A079147 -id:A079147 - cp's OEIS frontend

A079148 Primes p such that p-1 has at most 2 prime factors, counted with multiplicity; i.e., primes p such that bigomega(p-1) = A001222(p-1) <= 2.

2, 3, 5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459, 2579, 2819, 2879

Offset: 1

Cino Hilliard, Dec 27 2002

Sum of reciprocals ~ 1.477.

			83 is in the sequence because 83 - 1 = 2*41 has 2 prime factors.

Except for 2 and 3, this is identical to A005385.

Cf. A079147, A079149, A079151.

Select[Prime[Range[500]],PrimeOmega[#-1]<3&] (* Harvey P. Dale, May 17 2011 *)

s(n) = {sr=0; forprime(x=2,n, if(bigomega(x-1) < 3, print1(x" "); sr+=1.0/x; ); ); print(); print(sr); } \\ Lists primes p<=n such that p-1 has at most 2 prime factors.

A079149 Primes p such that either p-1 or p+1 has at most 2 prime factors, counted with multiplicity; i.e., primes p such that either bigomega(p-1) <= 2 or bigomega(p+1) <= 2, where bigomega(n) = A001222(n).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 23, 37, 47, 59, 61, 73, 83, 107, 157, 167, 179, 193, 227, 263, 277, 313, 347, 359, 383, 397, 421, 457, 467, 479, 503, 541, 563, 587, 613, 661, 673, 719, 733, 757, 839, 863, 877, 887, 983, 997, 1019, 1093, 1153, 1187, 1201, 1213, 1237

Offset: 1

Cino Hilliard, Dec 27 2002

There are only 2 primes such that both p-1 and p+1 have at most 2 prime factors - 3 and 5. Proof: If p > 5 then whichever of p-1 and p+1 is divisible by 4 has at least 3 prime factors.

Primes which are not the sum of two consecutive composite numbers. - Juri-Stepan Gerasimov, Nov 15 2009

Union of A079147 and A079148. Cf. A060254, A079152.

Select[Prime[Range[500]],MemberQ[PrimeOmega[{#-1,#+1}],2]&] (* Harvey P. Dale, Sep 04 2011 *)

s(n) = {sr=0; ct=0; forprime(x=2,n, if(bigomega(x-1) < 3 || bigomega(x+1) < 3, print1(x" "); sr+=1.0/x; ct+=1; ); ); print(); print(ct" "sr); } \\ Lists primes p<=n such that p+-1 has at most 2 prime factors.

A079150 Primes p such that p+1 has at most 3 prime factors, counted with multiplicity; i.e., primes p such that bigomega(p+1) = A001222(p+1) <= 3.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 29, 37, 41, 43, 61, 67, 73, 97, 101, 109, 113, 137, 157, 163, 173, 181, 193, 211, 229, 241, 257, 277, 281, 283, 313, 317, 331, 337, 353, 373, 397, 401, 409, 421, 433, 457, 523, 541, 547, 577, 601, 613, 617, 641, 653, 661, 673, 677

Offset: 1

Cino Hilliard, Dec 27 2002

			173 is in the sequence because 173+1 = 2*3*29 has 3 prime factors.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A001222, A079147, A079151, A079152, A079153.

Select[Prime[Range[200]],PrimeOmega[#+1]<4&] (* Harvey P. Dale, Feb 02 2012 *)

s(n) = {sr=0; ct=0; forprime(x=2,n, if(bigomega(x+1) < 4, print1(x" "); sr+=1.0/x; ct+=1; ); ); print(); print(ct" "sr); } \\ Lists primes p<=n such that p+1 has at most 3 prime factors.

A265396 Numerator of A265395(n)/A265394(n), record values / record positions in A265388.

Original entry on oeis.org

0, 3, 5, 11, 13, 29, 37, 61, 73, 157, 193, 277, 313, 397, 421, 457, 541, 613, 661, 673, 733, 757, 877, 997, 1093, 1153, 1201, 1213, 1237, 1321, 1381, 1453, 1621, 1657, 1753, 1873, 1933, 1993, 2017, 2137, 2341, 2473, 2557, 2593, 2797, 2857, 2917, 3061, 3217, 3253, 3313, 3517, 3733, 4021, 4057, 4177, 4261, 4273, 4357, 4441

Offset: 1

Antti Karttunen, Dec 11 2015

nonn

It seems that n=4 is the only case where A265395(n)/A265394(n) is not integral (as 33/6 = 11/2), thus for other n, a(n) actually seems to give the value of A265395(n)/A265394(n).

Antti Karttunen, Table of n, a(n) for n = 1..192

Cf. A265388, A265394, A265395.

For many terms coincides with A005383, A079147, A113733 and A256072.

(define (A265396 n) (numerator (/ (A265395 n) (A265394 n))))

A256072 Primes that cannot be represented as x*y + x + y, where x >= y > 1.

Original entry on oeis.org

2, 3, 5, 7, 13, 37, 61, 73, 157, 193, 277, 313, 397, 421, 457, 541, 613, 661, 673, 733, 757, 877, 997, 1093, 1153, 1201, 1213, 1237, 1321, 1381, 1453, 1621, 1657, 1753, 1873, 1933, 1993, 2017, 2137, 2341, 2473, 2557, 2593, 2797, 2857, 2917, 3061, 3217, 3253, 3313

Offset: 1

Alex Ratushnyak, Mar 14 2015

nonn

Primes in A254636.

Cf. A000040, A254636, A005383, A079147.

v=[];for(m=2,500,for(k=m,500,if(isprime(P=k*m+k+m),v=concat(v,P))));v=vecsort(v,,8);forprime(p=1,2000,if(!vecsearch(v,p),print1(p,", "))) \\ Derek Orr, Mar 21 2015

import sympy
from sympy import isprime
TOP = 10000
a = [0]*TOP
for y in range(2, TOP//2):
  for x in range(y, TOP//2):
    k = x*y + x + y
    if k>=TOP: break
    a[k]+=1
print([n for n in range(TOP) if a[n]==0 and isprime(n)])

{2, 7} UNION A005383 = {7} UNION A079147. - Chai Wah Wu, Oct 15 2024

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A079148 Primes p such that p-1 has at most 2 prime factors, counted with multiplicity; i.e., primes p such that bigomega(p-1) = A001222(p-1) <= 2.

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A079149 Primes p such that either p-1 or p+1 has at most 2 prime factors, counted with multiplicity; i.e., primes p such that either bigomega(p-1) <= 2 or bigomega(p+1) <= 2, where bigomega(n) = A001222(n).

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A079150 Primes p such that p+1 has at most 3 prime factors, counted with multiplicity; i.e., primes p such that bigomega(p+1) = A001222(p+1) <= 3.

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A265396 Numerator of A265395(n)/A265394(n), record values / record positions in A265388.

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A256072 Primes that cannot be represented as x*y + x + y, where x >= y > 1.

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