A262723 -id:A262723 - cp's OEIS frontend

A294751 Squarefree products of k primes that are symmetrically distributed around their average. Case k = 4.

2145, 4641, 4845, 5005, 9177, 11305, 13485, 13585, 17017, 21489, 21505, 23529, 26445, 31465, 31857, 33649, 35409, 35581, 36685, 42441, 43401, 46189, 46345, 49569, 50065, 53985, 60697, 61705, 63085, 63597, 65569, 67821, 69745, 77745, 80845, 83049, 87505, 88881

Offset: 1

Paolo P. Lava, Nov 08 2017

nonn

			2145 = 3*5*11*13. Prime factors average is (3 + 5 + 11 + 13)/4 = 8 and 3 + 5 = 8 = 13 - 5, 5 + 3 = 8 = 11 - 3.

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

Subsequence of A046386.

Cf. A006881 (k=2), A262723 (k=3), A294752 (k=5), A294776 (k=6).

with(numtheory): P:=proc(q,h) local a,b,k,n,ok;
for n from 2*3*5*7 to q do if not isprime(n) and issqrfree(n) then a:=ifactors(n)[2];
if nops(a)=h then b:=2*add(a[k][1],k=1..nops(a))/nops(a); ok:=1;
for k from 1 to trunc(nops(a)/2) do if a[k][1]+a[nops(a)-k+1][1]<>b then ok:=0; break; fi; od; if ok=1 then print(n); fi; fi; fi; od; end: P(10^9,4);
# Alternative:
N:= 10^5: # to get terms <= N
M:= floor(max(fsolve(3*5*(M-5)*(M-3) = N))):
P:= select(isprime, [seq(i,i=3..M/2,2)]): nP:= nops(P):
Res:= NULL:
for m from 10 by 2 to M do
  for ix from 1 to nP-2 do
    x:= P[ix];
    if x >= m/2 or (x*(m-x))^2 >= N then break fi;
    if not isprime(m-x) then next fi;
    for iy from ix+1 to nP-1 do
      y:= P[iy];
      if y >= m/2 or x*(m-x)*y*(m-y) >= N then break fi;
      if not isprime(m-y) then next fi;
      Res:= Res, x*(m-x)*y*(m-y);
od od od:
sort([Res]); # Robert Israel, May 19 2019

isok(n, nb=4) = {if (issquarefree(n) && (omega(n)==nb), f = factor(n)[, 1]~; avg = vecsum(f)/#f; for (k=1, #f\2, if (f[k] + f[#f-k+1] != 2*avg, return(0));); return (1););} \\ Michel Marcus, Nov 10 2017

More terms from Giovanni Resta, Nov 09 2017

A294752 Squarefree products of k primes that are symmetrically distributed around their average. Case k = 5.

Original entry on oeis.org

53295, 119301, 229245, 399993, 608235, 623645, 1462731, 2324495, 3696189, 3973145, 4482879, 5356445, 5920971, 6249633, 7588977, 8270385, 10160943, 10450121, 10505373, 13185969, 13630011, 13760929, 14935029, 19095395, 20280795, 22566271, 23131549, 23408259, 24778401

Offset: 1

Paolo P. Lava, Nov 08 2017

nonn

			53295 = 3*5*11*17*19. Prime factors average is (3 + 5 + 11 + 17 + 19)/5 = 11 and 3 + 8 = 11 = 19 - 8, 5 + 6 = 11 = 17 - 6.

Robert Israel, Table of n, a(n) for n = 1..3560

Subsequence of A046387, A203614.

