A301590 -id:A301590 - cp's OEIS frontend

A079151 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.

2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 43, 47, 53, 59, 67, 71, 79, 83, 103, 107, 131, 139, 149, 167, 173, 179, 191, 223, 227, 239, 263, 269, 283, 293, 311, 317, 347, 359, 367, 383, 389, 419, 431, 439, 443, 467, 479, 499, 503, 509, 557, 563, 587, 599, 607, 619, 643

Offset: 1

Cino Hilliard, Dec 27 2002

Can it be proved that this is a subsequence of A301590, except for a(5) = 13? (Checked up to A301591(10^4) = 427421.) - M. F. Hasler, Aug 14 2021

			149 is in the sequence because 149 - 1 = 2*2*37 has 3 prime factors.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A079148, A079150, A079152, A079153.

s[n_] := Reap[sr = 0; ct = 0; For[x = 2, x <= n, x = NextPrime[x], If[PrimeOmega[x - 1] < 4, Sow[x]; sr += 1.0/x; ct += 1]]][[2, 1]]; s[700] (* Jean-François Alcover, Jun 08 2013, translated and adapted from Pari *)
Select[Prime[Range[120]],PrimeOmega[#-1]<4&] (* Harvey P. Dale, Oct 02 2017 *)

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.

list(lim)=my(v=List([2,3]),t); forprime(p=2,(lim-1)\2, if(isprime(t=2*p+1), listput(v,t))); forprime(p=2,(lim-1)\4, forprime(q=2,min(p,(lim-1)\2\p), if(isprime(t=2*p*q+1), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Mar 06 2017

Typos in definition corrected by Harvey P. Dale, Oct 02 2017

A301591 Primes p that have other solutions x to A023900(x) = A023900(p) than a power of p.

Original entry on oeis.org

13, 37, 41, 61, 73, 89, 97, 109, 113, 157, 181, 193, 233, 241, 277, 281, 313, 337, 349, 353, 397, 401, 409, 421, 433, 449, 457, 461, 521, 541, 577, 593, 601, 613, 617, 641, 661, 673, 701, 733, 757, 761, 769, 821, 829, 877, 881, 929, 937, 953, 997, 1009, 1013, 1021, 1033, 1049

Offset: 1

Michel Marcus, Mar 24 2018

nonn

Contains A005383 \ {3, 5} as a subsequence, since if (p+1)/2 = q > 3 is prime, then A023900(2*3*q) = (1-2)*(1-3)*(1-q) = 1-p = A023900(p). - M. F. Hasler, Aug 14 2021

			13 is a term because A023900(42) = A023900(13), where 42 is not a power of 13.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001055, A023900, A301374.
Complement of A301590.
A005383 \ {3,5} is a subsequence.

f(n) = sumdivmult(n, d, d*moebius(d)); /* This is 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, ", ")); ); }

A301374 Values of A023900 which occur only at indices which are powers of a prime.

Original entry on oeis.org

-1, -2, -4, -6, -10, -16, -18, -22, -28, -30, -42, -46, -52, -58, -66, -70, -78, -82, -100, -102, -106, -126, -130, -136, -138, -148, -150, -162, -166, -172, -178, -190, -196, -198, -210, -222, -226, -228, -238, -250, -256, -262, -268, -270, -282, -292, -306

Offset: 1

Torlach Rush, Mar 19 2018

Terms are equal to A023900(p) = A023900(p^2) = A023900(p^3) = ... with p prime, but is never equal to A023900(m*p) with m not a power of p. [Corrected by M. F. Hasler, Sep 01 2021]
abs(a(n)) + 1 is prime (A301590).
For n > 1, if and only if n can't be factored into 2*m factors, m > 0, distinct factors f > 1 where f + 1 is prime then -n is a term. - David A. Corneth, Mar 25 2018
The values are of the form a(n) = 1 - p with prime p = A301590(n). These are exactly the values A023900(x) = 1 - p occurring only if x = p^j for some j >= 1. (See counterexample for p = 13 in EXAMPLE section.) - M. F. Hasler, Sep 01 2021

			a(1) = -1 = A023900(2^m), m > 0.
a(2) = -2 = A023900(3^m), m > 0.
a(3) = -4 = A023900(5^m), m > 0.
a(4) = -6 = A023900(7^m), m > 0.
a(5) = -10 = A023900(11^m), m > 0.
a(6) = -16 = A023900(17^m), m > 0.
A023900(13) = -12 is not a term as A023900(42) = -12, and 42 is the product of three prime factors.
From _David A. Corneth_, Mar 25 2018: (Start)
10 can't be factored in an even number of distinct factors f > 1 such that f + 1 is prime, so -10 is in the sequence.
12 can be factored in an even number of distinct factors f > 1; 12 = 2 * 6 and both 2 + 1 and 6 + 1 are prime, hence -12 is not a term. (End)

Cf. A000040, A001055, A023900, A301590, A301591.

Keys@ Select[Union /@ PrimeNu@ PositionIndex@ Array[DivisorSum[#, # MoebiusMu[#] &] &, 310], # == {1} &] (* Michael De Vlieger, Mar 26 2018 *)

f(n) = sumdivmult(n, d, d*moebius(d));
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(vp, ", ")););} \\ Michel Marcus, Mar 23 2018

cp's OEIS Frontend

A079151 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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A301591 Primes p that have other solutions x to A023900(x) = A023900(p) than a power of p.

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A301374 Values of A023900 which occur only at indices which are powers of a prime.

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PARI