n-previous_prime(n)>log(n)]

A151799(a(n)) + log(a(n)) < a(n) < A151800(a(n)) - log(a(n)).

A288907 Primes p whose distance from the next prime and from the previous prime is less than log(p).

Original entry on oeis.org

71, 101, 103, 107, 109, 193, 197, 227, 229, 281, 311, 313, 349, 433, 439, 443, 461, 463, 503, 563, 569, 571, 593, 599, 601, 607, 613, 617, 643, 647, 653, 659, 677, 733, 739, 757, 823, 827, 857, 859, 881, 883, 941, 947, 971, 977, 1013, 1019, 1033, 1063, 1091, 1093

Offset: 1

Giuseppe Coppoletta, Jun 19 2017

nonn

Primes preceded and followed by less-than-average prime gaps (by the Prime Number Theorem, see link).

This sequence is a subsequence of A381850 and of A383652. - Alain Rocchelli, May 07 2025

			n = 23 is not a term because 23 - 19 > log(23) = 3.13...
n = 71 is a term because log(71) = 4.71.. and 73 - log(71) < 71 < 67 + log(71).

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Number Theorem

Cf. A151799, A151800, A288908, A068996, A381850, A383652.

q:= p-> isprime(p) and is(max(nextprime(p)-p, p-prevprime(p))Alois P. Heinz, May 12 2025

Select[Range[2, 220] // Prime, Max[ Abs[# - NextPrime[#, {-1, 1}]]] < Log[#] &] (* Giovanni Resta, Jun 19 2017 *)

is(n) = ispseudoprime(n) && n-precprime(n-1) < log(n) && nextprime(n+1)-n < log(n) \\ Felix Fröhlich, Jun 19 2017

[n for n in prime_range(3,1300) if next_prime(n)-n

A151800(a(n)) - log(a(n)) < a(n) < A151799(a(n)) + log(a(n)).

Conjecture: Limit_{n->oo} n / PrimePi(a(n)) = (1-1/e)^2 (A068996). - Alain Rocchelli, May 07 2025

A284036 Positive integers n such that (n^2 - 3)/2 and (n^2 + 1)/2 are twin primes.

Original entry on oeis.org

3, 5, 11, 19, 25, 29, 65, 79, 101, 205, 209, 221, 245, 275, 289, 299, 349, 371, 409, 415, 449, 521, 535, 569, 571, 575, 595, 649, 661, 695, 739, 781, 791, 935, 949, 991, 1081, 1091, 1099, 1129, 1181, 1225, 1241, 1285, 1345, 1349, 1459, 1489, 1531, 1541, 1615

Offset: 1

Giuseppe Coppoletta, Mar 27 2017

nonn

All terms are obviously odd.

			25 is a term because (25^2 - 3)/2 = 311 and (25^2 + 1)/2 = 313 are twin primes.

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

Cf. A002731, A109358, A048161, A110589, A284034.

filter:= n -> isprime((n^2-3)/2) and isprime((n^2+1)/2):
select(filter, [seq(i,i=1..2000,2)]); # Robert Israel, Apr 24 2017

Select[Range[1, 1285, 2], Times @@ Boole@ Map[PrimeQ, (#^2 + {-3, 1})/2] == 1 &] (* Michael De Vlieger, Mar 28 2017 *)

isok(n) = isprime((n^2 - 3)/2) && isprime((n^2 + 1)/2); \\ Michel Marcus, Apr 04 2017

from sympy import isprime
print([n for n in range(3, 1700, 2) if isprime((n**2 - 3)//2) and isprime((n**2 + 1)//2)]) # Indranil Ghosh, Apr 04 2017

[n for n in range(3,1700,2) if is_prime((n^2 - 3)//2) and is_prime((n^2 + 1)//2)]

A284035 Upper twin primes which correspond to the hypotenuse in a Pythagorean triple whose short leg is also a prime.

Original entry on oeis.org

5, 13, 61, 181, 421, 3121, 5101, 60901, 83641, 100801, 135721, 161881, 163021, 218461, 273061, 491041, 595141, 637321, 697381, 1064341, 1108561, 1171981, 1806901, 2574181, 2601481, 2740141, 2763601, 2853661, 3248701, 3535141, 3567121, 3696481, 3723721, 3729181, 4832941

Offset: 1

Giuseppe Coppoletta, Mar 19 2017

nonn

A284034 gives the corresponding short leg primes in the definition.

