Alexandre Herrera - cp's OEIS frontend

A383672 Squarefree numbers k such that k^2+1 is not squarefree.

7, 38, 41, 43, 57, 70, 82, 93, 107, 118, 143, 157, 182, 193, 218, 239, 251, 257, 282, 293, 307, 318, 327, 357, 382, 393, 407, 418, 437, 443, 457, 482, 493, 515, 518, 543, 557, 577, 582, 593, 606, 607, 618, 643, 682, 707, 718, 743, 746, 757, 782, 793, 807, 818, 829, 843, 857, 893

Offset: 1

Alexandre Herrera, May 04 2025

nonn

			38 = 2*19 is squarefree but 38*38 + 1 = 1445 = 5*17*17 is not squarefree.

Intersection of A005117 and A049532.
Includes A141932 and A141941.
Cf. A080666, A224718.

filter:= proc(n) numtheory:-issqrfree(n) and not numtheory:-issqrfree(n^2+1) end proc:
select(filter, [$1..1000]); # Robert Israel, May 04 2025

Select[Range[900],SquareFreeQ[#] && !SquareFreeQ[#^2+1] &] (* Stefano Spezia, May 04 2025 *)

isok(k) = issquarefree(k) && !issquarefree(k^2+1); \\ Michel Marcus, May 04 2025

from sympy import factorint
def is_squarefree(n):
    return all(exponent == 1 for exponent in factorint(n).values())
print([a for a in range(1,900) if is_squarefree(a) and not(is_squarefree(a*a + 1))])

A379407 a(n) is the smallest semiprime > primorial(n).

Original entry on oeis.org

4, 9, 33, 213, 2315, 30031, 510515, 9699691, 223092871, 6469693233, 200560490134, 7420738134814, 304250263527221, 13082761331670031, 614889782588491414, 32589158477190044737, 1922760350154212639074, 117288381359406970983271, 7858321551080267055879091

Offset: 1

Alexandre Herrera, Dec 22 2024

nonn

			primorial(2) = 2*3 = 6 so a(2) = 9 because 9 = 3*3 is next semiprime > 6.

Cf. A001358, A002110, A089539, A106325.

a[n_] := Module[{m = Times @@ Prime[Range[n]] + 1}, While[PrimeOmega[m] != 2, m++]; m]; Array[a, 20] (* Amiram Eldar, Jan 01 2025 *)

import sympy
def ok(n): return sum(sympy.factorint(n).values()) == 2
primorial = 1
l = []
for i in range(1,20):
    primorial *= sympy.prime(i)
    next_sp = primorial + 1
    while not(ok(next_sp)):
        next_sp += 1
    l.append(next_sp)
print(l)

a(n) = A106325(A002110(n)+1).

A379036 Indices of zeros in binary concatenation of primes.

Original entry on oeis.org

1, 5, 11, 16, 19, 20, 21, 24, 25, 29, 36, 44, 45, 47, 50, 52, 53, 56, 58, 62, 69, 71, 76, 83, 86, 87, 88, 89, 93, 94, 95, 100, 101, 103, 104, 107, 108, 114, 116, 117, 121, 124, 125, 129, 130, 131, 132, 136, 137, 139, 143, 144, 150, 152, 157, 160, 165, 166, 167

Offset: 1

Alexandre Herrera, Dec 14 2024

The initial bit is labeled as bit 0.

			The primes, their binary expansions, and positions of successive zero bits, begin
   prime    2  3   5   7   11 ...
   binary  10 11 101 111 1011 ...
   zeros    ^     ^       ^
   a(n) =   1     5      11   ...

Cf. A003607, A191232.

seq[lim_] := -1 + Position[Flatten@ IntegerDigits[Prime[Range[lim]], 2], 0] // Flatten; seq[30] (* Amiram Eldar, Dec 31 2024 *)

import sympy
l = []
bin_primes = ""
for i in range(1,27):
    bin_primes += bin(sympy.prime(i))[2:]
for i in range(len(bin_primes)):
    if bin_primes[i] == '0':
        l.append(i)
print(l)

A373499 a(n) = Sum_{i=1..n-1} binomial(prime(n),prime(i)).

Original entry on oeis.org

0, 3, 20, 77, 1012, 3445, 41208, 166041, 2886776, 176545765, 707922076, 44154219471, 628182427994, 2318296787282, 32073418630027, 2032575090770969, 140272398486718041, 558946109921421607, 34092092791668401412, 554618378100523846567, 2286090868263899514704

Offset: 1

Alexandre Herrera, Jun 06 2024

nonn

			For n = 3, a(3) = binomial(prime(3),prime(1)) + binomial(prime(3),prime(2)) = binomial(5,2) + binomial(5,3) = 10 + 10 = 20.

Cf. A000040.

