A187073 -id:A187073 - cp's OEIS frontend

A262723 Products of three distinct primes that form an arithmetic progression.

105, 231, 627, 897, 935, 1581, 1729, 2465, 2967, 4123, 4301, 4715, 5487, 7685, 7881, 9717, 10707, 11339, 14993, 16377, 17353, 20213, 20915, 23779, 25327, 26331, 26765, 29341, 29607, 32021, 33335, 40587, 40807, 42911, 48635, 49321, 54739, 55581, 55637, 59563, 60297, 63017

Offset: 1

Antonio Roldán, Sep 28 2015

nonn

This sequence is subsequence of A046389, A088595, A187073, A203614 and A229094.
Obviously, the most repeated prime divisor for values of a(n) is 3. - Altug Alkan, Sep 30 2015
These are numbers 3(2k + 3)(4k + 3) where 2k + 3 and 4k + 3 are prime, together with numbers p(p - 6d)(p + 6d) where p, p - 6d, and p + 6d are prime. - Charles R Greathouse IV, Mar 16 2018

			627 is in this sequence because 627=3*11*19, and 3, 11, 19 form an arithmetic progression (11-3 = 19-11).

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

Cf. A046389, A088595, A187073, A203614, A229094.

Select[Range@ 64000, And[SquareFreeQ@ #, PrimeOmega@ # == 3, Subtract @@ Differences[First /@ FactorInteger@ #] == 0] &] (* Michael De Vlieger, Sep 30 2015 *)

for(i=2,10^5,if(issquarefree(i)&&omega(i)==3,f=factor(i);if(f[1, 1]+f[3, 1]==2*f[2,1],print1(i,", "))))

list(lim)=my(v=List()); lim\=1; forstep(d=6,sqrtint(lim\10),6, forprime(p=d+5, solve(x=sqrtn(lim,3),d*sqrtn(lim,3), x^3-d^2*x-lim)+.5, if(isprime(p-d) && isprime(p+d), listput(v, p*(p-d)*(p+d))))); forprime(p=5,(sqrt(24*lim+81)-27)/12+3.5, if(isprime(2*p-3), listput(v,p*(2*p-3)*3))); Set(v) \\ Charles R Greathouse IV, Mar 16 2018

New name from Peter Munn, Aug 27 2022

A192577 Numbers n such that the arithmetic mean of the unitary divisors of n is a prime number.

Original entry on oeis.org

3, 5, 6, 9, 12, 13, 25, 37, 48, 61, 73, 81, 121, 157, 193, 277, 313, 361, 397, 421, 457, 541, 613, 625, 661, 673, 733, 757, 768, 841, 877, 997, 1093, 1153, 1201, 1213, 1237, 1321, 1381, 1453, 1621, 1657, 1753, 1873, 1933, 1993, 2017, 2137, 2341, 2401, 2473

Offset: 1

Antonio Roldán, Jul 04 2011

nonn

Subsequence of A103826.
Similar to A187073, but considering unitary divisors, not prime divisors.
The odd terms of the sequence are: (1) the terms of A005383 (numbers n such that both n and (n+1)/2 are primes) and (2) the terms of A192618 (prime powers p^k with even exponents k>0 such that (1+p^k)/2 is prime).
[Note that A034448(n) and A034444(n) are multiplicative, so the arithmetic mean A034448(n)/A034444(n) is multiplicative with a(p^e) = (1+p^e)/2.]
The even terms of the sequence are 6, 12, 48, 768, 196608,... (no others < 10^10) with formula n = 3*2^(2^(k-1)) and averages 3, 5, 17, 257, 65537, ... (Fermat numbers, A000215).

			48 has unitary divisors 1, 3, 16, 48 and (1+3+16+48)/4 = 17 is prime, therefore 48 is in the sequence.

Klaus Brockhaus, Table of n, a(n) for n = 1..10000
A. Roldan Martinez, Numeros y hoja de calculo

Cf. A103826, A187073, A005383, A192618, A056798, A000215.

UnitaryDivisors:=func< n | [ d: d in Divisors(n) | Gcd(d, n div d) eq 1 ] >; [ n: n in [1..2500] | IsPrime(k) and s mod #U eq 0 where k is s div #U where s is &+U where U is UnitaryDivisors(n) ]; // Klaus Brockhaus, Jul 09 2011

usigma(n)= {local(f, u=1); f=factor(n); for(i=1, matsize(f)[1], u*=(1+ f[i, 1]^f[i, 2])); return(u)}
ud(n)= {local (f, u); f=factor(n); u=2^(matsize(f)[1]); return(u) }
{  for (n=2, 10^4, c=usigma(n)/ud(n); if (c==truncate(c),if(isprime(c), print1(n, ", ")))) }
\\ Antonio Roldán, Oct 08 2012

A271101 Squarefree semiprimes (A006881) whose average prime factor is prime.

Original entry on oeis.org

21, 33, 57, 69, 85, 93, 129, 133, 145, 177, 205, 213, 217, 237, 249, 253, 265, 309, 393, 417, 445, 469, 489, 493, 505, 517, 553, 565, 573, 597, 633, 669, 685, 697, 753, 781, 793, 813, 817, 865, 889, 913, 933, 949, 973, 985, 993, 1057, 1077, 1137, 1149, 1177, 1257, 1273, 1285, 1329

Offset: 1

Antonio Roldán, Mar 30 2016

nonn

Sum of factors of a(n) if semiprime (product 2*p, with p prime).
This sequence is subsequence of A006881, A089765, A187073, A108633 and A213015.
This sequence is also subsequence of A045835, because sopfr(omega(a(n))) = omega(sopfr(a(n))): sopfr(omega(a(n)))=sopfr(2)=2, and omega(sopfr(a(n)))=omega(2*p)=2  (p prime,  p>2, average prime factor).