Cf. A006881 (k=2), A262723 (k=3), A294751 (k=4), A294776 (k=6).

with(numtheory): P:=proc(q,h) local a,b,k,n,ok;
for n from 2*3*5*7*11 to q do if not isprime(n) and issqrfree(n) then a:=ifactors(n)[2];
if nops(a)=h then b:=2*add(a[k][1],k=1..nops(a))/nops(a); ok:=1;
for k from 1 to trunc(nops(a)/2) do if a[k][1]+a[nops(a)-k+1][1]<>b then ok:=0; break; fi; od; if ok=1 then print(n); fi; fi; fi; od; end: P(10^9,5);
# Alternative:
N:= 10^8: # to get all terms <= N
M:= floor((8*N/15)^(1/3)):
P:= select(isprime, [seq(i,i=3..M,2)]): nP:= nops(P):
Res:= NULL:
for i3 from 3 to nP-2 do
  p3:= P[i3];
  for i1 from 1 to i3-2 do
    if isprime(2*p3 - P[i1]) then
      for i2 from i1+1 to i3-1 do
        if isprime(2*p3 - P[i2]) then
          v:=P[i1]*P[i2]*p3*(2*p3-P[i2])*(2*p3-P[i1]);
          if v <= N then Res:= Res, v fi
        fi
      od
     fi
   od
od:
sort([Res]): # Robert Israel, Nov 10 2017

isok(n, nb=5) = {if (issquarefree(n) && (omega(n)==nb), f = factor(n)[, 1]~; avg = vecsum(f)/#f; for (k=1, #f\2, if (f[k] + f[#f-k+1] != 2*avg, return(0));); return (1););} \\ Michel Marcus, Nov 10 2017

More terms from Giovanni Resta, Nov 09 2017

Missing term 23131549 inserted by Robert Israel, Nov 10 2017

A294776 Squarefree products of k primes that are symmetrically distributed around their average. Case k = 6.

Original entry on oeis.org

1616615, 3411705, 7436429, 9408035, 10163195, 12838371, 13037385, 13844919, 14969435, 19605131, 20414121, 23783045, 24997749, 25113935, 27568145, 30478565, 31473255, 32518535, 33999455, 39946569, 43134015, 46115135, 48215255, 50907855, 56179409, 61558343

Offset: 1

Paolo P. Lava, Nov 09 2017

nonn

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

Subsequence of A067885.

Cf. A006881 (k=2), A262723 (k=3), A294751 (k=4), A294752 (k=5).

with(numtheory): P:=proc(q,h) local a,b,k,n,ok;
for n from 2*3*5*7*11*13 to q do if not isprime(n) and issqrfree(n) then a:=ifactors(n)[2];
if nops(a)=h then b:=2*add(a[k][1],k=1..nops(a))/nops(a); ok:=1;
for k from 1 to trunc(nops(a)/2) do if a[k][1]+a[nops(a)-k+1][1]<>b then ok:=0; break; fi; od; if ok=1 then print(n); fi; fi; fi; od; end: P(10^9,6);
# Alternative:
N:= 10^8: # to get all terms <= N
M:= floor(fsolve(3*5*7*(M-7)*(M-5)*(M-3) = N)):
P:= select(isprime, [seq(i,i=3..M/2,2)]): nP:= nops(P):
Res:= NULL:
for m from 10 by 2 to M do
  for ix from 1 to nP-2 do
    x:= P[ix];
    if x >= m/2 or (x*(m-x))^3 >= N then break fi;
    if not isprime(m-x) then next fi;
    for iy from ix+1 to nP-1 do
      y:= P[iy];
      if y >= m/2 or x*(m-x)*(y*(m-y))^2 >= N then break fi;
      if not isprime(m-y) then next fi;
      for iz from iy+1 to nP do
        z:= P[iz];
        if z >= m/2 then break fi;
        v:= x*(m-x)*y*(m-y)*z*(m-z);
        if v > N then break fi;
        if isprime(m-z) then Res:= Res, v fi;
od od od od:
sort([Res]); # Robert Israel, May 19 2019

isok(n, nb=6) = {if (issquarefree(n) && (omega(n)==nb), f = factor(n)[, 1]~; avg = vecsum(f)/#f; for (k=1, #f\2, if (f[k] + f[#f-k+1] != 2*avg, return(0));); return (1););} \\ Michel Marcus, Nov 10 2017

