A075819 -id:A075819 - cp's OEIS frontend

A081400 a(n) = d(n) - bigomega(n) - A005361(n).

0, 0, 0, -1, 0, 1, 0, -2, -1, 1, 0, 1, 0, 1, 1, -3, 0, 1, 0, 1, 1, 1, 0, 1, -1, 1, -2, 1, 0, 4, 0, -4, 1, 1, 1, 1, 0, 1, 1, 1, 0, 4, 0, 1, 1, 1, 0, 1, -1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 6, 0, 1, 1, -5, 1, 4, 0, 1, 1, 4, 0, 1, 0, 1, 1, 1, 1, 4, 0, 1, -3, 1, 0, 6, 1, 1, 1, 1, 0, 6, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 4, 0, 1, 4, 1, 0, 1, 0, 4, 1, 1, 0, 4, 1, 1, 1, 1, 1, 8, -1, 1

Offset: 1

Labos Elemer, Mar 28 2003

sign

			Negative for true prime powers; zero for 1 and primes; see also A030231, A007304, A034683, A075819 etc. to judge about positivity or magnitude.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A001222, A005361, A030231, A030230, A007304, A034683, A075819.

a(n) = my(f=factor(n)); numdiv(n) - bigomega(n) - prod(k=1, #f~, f[k,2]); \\ Michel Marcus, May 25 2017

from sympy import primefactors, factorint, divisor_count
from operator import mul
def bigomega(n): return 0 if n==1 else bigomega(n/primefactors(n)[0]) + 1
def a005361(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [f[i] for i in f])
def a(n): return divisor_count(n) - bigomega(n) - a005361(n) # Indranil Ghosh, May 25 2017

a(n) = A000005(n) - A001222(n) - A005361(n).

A291446 Squarefree triprimes of the form pqr such that p + q + r + 1 is prime.

Original entry on oeis.org

30, 42, 66, 78, 102, 110, 138, 182, 186, 222, 230, 246, 266, 282, 290, 318, 366, 374, 402, 434, 438, 498, 506, 518, 530, 582, 590, 602, 606, 618, 638, 642, 710, 782, 786, 806, 854, 890, 906, 942, 962, 1002, 1010, 1022, 1034, 1038, 1106, 1118, 1146, 1158, 1166, 1178, 1298

Offset: 1

Vincenzo Librandi, Aug 24 2017

nonn

All terms are even. - Muniru A Asiru, Aug 29 2017

			42 = 2*3*7 and 2 + 3 + 7 + 1 is prime, so 42 is a term.
402 = 2*3*67 and 2 + 3 + 67 + 1 is prime, so 402 is a term.

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

Subsequence of A075819, and hence of A007304.

Cf. A291319.

A291446:=List(Filtered(Filtered(List(Filtered(List([1..10^6],Factors),i->Length(i)=3),Set),j->Length(j)=3),i->IsPrime(Sum(i)+1)),Product); # Muniru A Asiru, Aug 29 2017

With[{nnn=80}, Take[Times@@@Select[Subsets[Prime[Range[nnn]], {3}], PrimeQ[Total[#] + 1] &]//Union, nnn]]

list(lim)=my(v=List()); forprime(p=5, lim\6, forprime(q=3, min(lim\(2*p),p-2), if(isprime(p+q+3), listput(v, 2*p*q)))); Set(v) \\ Charles R Greathouse IV, Aug 29 2017

A296117 Base-2 pseudoprimes of the form 2pq where p and q are primes.

Original entry on oeis.org

161038, 49699666, 760569694, 4338249646, 357647681422, 547551530002, 3299605275646, 22999986587854, 42820164121582, 55173914702146, 69345154539266, 353190859033982

Offset: 1

Max Alekseyev, Dec 05 2017

a(5) and a(10) are found by McDaniel (1989).

Terms in this sequence are of the form 2pq where p and q are distinct odd primes (A075819). - Charles R Greathouse IV, Dec 05 2017

W. L. McDaniel, Some pseudoprimes and related numbers having special forms, Math. Comp. 53:187 (1989), 407-409.

Subsequence of A006935 and hence of A015919.
The even terms of A215672.
Intersection of A006935 and A215672. - Felix Fröhlich, Dec 05 2017
Cf. A270973, A075819, A001220.

list(lim)=my(v=List(),pq); forprime(p=3,lim\6, forprime(q=3,min(lim\(2*p),p), pq=p*q; if(Mod(4,pq)^pq==2, listput(v,2*pq)))); Set(v) \\ Charles R Greathouse IV, Dec 05 2017

A375908 Sphenic numbers that are sandwiched between products of exactly 4 distinct primes (or tetraprimes).

Original entry on oeis.org

18446, 39766, 74306, 83434, 94106, 100346, 107966, 111154, 111814, 113366, 140834, 144754, 145606, 146014, 147406, 149854, 154946, 155702, 156146, 165346, 171786, 189034, 190618, 191806, 197354, 201686, 203314, 206194, 211394, 211946, 219386, 231286, 234394, 253114, 258266, 262294, 263966

Offset: 1

Massimo Kofler, Sep 02 2024

nonn

Terms are of the form 4*k+2.

			18446 = 2 * 23 * 401  (between 18445 = 5*7*17*31 and 18447 = 3*11*13*43).
39766 = 2 * 59 * 337  (between 39765 = 3*5*11*241 and 39767 = 7*13*19*23).
74306 = 2 * 53 * 701  (between 74305 = 5*7*11*193 and 74307 = 3*17*31*47).

Cf. A007304, A046386. Subsequence of A075819.

N:= 5*10^5: # for terms <= N
P:= select(isprime,[seq(i,i=3..N/3,2)]): nP:= nops(P):
R:= NULL:
for i from 1 to nP while 2*P[i]*P[i+1] <= N do
  for j from i+1 to nP do
    x:= 2*P[i]*P[j];
    if x > N then break fi;
    if numtheory:-bigomega(x-1) = 4 and numtheory:-bigomega(x+1) = 4 and
      numtheory:-issqrfree(x-1) and numtheory:-issqrfree(x+1) then
        R:= R,x
    fi
od od:
sort([R]); # Robert Israel, Sep 02 2024

e[n_] := FactorInteger[n][[;; , 2]]; SequencePosition[e /@ Range[300000], {{1, 1, 1, 1}, {1, 1, 1}, {1, 1, 1, 1}}][[;; , 1]] + 1 (* Amiram Eldar, Sep 02 2024 *)

cp's OEIS Frontend

A081400 a(n) = d(n) - bigomega(n) - A005361(n).

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A291446 Squarefree triprimes of the form p*q*r such that p + q + r + 1 is prime.

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A296117 Base-2 pseudoprimes of the form 2*p*q where p and q are primes.

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PARI

A375908 Sphenic numbers that are sandwiched between products of exactly 4 distinct primes (or tetraprimes).

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Maple

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A291446 Squarefree triprimes of the form pqr such that p + q + r + 1 is prime.

A296117 Base-2 pseudoprimes of the form 2pq where p and q are primes.