			The prime q = 3121 is in the sequence because q - 2 = 3119 is prime and {79, 3120, 3121} is a Pythagorean triple with prime short leg (see example in A284034).

Cf. A051859, A067756, A284034.

lista(nn) = forprime(p=3, nn, if (isprime(p) && isprime((p^2-3)/2) && isprime(q=(p^2+1)/2), print1(q, ", "))); \\ Michel Marcus, Apr 01 2017

A284034(n)^2 + (a(n) - 1)^2 = a(n)^2, i.e., a(n) = (A284034(n)^2 + 1)/2.

More terms from Michel Marcus, Apr 01 2017

A284034 Primes p such that (p^2 - 3)/2 and (p^2 + 1)/2 are twin primes.

Original entry on oeis.org

3, 5, 11, 19, 29, 79, 101, 349, 409, 449, 521, 569, 571, 661, 739, 991, 1091, 1129, 1181, 1459, 1489, 1531, 1901, 2269, 2281, 2341, 2351, 2389, 2549, 2659, 2671, 2719, 2729, 2731, 3109, 4049, 4349, 5279, 5431, 5471, 5531, 5591, 5669, 6329, 6359, 6871, 7559, 7741

Offset: 1

Giuseppe Coppoletta, Mar 19 2017

nonn

Primes which correspond to the short leg of an integral right triangle whose hypotenuse is part of a twin prime pair.

Each term p of the sequence must be part of a Pythagorean triple of the form {p, (p^2 - 1)/2, (p^2 + 1)/2} corresponding to {a(n), A284035(n) - 1, A284035(n)}.

			The prime p = 79 is in the sequence because (p^2-3)/2 = 3119 and (p^2+1)/2 = 3121 are twin primes. Remark that {79, 3120, 3121} is a Pythagorean triple.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A284035, A048161, A165635, A284036, A051642.

Select[Prime@ Range[10^3], Function[p, Times @@ Boole@ Map[PrimeQ[(p^2 + #)/2 ] &, {-3, 1}] == 1]] (* Michael De Vlieger, Mar 20 2017 *)
Select[Prime[Range[1000]],AllTrue[{(#^2-3)/2,(#^2+1)/2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 04 2017 *)

isok(p) = isprime(p) && isprime((p^2-3)/2) && isprime((p^2+1)/2); \\ Michel Marcus, Mar 31 2017

[p for p in prime_range(10000) if is_prime((p^2-3)//2) and is_prime((p^2+1)//2)]

A284037 Primes p such that p-1 and p+1 have two distinct prime factors.

Original entry on oeis.org

11, 13, 19, 23, 37, 47, 53, 73, 97, 107, 163, 193, 383, 487, 577, 863, 1153, 2593, 2917, 4373, 8747, 995327, 1492993, 1990657, 5308417, 28311553, 86093443, 6879707137, 1761205026817, 2348273369087, 5566277615617, 7421703487487, 21422803359743, 79164837199873

Offset: 1

Giuseppe Coppoletta, Mar 28 2017

nonn

Either p-1 or p+1 must be of the form 2^i * 3^j, since among three consecutive numbers exactly one is a multiple of 3. - Giovanni Resta, Mar 29 2017

Subsequence of A219528. See the previous comment. - Jason Yuen, Mar 08 2025

			7 is not a term because n + 1 = 8 has only one prime factor.
23 is a term because it is prime and n - 1 = 22 has two distinct prime factors (2, 11) and n + 1 = 24 has two distinct prime factors (2, 3).
43 is not a term because n - 1 = 42 has three distinct prime factors (2, 3, 7).

Robert Israel, Table of n, a(n) for n = 1..51 (all terms < 10^75)

Cf. A000668, A001221, A005105, A005109, A033845, A058383, A067386, A092506, A215504, A219528, A275598.