More terms from Giovanni Resta, Nov 09 2017

A141807 Numbers k such that the maximal prime power divisors of k form a run of integers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 16, 17, 19, 20, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 56, 59, 60, 61, 64, 67, 71, 72, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191

Offset: 1

Leroy Quet, Jul 07 2008

nonn

Old name and expanded definition: If p^b(n,p) is the largest power of the prime p to divide n, then the positive integer n is included in the sequence if p(1)^b(n,p(1)) = p(2)^b(n,p(2))+1 = p(3)^b(n,p(3))+2 =...= p(k)^b(n,p(k))+k-1, where (p(1),p(2),p(3),...,p(k)) is some permutation of the distinct primes that divide n.
All prime powers (A000961) are included in this sequence.
Sequence A141808 consists of the terms of this sequence that are not prime powers.

			The prime factorization of 60 is 2^2 * 3^1 * 5^1. Since 5^1 = 2^2 + 1 = 3^1 + 2 (i.e., the prime powers, in some order, occur in an arithmetic progression with a difference of 1 between consecutive terms), then 60 is included in the sequence.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000961, A141808, A262723.

Select[Range[192], (pp = Sort[#[[1]]^#[[2]] & /@ FactorInteger@#]) - pp[[1]] + 1 == Range@Length@pp &] (* Ivan Neretin, Aug 13 2015 *)

Extended by Ray Chandler, Jun 21 2009

New name from Peter Munn, Aug 31 2022

A141808 Numbers k such that the maximal prime power divisors of k form a nontrivial run of integers.

Original entry on oeis.org

6, 12, 20, 56, 60, 72, 272, 504, 992, 16256, 65792, 67100672, 4295032832, 17179738112, 274877382656, 4611686016279904256, 5316911983139663489309385231907684352, 383123885216472214589586756168607276261994643096338432

Offset: 1

Leroy Quet, Jul 07 2008

nonn

Old name and expanded definition: If p^b(n,p) is the largest power of the prime p to divide n, then the positive integer non-prime-power n is included in the sequence if p(1)^b(n,p(1)) = p(2)^b(n,p(2))+1 = p(3)^b(n,p(3))+2 = ... = p(k)^b(n, p(k))+k-1, where (p(1),p(2),p(3),...,p(k)) is some permutation of the distinct primes that divide n.
Sequence A141807 is the union of the prime powers (A000961) and this sequence.
Terms with two distinct prime factors occur where either 2^m+1 or 2^m-1 is a prime power. Terms with three distinct prime factors (60, 504) occur where both 2^m+1 and 2^m-1 are prime powers. There are no terms with more than three distinct prime factors. For every Mersenne prime p (A000668), p*(p+1) is in this sequence. For every prime p in A000043, 2^p*(2^p-1) is in this sequence. - Ray Chandler, Jun 21 2009

			The prime factorization of 60 is 2^2 * 3^1 * 5^1. Since 60 is not a prime power and since 5^1 = 2^2 + 1 = 3^1 + 2 (i.e., the prime powers, in some order, occur in an arithmetic progression with a difference of 1 between consecutive terms), 60 is included in the sequence.

Cf. A000961, A141807.

Cf. A000043, A000668, A262723.

Extended by Ray Chandler, Jun 21 2009

New name from Peter Munn, Aug 31 2022

A300949 Carmichael numbers whose prime factors form an arithmetic progression.

