A079151 -id:A079151 - 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.

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.

A079152 Primes p such that either p-1 or p+1 has at most 3 prime factors, counted with multiplicity; i.e., primes p such that either bigomega(p-1) <= 3 or bigomega(p+1) <= 3, where bigomega(n) = A001222(n).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 157, 163, 167, 173, 179, 181, 191, 193, 211, 223, 227, 229, 239, 241, 257, 263, 269, 277, 281, 283, 293, 311, 313, 317, 331

Offset: 1

Cino Hilliard, Dec 27 2002

nonn

Up to 83, this is the sequence of prime numbers A000040. 89 is not in the sequence because both 89-1 = 88 = 2*2*2*11 and 89+1 = 90 = 2*3*3*5 have 4 prime factors.

			97 is in the sequence because 97+1 = 98 = 2*7*7 has 3 prime factors.

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

Union of A079150 and A079151. Cf. A079149, A079153.

bg:=func; [2] cat [p: p in PrimesInInterval(3,340)| bg(p-1) le 3 or bg(p+1) le 3]; // Marius A. Burtea, Jan 16 2020

Select[Prime /@ Range[70], PrimeOmega[# - 1] <= 3 || PrimeOmega[# + 1] <= 3 & ] (* Jean-François Alcover, Jul 02 2013 *)

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

A079153 Primes p such that both p-1 and p+1 have at most 3 prime factors, counted with multiplicity; i.e., primes p such that bigomega(p-1) <= 3 and bigomega(p+1) <= 3, where bigomega(n) = A001222(n).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 29, 43, 67, 173, 283, 317, 653, 787, 907, 1867, 2083, 2693, 2803, 3413, 3643, 3677, 4253, 4363, 4723, 5443, 5717, 6197, 6547, 6653, 8563, 8573, 9067, 9187, 9403, 9643, 10733, 11443, 11587, 12163, 12917, 13997, 14107, 14683, 15187

Offset: 1

Cino Hilliard, Dec 27 2002

Sum of reciprocals ~ 1.495. There are 3528 primes of this kind <= 10^7.

From a(7) = 19 onward, this sequence is identical to A063644(n-6). - Robin Saunders, Sep 22 2014

			907 is in the sequence because both 907-1 = 2*3*151 and 907+1 = 2*2*227 have 3 prime factors.

Vincenzo Librandi, Table of n, a(n) for n = 1..1100

Intersection of A079150 and A079151. Cf. A079152.

filter:= p -> isprime(p) and numtheory:-bigomega(p-1) <= 3 and numtheory:-bigomega(p+1) <= 3:
select(filter, [2,seq(2*i+1, i=1..10^4)]); # Robert Israel, Nov 11 2014

Select[Prime[Range[2000]],Max[PrimeOmega[#+{1,-1}]]<4&] (* Harvey P. Dale, Oct 07 2015 *)

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

A301590 Primes p such that there are no other solutions to A023900(x) = A023900(p) than a power of p.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 19, 23, 29, 31, 43, 47, 53, 59, 67, 71, 79, 83, 101, 103, 107, 127, 131, 137, 139, 149, 151, 163, 167, 173, 179, 191, 197, 199, 211, 223, 227, 229, 239, 251, 257, 263, 269, 271, 283, 293, 307, 311, 317, 331, 347, 359, 367, 373, 379, 383, 389

Offset: 1

Michel Marcus, Mar 24 2018

nonn

In the definition, A023900(p) = 1-p. One has sign(A023900(n)) = (-1)^A001221(n), so a different solution x can only exist if x has at least 3 distinct prime factors. The smallest number of the form p*q*r such that (p-1)*(q-1)*(r-1) = P-1 for primes p, q, r, P is 2*3*7 = 42, eliminating P = 13 = A301591(1) from this sequence. This is the case whenever (P+1)/2 = p > 3 is a prime (in A005382), whence P-1 = (2-1)*(3-1)*(p-1), which eliminates all P > 5 in A005383 from this sequence. - M. F. Hasler, Aug 14 2021

			2 is a term because there are no other solutions to A023900(x) = A023900(2) = -1 than other powers of 2.
13 is not a term because A023900(42) = -12 = A023900(13). Similarly, no P > 5 in A005383 is a term because A023900(P) = 1-P = (1-2)*(1-3)*(1-p) = A023900(2*3*p) with p = (P+1)/2. - _M. F. Hasler_, Aug 14 2021

M. F. Hasler, Table of n, a(n) for n = 1..10000, Sep 01 2021

Cf. A023900, A301374.

Complement (within the primes) of A301591, which has A005383 \ {3, 5} as a subsequence. Appears to have A079151 \ {13} as subsequence.

f(n) = sumdivmult(n, d, d*moebius(d)); /* A023900 */
isok(p, vp) = {for (k=p+1, p^2-1, if (f(k) == vp, return (0)); ); return (1); }
lista(nn) = {forprime(p=2, nn, vp = f(p); if (isok(p, vp), print1(p, ", ")); ); }

select( {is_A301590(p)=!forcomposite(k=p+1, p^2-1, A023900(k)!=1-p|| return)&& isprime(p)}, primes([1,399])) \\ M. F. Hasler, Aug 14 2021

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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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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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A079152 Primes p such that either p-1 or p+1 has at most 3 prime factors, counted with multiplicity; i.e., primes p such that either bigomega(p-1) <= 3 or bigomega(p+1) <= 3, where bigomega(n) = A001222(n).

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A079153 Primes p such that both p-1 and p+1 have at most 3 prime factors, counted with multiplicity; i.e., primes p such that bigomega(p-1) <= 3 and bigomega(p+1) <= 3, where bigomega(n) = A001222(n).

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A301590 Primes p such that there are no other solutions to A023900(x) = A023900(p) than a power of p.

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