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.

A124785 Semiprime sopfr of semiprimes A115585.

Original entry on oeis.org

4, 6, 9, 10, 10, 15, 14, 21, 25, 14, 22, 33, 26, 39, 22, 34, 49, 55, 22, 46, 26, 69, 34, 85, 26, 62, 91, 46, 74, 38, 111, 115, 82, 86, 34, 129, 133, 58, 141, 34, 106, 159, 169, 38, 183, 134, 201, 142, 213, 94, 235, 74, 166, 46, 253, 106, 259, 58, 265, 46, 86, 118, 194
Offset: 1

Views

Author

Zak Seidov, Nov 07 2006

Keywords

Comments

Cf. A001358 semiprimes, A001414 sopfr, A115585 Semiprimes with a semiprime sum of factors, A124786 Intersection of A115585 and A124785.

Examples

			a(3)=9 because A115585(3)=14=2*7=>2+7=9=3*3,
a(6)=15 because A115585(6)=26=2*13=>2+13=15=3*5.

Crossrefs

Cf. A001358, A001414, A115585, A124786.

Formula

a(n)=sopfr(A115585(n))

A124786 Numbers n such that n, sopfr(n) and sopfr(sopfr(n)) are all semiprimes.

Original entry on oeis.org

4, 14, 33, 38, 46, 49, 62, 69, 94, 129, 133, 134, 166, 169, 177, 205, 213, 217, 254, 262, 309, 334, 361, 393, 422, 445, 469, 489, 493, 502, 505, 526, 529, 614, 669, 718, 753, 793, 817, 865, 886, 889, 913, 933, 934, 961, 974, 982, 993, 1006, 1077, 1126, 1142
Offset: 1

Views

Author

Zak Seidov, Nov 07 2006

Keywords

Comments

Cf. A001358 semiprimes, A001414 sopfr, A115585 Semiprimes with a semiprime sum of factors, A124785(n)=sopfr(A115585(n)).

Examples

			Full table of {n, sopfr(n), sopfr(sopfr(n))}:
{4, 4, 4}, {14, 9, 6}, {33, 14, 9}, {38, 21, 10}, {46, 25, 10}, {62, 33, 14}, {69, 26, 15}, {94, 49, 14}, {129, 46, 25}, {134, 69, 26}, {166, 85, 22}, {177, 62, 33}, {213, 74, 39}, {217, 38, 21}, {254, 129, 46}, {262, 133, 26}, {309, 106, 55}, {334, 169, 26}, {393, 134, 69}, {422, 213, 74}, {445, 94, 49}, {489, 166, 85}, {502, 253, 34}, {526, 265, 58}.

Crossrefs

Cf. A001358, A001414, A115585, A124785.

Programs

  • Maple
    isA001358 := proc(n) if numtheory[bigomega](n) = 2 then true ; else false ; fi ; end: A001414 := proc(n) local ifs; if n = 1 then 0; else ifs := ifactors(n)[2] ; add( op(1,i)*op(2,i),i=ifs) ; fi ; end: A081758 := proc(n) A001414(A001414(n)) ; end: isA124786 := proc(n) if isA001358(n) and isA001358(A001414(n)) and isA001358(A081758(n)) then true ; else false ; fi ; end: for n from 2 to 2000 do if isA124786(n) then printf("%d, ",n) ; fi : od: # R. J. Mathar, Sep 23 2007
  • Mathematica
    semiprimeQ[n_] := PrimeOmega[n] == 2;
sopfr[n_] := Total[Times @@@ FactorInteger[n]];
okQ[n_] := semiprimeQ[n] && semiprimeQ[ sopfr[n]] && semiprimeQ[ sopfr@ sopfr@n];
Select[Range[2000], okQ] (* Jean-François Alcover, Jul 20 2024 *)

Extensions

Corrected and extended by R. J. Mathar, Sep 23 2007

A187400 Semiprimes with a semiprime average of the two factors.

Original entry on oeis.org

15, 35, 51, 65, 77, 91, 115, 123, 141, 161, 185, 187, 201, 209, 219, 221, 235, 259, 267, 301, 305, 321, 339, 341, 355, 365, 377, 381, 403, 413, 427, 437, 451, 453, 481, 485, 497, 501, 537, 545, 589, 649, 667, 681, 689, 699, 717, 721, 723, 737, 745, 749, 763, 789, 835, 843, 849, 893, 901, 905
Offset: 1

Views

Author

Antonio Roldán, Mar 09 2011

Keywords

Comments

The definition is similar to A115585, but considering the arithmetic mean, not the sum of the factors.
Even semiprimes, A100484, are not in the sequence, because (with the exception of 4) the average of their factors is not an integer.

Examples

			The semiprime 187 = 11*17 is in the sequence, because the average (11+17)/2 = 14 = 2*7 is semiprime.
The semiprime 267 = 3*89 is in the sequence because the average (3+89)/2 = 46 = 2*23 is semiprime.

Links

Crossrefs

Cf. A001358, A100484, A115585.

Programs

  • Mathematica
    semiPrimeQ[n_] := Total[FactorInteger[n]][[2]] == 2; Reap[Do[{p, e} = Transpose[FactorInteger[n]]; If[Total[e] == 2 && semiPrimeQ[Total[p]/2], Sow[n]], {n, 1000}]][[2, 1]]
  • PARI
    sopf(n)= { local(f, s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(s) }
averg(n)={local(s); s=sopf(n)/omega(n);return(s)}
{  for (n=4, 10^3, m=averg(n);if(bigomega(n)==2,if(m==floor(m)&&bigomega(m)==2,print1(n, ", ")))) }
\\ Antonio Roldán, Oct 15 2012
  • PARI
    list(lim)=my(v=List()); forprime(p=3,lim\3, forprime(q=3,min(p-2,lim\p), if(bigomega((p+q)/2)==2, listput(v,p*q)))); Set(v) \\ Charles R Greathouse IV, Oct 16 2014

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

Views

Author

Antonio Roldán, Mar 30 2016

Keywords

Comments

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).

Examples

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

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    Select[Select[Range@ 1330, SquareFreeQ@ # && PrimeOmega@ # == 2 &], PrimeQ@ Mean[First /@ FactorInteger@ #] &] (* Michael De Vlieger, Mar 30 2016 *)
  • PARI
    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, ", ")))))}
Showing 1-4 of 4 results.

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