cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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.

Original entry on oeis.org

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

Views

Author

Cino Hilliard, Dec 27 2002

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A079148, A079150, A079152, A079153.

Programs

  • Mathematica
    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 *)
  • PARI
    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.
  • PARI
    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

Extensions

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

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

Views

Author

Michel Marcus, Mar 24 2018

Keywords

Comments

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

Examples

			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

Links

Crossrefs

Cf. A023900, A301374.
Complement (within the primes) of A301591, which has A005383 \ {3, 5} as a subsequence. Appears to have A079151 \ {13} as subsequence.

Programs

  • PARI
    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, ", ")); ); }
  • PARI
    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

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

Views

Author

Torlach Rush, Mar 19 2018

Keywords

Comments

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

Examples

			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)

Crossrefs

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

Programs

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

A301857 The lesser of j,k, the smallest pair such that A023900(j) + A023900(k) = 0 and j,k are coprime.

Original entry on oeis.org

1, 13, 37, 41, 61, 73, 89, 97, 109, 113, 185, 157, 181, 193, 233, 241, 277, 281, 323, 313, 337, 349, 353, 403, 397, 401, 409, 421, 433, 449, 457, 461, 527, 551, 521, 541, 577, 593, 601, 613, 617, 641, 661, 673, 701, 779, 733, 757, 761, 769, 959, 821, 829, 899, 877, 881, 929, 937, 953, 997, 1009, 1013, 1021, 1033, 1049
Offset: 1

Views

Author

Torlach Rush, Mar 27 2018

Keywords

Comments

a(11) is the first term where neither j or k is prime.
A301591 is contained in this sequence.

Examples

			1 is a term since A023900(1) + A023900(2) = 0 and gcd(1,2) = 1.
185 is a term since A023900(185) + A023900(273) = 0 and gcd(185,273) = 1.

Crossrefs

Cf. A023900, A301591, A301374.
Showing 1-4 of 4 results.

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