N:= 10^20: # To get all terms <= N
Res:= {}:
for i from 1 to ilog2(N) do
  for j from 1 to floor(log[3](N/2^i)) do
    q:= 2^i*3^j;
    if isprime(q-1) and nops(numtheory:-factorset((q-2)/2^padic:-ordp(q-2,2)))=1 then Res:= Res union {q-1} fi;
    if isprime(q+1) and nops(numtheory:-factorset((q+2)/2^padic:-ordp(q+2,2)))=1 then Res:= Res union {q+1} fi
od od:
sort(convert(Res,list)); # Robert Israel, Apr 16 2017

mx = 10^30; ok[t_] := PrimeQ[t] && PrimeNu[t-1]==2==PrimeNu[t+1]; Sort@ Reap[Do[ w = 2^i 3^j; Sow /@ Select[ w+ {1,-1}, ok], {i, Log2@ mx}, {j, 1, Log[3, mx/2^i]}]][[2, 1]] (* terms up to mx, Giovanni Resta, Mar 29 2017 *)

isok(n) = isprime(n) && (omega(n-1)==2) && (omega(n+1)==2); \\ Michel Marcus, Apr 17 2017

omega=sloane.A001221; [n for n in prime_range(10^6) if 2==omega(n-1)==omega(n+1)]

sorted([2^i*3^j+k for i in (1..40) for j in (1..20) for k in (-1,1) if is_prime(2^i*3^j+k) and sloane.A001221(2^i*3^j+2*k)==2])

A001221(a(n)) = 1 and A001221(a(n) - 1) = A001221(a(n) + 1) = 2.

a(33)-a(34) from Giovanni Resta, Mar 29 2017

A273286 Positive integers n such that n=p+q for some primes p,q with pi(p)*pi(q) = sigma(n).

Original entry on oeis.org

92, 130, 132, 136, 154, 270, 286, 384, 398, 456, 546, 608, 630, 636, 702, 934, 944, 2730, 4394, 4470, 4556, 5544, 12084, 14320, 17572, 22632, 27808, 27930, 31150, 31284, 32534, 36346, 41004, 41544, 42274, 56916, 58552, 61680, 66654, 74826, 86200

Offset: 1

Giuseppe Coppoletta, Jun 20 2016

nonn

Equivalently, integers n such that sigma(n) = i*j for some i,j with prime(i)+prime(j) = n. Each term is necessarily even, otherwise if n is odd n=2+q < sigma(2+q) = pi(2)*pi(q) = pi(q) < q which is absurd. Also p and q cannot be equal, otherwise sigma(2*p) = 3*(p+1) = pi(p)^2 with no solution.
Conjecture: the sequence is infinite and each term has only one decomposition into a sum of suitable primes p,q.
Using Rosser's theorem we can show that the primes p,q >= 19 and each of them can only occur for a finite number of terms n. - Robert Israel, Jun 30 2016

			92 = 19 + 73 with pi(19) * pi(73) = 8 * 21 = 168 = sigma(92).

Eric Weisstein's World of Mathematics, Rosser's Theorem

Cf. A000203, A000720, A272860, A272862.

N:= 10^6: # to use primes up to N
Primes:= select(isprime, [2,seq(i,i=3..N,2)]):
filter:= proc(n) local s,i,j;
  s:= numtheory:-sigma(n);
  for i in select(`>=`,numtheory:-divisors(s), ceil(sqrt(s))) minus {s} do
     if i > nops(Primes) then return FAIL
     elif Primes[i] + Primes[s/i] = n then return true fi
  od:
  false
end proc:
A:= NULL:
for n from 2 by 2 do
  v:= filter(n);
  if v = FAIL then break
  elif v then A:= A, n
  fi
od:
A; # Robert Israel, Jun 30 2016