Original entry on oeis.org

1729, 2465, 29341, 294409, 1152271, 1857241, 6189121, 19384289, 56052361, 64377991, 118901521, 172947529, 216821881, 228842209, 625482001, 775368901, 1213619761, 1299963601, 2301745249, 4562359201, 8346731851, 9293756581, 9624742921, 9701285761, 11346205609, 13079177569, 13946829751, 14386156093

Offset: 1

Robert Israel and Thomas Ordowski, Mar 16 2018

nonn

All terms < 10^18 have three prime factors.  There are terms with more, e.g., 97888020200929464481 = 34471 * 91921 * 149371 * 206821, 147681255946700149193521 = 214831 * 572881 * 930931 * 1288981, and 2393527068197020059464161 = 431047 * 1149457 * 1867867 * 2586277.
A term with 3 prime factors is of the form (p-d)p(p+d), where p-d, p and p+d are prime, and p-d-1 | d(2d+3), p-1 | d^2, and p+d-1 | d(2d-3). Thus for each d there are only finitely many possible p that make this work. Note that 6|d, see A262723.
Conjecture: if n is a Carmichael number and lpf(n)gpf(n)(lpf(n)+gpf(n))/2 = n, then (lpf(n)+gpf(n))/2 is prime; and thus n has exactly three prime factors. Such numbers n form a proper subsequence of this sequence, also subsequence of A262723. - Charles R Greathouse IV and Thomas Ordowski, Mar 17 2018. Edited by Max Alekseyev, Mar 17 2018
Proof of the above conjecture: Say n = paq with 2 < p < q being primes and a = (p+q)/2, with (a,p!)=1. If n is a Carmichael number, then pa == 1 (mod q-1), so p^2 + pq == 2 (mod q-1), so p^2 + p == 2 (mod q-1). In particular, p^2 + p - 2 >= q-1, which implies that (p+1)^2 > q. Say a has k prime factors, so that a >= (p+2)^k. But a < q, so q > (p+2)^k. Thus, (p+1)^2 > q > (p+2)^k. This implies k=1. - Carl Pomerance (in a letter to the second author), Mar 18 2018
Note: this does not exclude the existence of the Carmichael numbers m = pq(p+q)/2 with more than three prime factors, where p and q are prime. - Thomas Ordowski, Mar 19 2018

			29341 = 13*37*61 is a Carmichael number, and [13, 37, 61] is an arithmetic progression of length 3 and with common difference of 37 - 13 = 61 - 37 = 24. We have 37 = (13 + 61)/2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier; terms 1..3841 from Max Alekseyev)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
R. G. E. Pinch, Tables relating to Carmichael numbers.

Cf. A002997, A087788, A262723.

# using Richard Pinch data file carmichael-16
cfile:= "carmichael-16":
Res:= NULL:
do
  S:= readline(cfile);
  if S = 0 then break fi;
  L:= map(parse, StringTools:-Split(S));
  R:= sort(L[2..-1]);
  d:= (R[-1]-R[1])/(nops(R)-1);
  if R = [seq(i,i=R[1]..R[-1],d)] then
    Res:= Res, L[1];
  fi
od:
fclose(cfile):
Res;

Select[Cases[Range[1, 10^7, 2], ?(And[Mod[#, CarmichaelLambda@ #] == 1, ! PrimeQ[#]] &)], Length@ Union@ Differences@ FactorInteger[#][[All, 1]] == 1 &] (* _Michael De Vlieger, Mar 17 2018, after Artur Jasinski at A002997 *)

A307108 Numbers x that are equal to lpf(x)gpf(x)(lpf(x)+gpf(x))/2, where lpf(x) < gpf(x) are the least and the greatest prime factors of x: A020639 and A006530.

Original entry on oeis.org

105, 231, 627, 897, 935, 1581, 1729, 2465, 2967, 3525, 4123, 4301, 4715, 5487, 7035, 7685, 7881, 9717, 10707, 11339, 14993, 16377, 17353, 17655, 20213, 20915, 23779, 24765, 25327, 26331, 26765, 29341, 29607, 32021, 33335, 34881, 40587, 40807, 42585, 42911, 48635

Offset: 1

Alex Ratushnyak, Mar 25 2019

nonn

			105 = 3*7*5 is a term, 5 = (3+7) / 2.
231 = 3*11*7 is a term, 7 = (3+11) / 2.
3525 = 3*47*25 is a term, 25 = (3+47) / 2.