			133 is in the sequence because 133 is a squarefree semiprime: 133=7*19, and (7+19)/2=13, a prime number.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006881, A046315, A046388, A115585, A187073, A089765, A108633, A213015.

N:= 10000: # for terms <= N
Primes:= select(isprime, [seq(i, i=3..N/3)]):
SP:= [seq(seq([p, q], q = select(`<=`, Primes, min(p-1, N/p))), p=Primes)]:
B:= select(t -> isprime((t[1]+t[2])/2), SP):
sort(map(t -> t[1]*t[2], B)); # Robert Israel, Dec 14 2019

Select[Select[Range@ 1330, SquareFreeQ@ # && PrimeOmega@ # == 2 &], PrimeQ@ Mean[First /@ FactorInteger@ #] &] (* Michael De Vlieger, Mar 30 2016 *)

sopf(n)= { local(f, s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(s) }
{for (n=6, 2*10^3,  if(bigomega(n)==2&&omega(n)==2, m=sopf(n)/2;if(m==truncate(m),if(isprime(m), print1(n, ", ")))))}

A275384 Composite squarefree numbers such that the arithmetic mean of its prime factors is an integer.

Original entry on oeis.org

15, 21, 33, 35, 39, 42, 51, 55, 57, 65, 69, 77, 78, 85, 87, 91, 93, 95, 105, 110, 111, 114, 115, 119, 123, 129, 133, 141, 143, 145, 155, 159, 161, 170, 177, 183, 185, 186, 187, 195, 201, 203, 205, 209, 213, 215, 217, 219, 221, 222, 230, 231, 235, 237, 247, 249, 253, 258, 259, 265, 267

Offset: 1

Antonio Roldán, Jul 25 2016

nonn

Sopf(a(n)) is multiple of omega(a(n)) (sopf(n) is the sum of the distinct prime factors of n, and omega(n) is the number of distinct primes dividing n).

This sequence is subsequence of A078177 and supersequence of A187073.

			170 is in the sequence because 170 = 17*2*5 (squarefree number) and (17+2+5)/3 = 8 is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A078177, A134344, A187073, A229094.

Select[Range@ 270, And[CompositeQ@ #, SquareFreeQ@ #, IntegerQ@ Mean@ FactorInteger[#][[All, 1]]] &] (* Michael De Vlieger, Jul 25 2016 *)

sopf(n)= my(f, s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); s
for(i=2,500,if(issquarefree(i)&&!isprime(i),m=sopf(i)/omega(i);if(m==truncate(m),print1(i,", "))))

A191683 Smallest representative squarefree composite n with prescribed number of prime factors and prescribed, prime arithmetic average of these factors.

Original entry on oeis.org

21, 33, 57, 69, 93, 105, 129, 177, 195, 213, 217, 237, 249, 265, 309, 393, 417, 445, 465, 483, 489, 565, 573, 597, 633, 645, 669, 753, 813, 865, 915, 933, 973, 987, 993, 1057, 1077, 1137, 1149, 1185, 1257, 1285, 1329, 1365, 1389, 1393, 1417, 1437, 1465, 1477, 1497, 1545, 1569, 1689, 1743, 1765, 1857, 1893, 1897, 1945, 1977

Offset: 1

Rafael Parra Machio, Jun 11 2011

nonn

A187073 contains numbers n = q_1*q_2*q_3*... *q_k with k distinct prime factors q subject to the condition that the arithmetic average (q_1+q_2+...+q_k)/k is some prime p.

This sequence here is a subsequence of A187073 and lists only the smallest n associated with the two parameters k and p. If a larger/later number in A187073 represents the same prime p with the same number k, it is not copied into this sequence here.

			195 and 231 are representatives of the prime average p=7 with k=3 primes in A187073. The smaller 195 is, but the larger 231 is not in this sequence here.
57 and 85 are representatives of p=11 with k=2 primes in A187073. Only the smaller 57 is in here.
93, 145 and 253 are representatives of p=17 with k=2 primes in A187073. Only the smallest representative 93 is in this sequence here.

Carlos Sánchez y Rita Roldán, Goldbach: Una Conjetura Indómita, Nivola, 2009, p. 105

Antonio Roldán, Numeros y hoja de calculo: primos por tadas partes
Antonio Roldán, Numeros y hoja de calculo: números Arolmar
Rafael Parra Machío, Números Arolmar (PDF)

Cf. A187073

cp's OEIS Frontend

A262723 Products of three distinct primes that form an arithmetic progression.

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A192577 Numbers n such that the arithmetic mean of the unitary divisors of n is a prime number.

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Magma

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A271101 Squarefree semiprimes (A006881) whose average prime factor is prime.

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Maple

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A275384 Composite squarefree numbers such that the arithmetic mean of its prime factors is an integer.

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A191683 Smallest representative squarefree composite n with prescribed number of prime factors and prescribed, prime arithmetic average of these factors.

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