Select[Range[10^3], Function[n, Length@ Select[Transpose@ {#, n - #} &@ Range[Floor[n/2]], And[Times @@ Boole@ PrimeQ@ {First@ #, Last@ #} == 1, DivisorSigma[1, First@ # + Last@ #] == PrimePi[First@ #] PrimePi[Last@ #]] &] > 0]] (* Michael De Vlieger, Jun 30 2016 *)

is(n) = if(n%2==1, return(0), my(x=n-1, y=1); while(x > y, if(ispseudoprime(x) && ispseudoprime(y) && sigma(x+y)==primepi(x)*primepi(y), return(1)); x--; y++); return(0)) \\ Felix Fröhlich, Jun 28 2016

is(n) = my( d=divisors(sigma(n))); for(i=1,ceil(#d/2), if(prime(d[i]) + prime(d[#d + 1-i]) == n, return(1))); return(0) \\ David A. Corneth, Jun 30 2016

def sol(n): return [j for j in divisors(sigma(n)) if j^2<= sigma(n) and is_prime(n-nth_prime(j)) and j * prime_pi(n-nth_prime(j))==sigma(n)]
v=[n for n in range(2,100000,2) if sol(n)]
print('list_n =',v)
w=[sigma(n) for n in v]; print('list_sigma(n) =',w)
list_pi(p)=flatten([sol(n) for n in range(2,100000,2) if sol(n)])
print('list_pi(p) =',list_pi(p))
list_pi(q)=[w[n]/list_pi[n] for n in range(len(v))]
print('list_pi(q) =',list_pi(q))

Integers n such that sigma(n) = pi(q) * pi(n-q) for some prime q.

A272862 Positive integers j such that prime(i) + prime(j) = i*j for some i <= j.

Original entry on oeis.org

4, 6, 8, 24, 29, 30, 164, 165, 166, 1051, 2624, 2638, 2650, 2670, 2674, 2676, 40027, 40028, 40112, 251701, 251703, 251706, 251751, 637144, 637202, 637216, 637220, 1617162, 1617165, 4124694, 10553383, 10553408, 10553464, 10553533, 10553535, 10553839, 69709686

Offset: 1

Giuseppe Coppoletta, Jul 25 2016

nonn

Also pi(q) for primes q verifying p+q = pi(p)*pi(q) for some prime p <= q.

The list of products i*j gives A272860. See also comments there.

			8 is a term as prime(3) + prime(8) = 3*8.

Giuseppe Coppoletta, Table of n, a(n) for n = 1..43

Cf. A272860, A272861, A000720, A000040.

Select[Range[3000], Function[j, Total@ Boole@ Map[Prime@ # + Prime@ j == # j &, Range@ j] > 0]] (* Michael De Vlieger, Jul 28 2016 *)

is(n) = for(i=1, n, if(prime(i)+prime(n)==i*n, return(1))); return(0) \\ Felix Fröhlich, Jul 27 2016

is(n,p=prime(n))=my(i); forprime(q=2,p, if(i++*n==p+q, return(1))); 0
v=List(); n=0; forprime(p=2,1e6, if(is(n++,p), listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Jul 28 2016

def sol(n):
    if n<5: a=n
    else: a=exp(n+1)/(n+1)
    b=(n-1)/n^2*exp(n^2/(n-1.1))
    return [j for j in range(a,b) if is_prime(n*j-nth_prime(n)) and prime_pi(n*j-nth_prime(n))==j]
flatten([sol(i) for i in (1..15) if len(sol(i))>0]) #

A272860 Sums of two primes (in increasing order) when equal to the product of their prime-counting functions.

Original entry on oeis.org

12, 18, 24, 96, 116, 120, 984, 990, 996, 8408, 23616, 23742, 23850, 24030, 24066, 24084, 480324, 480336, 481344, 3523814, 3523842, 3523884, 3524514, 9557160, 9558030, 9558240, 9558300, 25874592, 25874640, 70119798, 189960894, 189961344, 189962352, 189963594, 189963630, 189969102

Offset: 1

Giuseppe Coppoletta, Jun 19 2016

nonn

Each term is necessarily even and 3 < p < q in the formula n = p+q = pi(p)*pi(q). Indeed, assuming p<=q, if p=2 then n = 2+q = pi(2)*pi(q) = pi(q) < q. Inequality p > 3 easily follows from prime(k) > k*log(k) and if p=q then 2*p = pi(p)^2 with no solution.
Primes p,q can only occur for a finite number of terms n (see comments in A273286).
Conjecture: the sequence is infinite and each term has only one decomposition into a sum of suitable primes p,q.
From David A. Corneth, Jun 28 2016: (Start)
Pi(p) and pi(q) seem dependent on each other. Below is a small list of pi(p), the least corresponding pi(q) and the largest corresponding pi(q). If a value of pi(p) isn't listed, no terms are formed with it.
3, 4, 8
4, 24, 30
6, 164, 166
8, 1051, 1051
9, 2624, 2676
12, 40027, 40112
Can these bounds on pi(q) be expressed in terms of pi(p)? (End)

			12 is a term because 12 = 5 + 7 = pi(5) * pi(7).