Cf. A020639, A006530.

Cf. A262723 (a squarefree subsequence).

is(n) = my(f = factor(n)); omega(f) > 2 && (f[1, 1] * f[#f~, 1]) * (f[1, 1]+f[#f~, 1]) == n << 1 \\ David A. Corneth, Mar 25 2019

A294906 a(n) is the least squarefree integer, product of n primes that are symmetrically distributed around their average.

Original entry on oeis.org

2, 6, 105, 2145, 53295, 1616615, 57998985, 3038795305, 3907126810041, 7292509103495, 66240019730740785, 82246340873964085, 1870667082297874652055, 343515424581301546805, 9160656702012692171113335, 2356596317899272514936585, 1903895854998638367242867256645

Offset: 1

Michel Marcus, Nov 10 2017

nonn

			The prime factors of the first terms are: [2], [2, 3], [3, 5, 7], [3, 5, 11, 13], [3, 5, 11, 17, 19], [5, 7, 11, 13, 17, 19], [3, 5, 11, 17, 23, 29, 31], ...

Cf. A262723, A294751, A294752, A294776.

isok(n, nb) = {if (issquarefree(n) && (omega(n)==nb), f = factor(n)[, 1]~; avg = vecsum(f)/#f; for (k=1, #f\2, if (f[k] + f[#f-k+1] != 2*avg, return(0));); return (1););}
a(n) = {my(k = prod(i=1, n, prime(i))); while (! isok(k, n), k++); k;}

a(8)-a(17) from Giovanni Resta, Nov 10 2017

A307117 Numbers x that are equal to lpf(x)gpf(x)(lpf(x)+gpf(x))/2, where lpf(x) and gpf(x) are the least and the greatest prime factors of x: A020639 and A006530.

Original entry on oeis.org

8, 27, 105, 125, 231, 343, 627, 897, 935, 1331, 1581, 1729, 2197, 2465, 2967, 3525, 4123, 4301, 4715, 4913, 5487, 6859, 7035, 7685, 7881, 9717, 10707, 11339, 12167, 14993, 16377, 17353, 17655, 20213, 20915, 23779, 24389, 24765, 25327, 26331, 26765, 29341, 29607

Offset: 1

Alex Ratushnyak, Mar 25 2019

nonn

This is A307108 with terms such that lpf(x)=gpf(x), that is, with cubes of primes.

Cf. A000040, A020639, A006530, A307108.

Cf. A262723 is the subsequence with terms such that lpf(x)!=gpf(x), and (lpf(x)+gpf(x))/2 a prime.

cp's OEIS Frontend

A294751 Squarefree products of k primes that are symmetrically distributed around their average. Case k = 4.

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A294752 Squarefree products of k primes that are symmetrically distributed around their average. Case k = 5.

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A294776 Squarefree products of k primes that are symmetrically distributed around their average. Case k = 6.

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A141807 Numbers k such that the maximal prime power divisors of k form a run of integers.

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Mathematica

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A141808 Numbers k such that the maximal prime power divisors of k form a nontrivial run of integers.

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Extensions

A300949 Carmichael numbers whose prime factors form an arithmetic progression.

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Maple

Mathematica

A307108 Numbers x that are equal to lpf(x)*gpf(x)*(lpf(x)+gpf(x))/2, where lpf(x) < gpf(x) are the least and the greatest prime factors of x: A020639 and A006530.

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PARI

A294906 a(n) is the least squarefree integer, product of n primes that are symmetrically distributed around their average.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Extensions

A307108 Numbers x that are equal to lpf(x)gpf(x)(lpf(x)+gpf(x))/2, where lpf(x) < gpf(x) are the least and the greatest prime factors of x: A020639 and A006530.

A307117 Numbers x that are equal to lpf(x)gpf(x)(lpf(x)+gpf(x))/2, where lpf(x) and gpf(x) are the least and the greatest prime factors of x: A020639 and A006530.