Table[Sum[Binomial[Prime[n], Prime[i]], {i, n-1}], {n, 25}] (* Paolo Xausa, Jun 29 2024 *)

a(n) = sum(i=1, n-1, binomial(prime(n), prime(i))); \\ Michel Marcus, Jun 25 2024

from sympy import binomial
from sympy import prime
def a(n): return sum(binomial(prime(n),prime(i)) for i in range(1,n))
print([a(n) for n in range(1,22)])

a(1) = 0, a(n) = Sum_{i=1..n-1} binomial(A000040(n),A000040(i)).

A373299 Numbers prime(k) such that prime(k) - prime(k-1) = prime(k+2) - prime(k+1).

Original entry on oeis.org

7, 11, 13, 17, 29, 41, 59, 79, 101, 103, 107, 113, 139, 163, 181, 193, 227, 257, 269, 311, 359, 379, 397, 419, 421, 439, 461, 487, 491, 547, 569, 577, 599, 691, 701, 709, 761, 811, 823, 857, 863, 881, 887, 919, 983, 1021, 1049, 1051, 1091, 1109, 1163

Offset: 1

Alexandre Herrera, May 31 2024

nonn

			7 is in the list because the prime previous to 7 is 5 and the next primes after 7 are 11 and 13, so we have 7 - 5 = 13 - 11 = 2.

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

Cf. A001223, A022885, A151800, A263674.

P:= select(isprime,[seq(i,i=3..10^4,2)]):
G:= P[2..-1]-P[1..-2]: nG:= nops(G):
J:= select(t -> G[t-1]=G[t+1],[$2..nG-1]):
P[J]; # Robert Israel, May 31 2024

Select[Partition[Prime[Range[200]], 4, 1], #[[2]] - #[[1]] == #[[4]] - #[[3]] &][[;; , 2]] (* Amiram Eldar, May 31 2024 *)

from sympy import prime
def ok(k):
    return prime(k)-prime(k-1) == prime(k+2)-prime(k+1)
print([prime(k) for k in range(2,200) if ok(k)])

from sympy import nextprime
from itertools import islice
def agen(): # generator of terms
    p, q, r, s = [2, 3, 5, 7]
    while True:
        if q-p == s-r: yield q
        p, q, r, s = q, r, s, nextprime(s)
print(list(islice(agen(), 60))) # Michael S. Branicky, May 31 2024

a(n) = A151800(A022885(n)).

A371981 Number of primes between two successive Sophie Germain primes, with Sophie Germain primes not themselves included in the count.

Original entry on oeis.org

0, 0, 1, 3, 0, 2, 2, 6, 0, 5, 1, 7, 0, 1, 7, 0, 1, 5, 1, 9, 8, 1, 2, 7, 2, 10, 7, 2, 0, 3, 3, 3, 2, 4, 15, 5, 7, 0, 1, 2, 8, 14, 0, 7, 13, 4, 1, 3, 4, 0, 5, 3, 1, 17, 9, 9, 0, 2, 3, 5, 4, 1, 0, 7, 2, 14, 7, 2, 6, 0, 6, 7, 0, 18, 0, 6, 1, 7, 9, 3, 2, 0, 5, 28, 5, 3, 3, 2, 1, 5, 6, 7, 3, 15, 2

Offset: 1

Alexandre Herrera, Apr 15 2024

nonn

Number of primes between A005384(n) and A005384(n+1).

			a(4) = 3 because there are 3 primes between 11 and 23: 13, 17 and 19.

Cf. A000720, A005384.

-1 + Subtract @@ Map[PrimePi, {Last[#], First[#]}] & /@ Partition[Select[Prime[Range[500]], PrimeQ[2 # + 1] &], 2, 1] (* Michael De Vlieger, Apr 19 2024 *)

lista(nn) = my(vp = select(p->isprime(2*p+1), primes(nn)), wp = apply(primepi, vp)); vector(#wp-1, k, wp[k+1]-wp[k]-1); \\ Michel Marcus, May 21 2024

from sympy import isprime
l = []
s = 0
for i in range(3,3800):
    if isprime(i):
        if isprime(2*i + 1):
            l.append(s)
            s = 0
        else:
            s += 1
print(l)

a(n) = A000720(A005384(n+1)) - A000720(A005384(n)) - 1. - Michael De Vlieger, Apr 19 2024

A372730 Number of primes <= A005867(n).

Original entry on oeis.org

0, 0, 1, 4, 15, 92, 757, 8899, 125261, 2232782, 51902553, 1327191561, 41351244491, 1452937916515, 54332144724834, 2246960940148460, 105818707666943651, 5595105626396158784, 308241771351984486729, 18772520681296116861073

Offset: 0

Alexandre Herrera, May 11 2024

			a(3) = 4 because there are 4 primes less than A005867(3) = 8: 2, 3, 5 and 7.

Cf. A000720, A005867, A000849.