Giuseppe Coppoletta, Table of n, a(n) for n = 1..43
Eric Weisstein's World of Mathematics, Rosser's Theorem
Pierre Dusart, Estimates of some functions over primes without R.H., arXiv:1002.0442 [math.NT], 2010.

Cf. A272861, A272862, A273286, A000040, A000720.

Select[Range[10^3], Function[n, MemberQ[Times @@ # & /@ PrimePi@ Select[Transpose@ {#, n - #} &@ Range[Floor[n/2]], Times @@ Boole@ PrimeQ@ {First@ #, Last@ #} == 1 &], n]]] (* Michael De Vlieger, Jun 29 2016 *)

def sol(n):
    return [k for k in divisors(n) if k^2<= n and is_prime(n-nth_prime(k)) and k*prime_pi(n-nth_prime(k))==n]
N=25000
v=[n for n in range(2,N,2) if len(sol(n))>0]
print('A272862 =',v)
list_pi=flatten([sol(n) for n in range(2,N,2) if sol(n)])
print('list_pi(p) =',list_pi)

Numbers n = p+q = pi(p)*pi(q) for some primes p and q.
Equivalently, n = i*j = prime(i) + prime(j) for some i,j.
A272862 gives the corresponding terms pi(q) (with q>p). The terms pi(p) are given by A272860 / A272862

More terms from David A. Corneth, Jun 28 2016

A272861 Sum of two integers when equal to the product of their prime-counting functions.

Original entry on oeis.org

12, 16, 18, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 116, 120, 280, 310, 325, 330, 942, 948, 954, 960, 966, 972, 984, 990, 996, 2968, 3003, 8224, 8232, 8240, 8248, 8280, 8288, 8304, 8312, 8360, 8408, 23499, 23508, 23589

Offset: 1

Giuseppe Coppoletta, Jun 18 2016

nonn

The sums are listed in increasing order. The only term with equal addends is a(2)= 16 = 8 + 8 = pi(8)^2, Indeed j=8 is the only solution to pi(j)^2 = 2*j, which is easily seen using pi(j) > j/log(j) for j>16.

			32 is a term because 32 = 10 + 22 = 4 * 8 = pi(10) * pi(22).

Giovanni Resta, Table of n, a(n) for n = 1..320 (terms < 4*10^9)
Eric Weisstein's World of Mathematics, Prime Counting Function

Cf. A272860, A273286, A000720.

nn = 10^3; Select[Range@ nn, Function[k, IntegerQ@ SelectFirst[Range@ nn, k == PrimePi[#] PrimePi[k - #] &]]] (* Version 10, or *)
Select[Range[10^3], Function[n, Length@ Select[Transpose@ {#, n - #} &@ Range[Floor[n/2]], n == PrimePi[First@ #] PrimePi[Last@ #] &] > 0]] (* Michael De Vlieger, Jun 30 2016 *)

def g(n): return [k for k in (3..n/2) if n==prime_pi(k)*prime_pi(n-k)]
[n for n in range(2,1000) if len(g(n))>0]

Positive integers n such that n = pi(j) * pi(n-j) for some j.

a(50)-a(53) from Giovanni Resta, Jun 29 2016

cp's OEIS Frontend

User: Giuseppe Coppoletta

A288908 Primes p whose distance from next prime and from previous prime is greater than log(p).

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A288907 Primes p whose distance from the next prime and from the previous prime is less than log(p).

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A284036 Positive integers n such that (n^2 - 3)/2 and (n^2 + 1)/2 are twin primes.

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A284035 Upper twin primes which correspond to the hypotenuse in a Pythagorean triple whose short leg is also a prime.

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A284034 Primes p such that (p^2 - 3)/2 and (p^2 + 1)/2 are twin primes.

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Mathematica

PARI

Sage

A284037 Primes p such that p-1 and p+1 have two distinct prime factors.

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Maple

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