A372730(n) = primepi(prod(k=1, n, prime(k)-1)); \\ Antti Karttunen, May 22 2024

from sympy import prime,primepi
p = 1
l = [0]
for i in range(1,12):
    p *= (prime(i) - 1)
    l.append(primepi(p))
print(l)

a(n) = A000720(A005867(n)).

a(9)-a(11) from Antti Karttunen, May 22 2024
a(12)-a(16) from Amiram Eldar, May 22 2024
a(17)-a(18) from Chai Wah Wu, Jun 04 2024
a(19) from Chai Wah Wu, Jun 05 2024

A372113 Numbers k for which (k-1)/2 and 2*k+1 are both primes.

Original entry on oeis.org

5, 11, 15, 23, 35, 39, 63, 75, 83, 95, 119, 135, 179, 215, 219, 299, 303, 315, 359, 363, 455, 459, 483, 515, 543, 615, 663, 699, 719, 735, 779, 803, 879, 915, 923, 935, 975, 999, 1019, 1043, 1143, 1155, 1175, 1199, 1295, 1323, 1355, 1383, 1439, 1539, 1595, 1659, 1679, 1755, 1763, 1815, 1859, 1883

Offset: 1

Alexandre Herrera, Apr 19 2024

nonn

Intersection of A072055 and A104635.

			5 is a term because (5-1)/2 = 2 is prime and 2*5+1 = 11 is prime.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A072055, A104635.

Cf. A023213, A126330.

Select[Range[1, 2000, 2], AllTrue[{(# - 1)/2, 2 # + 1}, PrimeQ] &] (* Michael De Vlieger, Apr 19 2024 *)

from sympy import isprime
def a(n): return n%2 == 1 and isprime((n-1)>>1) and isprime(2*n+1)
print([n for n in range(2, 1900) if a(n)])

a(n) = 2*A023213(n) + 1.

a(n) = (A126330(n)-1)/2.

A371980 Sophie Germain primes p such that 4*p + 3 is a composite number.

Original entry on oeis.org

3, 23, 29, 53, 83, 113, 131, 173, 191, 233, 239, 251, 281, 293, 419, 431, 443, 491, 593, 641, 653, 659, 683, 743, 761, 809, 911, 953, 1013, 1049, 1103, 1223, 1289, 1439, 1499, 1559, 1583, 1601, 1733, 1973, 2003, 2039, 2063, 2069, 2129, 2141, 2273, 2339, 2351, 2393, 2399, 2543, 2549, 2693, 2741, 2753

Offset: 1

Alexandre Herrera, Apr 15 2024

nonn

			a(1) = 3 is prime and 2*3 + 1 = 7 also but not 4*3 + 3 = 15.

Cf. A005384.

Select[Prime[Range[410]], And[PrimeQ[2 # + 1], CompositeQ[4 # + 3]] &] (* Michael De Vlieger, Apr 19 2024 *)

import sympy as sp
l = []
for i in range(2,2800):
    if sp.isprime(i) and sp.isprime(2*i + 1) and not(sp.isprime(4*i + 3)):
        l.append(i)
print(l)

A371035 a(n) = A086330(prime(n)).

Original entry on oeis.org

0, 2, 7, 18, 43, 73, 113, 159, 203, 334, 496, 706, 863, 874, 1097, 1124, 1560, 2033, 2073, 2409, 2462, 3336, 3345, 3634, 3958, 4657, 5198, 5284, 5186, 6096, 7801, 8594, 9270, 9167, 10659, 10578, 12375, 12227, 13221, 13769, 15958, 16458, 18820, 17919, 18722

Offset: 1

Alexandre Herrera, Apr 10 2024

The sequence sometimes decreases, as for example at a(29) = 5186 < 5284 = a(28).

			For n = 3, a(n) = A086330(prime(3)) = A086330(5) = (2! mod 5) + (3! mod 5) + (4! mod 5) = 2 + 1 + 4 = 7.

Cf. A086330, A000040.

a(n) = my(p=prime(n)); sum(m=2, p, m! % p); \\ Michel Marcus, Apr 11 2024

from sympy import isprime
l = []
for i in range(2,185):
    if isprime(i):
        sum = 0
        reminder = 1
        for j in range(2, i):
            reminder = (reminder * j) % i
            sum += reminder
        l.append(sum)
print(l)

from sympy import prime
def A371035(n):
    a, c, p = 0, 1, prime(n)
    for m in range(2,p):
        c = c*m%p
        a += c
    return a # Chai Wah Wu, Apr 16 2024

cp's OEIS Frontend

User: Alexandre Herrera

A383672 Squarefree numbers k such that k^2+1 is not squarefree.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Python

A379407 a(n) is the smallest semiprime > primorial(n).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Python

Formula

A379036 Indices of zeros in binary concatenation of primes.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python

A373499 a(n) = Sum_{i=1..n-1} binomial(prime(n),prime(i)).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A373299 Numbers prime(k) such that prime(k) - prime(k-1) = prime(k+2) - prime(k+1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Python

Formula

A371981 Number of primes between two successive Sophie Germain primes, with Sophie Germain primes not themselves included in the count.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A372730 Number of primes <= A005867(n).

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Python

Formula

Extensions