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.

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A106554 Concatenation of the two prime divisors of a semiprime, smallest divisor first.

Original entry on oeis.org

22, 23, 33, 25, 27, 35, 37, 211, 55, 213, 311, 217, 57, 219, 313, 223, 77, 317, 511, 319, 229, 231, 513, 323, 237, 711, 241, 517, 243, 329, 713, 331, 247, 519, 253, 337, 523, 259, 717, 1111, 261, 341, 343, 719, 267, 347, 271, 1113
Offset: 1

Views

Author

Eric Angelini, May 09 2005

Keywords

Comments

Concatenation of the divisors starting with the largest one leads to another sequence.

Examples

			First semiprime is 4; 4 is 2*2 -> 22.
Second semiprime is 6; 6 is 2*3 -> 23.
Third semiprime is 9; 9 is 3*3 -> 33.
Fourth semiprime is 10; 10 is 2*5 -> 25.

Links

Crossrefs

Cf. A106550, A084126, A084127, A001358.

Programs

  • Maple
    read("transforms") :
A106554 := proc(n)
    A037276(A001358(n)) ;
end proc: # R. J. Mathar, Oct 29 2012
  • Mathematica
    FromDigits@ Flatten[IntegerDigits@ Table[#1, {#2}] & @@@ FactorInteger@ #] & /@ Select[Range@ 144, PrimeOmega@ # == 2 &] (* Michael De Vlieger, Oct 01 2015 *)
  • PARI
    do(x)=my(v=List()); forprime(p=2, sqrt(x), forprime(q=p, x\p, listput(v, [p*q, eval(Str(p, q))]))); Vec(apply(u->u[2], vecsort(v, 1))) \\ Charles R Greathouse IV, Sep 30 2015

Formula

a(n) = A037276(A001358(n)). - R. J. Mathar, Oct 29 2012

Extensions

More terms from Reinhard Zumkeller, May 19 2005
Data corrected by Giovanni Teofilatto and Altug Alkan, Oct 01 2015

A106550 a(n) = n-th semiprime + (concatenation of its two prime factors, smallest factor first).

Original entry on oeis.org

26, 29, 42, 35, 41, 50, 58, 233, 80, 239, 344, 251, 92, 257, 352, 269, 126, 368, 566, 376, 287, 293, 578, 392, 311, 788, 323, 602, 329, 416, 804, 424, 341, 614, 359, 448, 638, 377, 836, 1232, 383, 464, 472, 852, 401, 488, 413, 1256, 674, 419, 686, 437, 512, 884
Offset: 1

Views

Author

Eric Angelini, May 09 2005

Keywords

Examples

			First semiprime is 4; 4 is 2*2; 26=4+22.
Second semiprime is 6; 6 is 2*3; 29=6+23.
Third semiprime is 9; 9 is 3*3; 41=9+33.
Fourth semiprime is 10; 10 is 2*5; 35=10+25.

Crossrefs

Cf. A084126, A084127, A001358, A106554.

Programs

  • Mathematica
    R=165;SP=Select[Range[R], PrimeOmega[#]==2&]; FromDigits@ Flatten[IntegerDigits@ Table[#1, {#2}] & @@@ FactorInteger@ #] & /@ SP +SP (* James C. McMahon, Feb 18 2024 *)

Formula

a(n) = A001358(n) + A106554(n). - Reinhard Zumkeller, May 19 2005

Extensions

More terms from Reinhard Zumkeller, May 19 2005
Typo corrected by James C. McMahon, Feb 06 2024

A126663 Absolute difference between largest prime factors of two successive semiprimes.

Original entry on oeis.org

1, 0, 2, 2, 2, 2, 4, 6, 8, 2, 6, 10, 12, 6, 10, 16, 10, 6, 8, 10, 2, 18, 10, 14, 26, 30, 24, 26, 14, 16, 18, 16, 28, 34, 16, 14, 36, 42, 6, 50, 20, 2, 24, 48, 20, 24, 58, 16, 44, 42, 48, 26, 30, 60, 70, 46, 30, 28, 24, 20, 80, 30, 34, 72, 12, 62, 84, 52, 36, 64, 12, 78, 36, 56, 96
Offset: 1

Views

Author

Giovanni Teofilatto, Mar 23 2007

Keywords

Comments

Absolute first difference of A084127.

Examples

			a(1) = 1 because 4 = 2*2 and 6 = 2*3 so 3 - 2 = 1;
a(2) = 0 because 6 = 2*3 and 9 = 3*3 so 3 - 3 = 0;
a(3) = 2 because 9 = 3*3 snd 10 = 2*5 so 5 - 3 = 2.

Crossrefs

Cf. A001358, A006530, A084127.

Programs

  • Mathematica
    Select[ FactorInteger[ Range[300]], Total[#[[All, 2]]] == 2 &][[All, -1, 1]] // Differences // Abs (* Jean-François Alcover, Dec 03 2013 *)
Abs[#[[1]]-#[[2]]]&/@(Partition[FactorInteger[#][[-1,1]]&/@Select[ Range[ 300],PrimeOmega[#]==2&],2,1]) (* Harvey P. Dale, Nov 07 2020 *)

Extensions

Edited and extended by Ray Chandler, Mar 25 2007

A131284 Numbers n such that difference between prime factors of n-th semiprime is n.

Original entry on oeis.org

5, 80, 86, 613668, 6384425704
Offset: 1

Views

Author

Zak Seidov, Sep 25 2007

Keywords

Comments

The 6384425704th semiprime is 44690979977 = 7*6384425711. 6384425711 - 7 = 6384425704. - Donovan Johnson, Jul 11 2010

Examples

			sp(5) = 14 = 2*7 and 7 - 2 = 5, sp(80) = 249 = 3*83 and 83 - 3 = 80, sp(86) = 267 = 3*89 and 89 - 3 = 86; sp(n) = n-th semiprime.

Crossrefs

Cf. A064910, A084126, A084127, A109313.

Programs

  • PARI
    { n=0; j=1; /* n=3068365-1; j=613668;*/
while( l=(j\10^4+1)*10^4, until( l < j++, until(bigomega(n+=1)==2,);
if(2!=#f=factor(n)[,1],next); if(j==f[2]-f[1],print("\n",[j,n,f])));
print1(j-1,":"n", "))} \\ M. F. Hasler, Sep 28 2007

Extensions

a(4) = 613668 (p=5, q=613673) from M. F. Hasler, Sep 28 2007
a(5) from Donovan Johnson, Jul 11 2010

A239586 Prime factor >= other prime factor of n-th brilliant number, cf. A078972.

Original entry on oeis.org

2, 3, 3, 5, 7, 5, 7, 5, 7, 7, 11, 13, 13, 17, 19, 17, 19, 23, 17, 23, 29, 19, 31, 19, 29, 23, 31, 37, 23, 41, 43, 37, 29, 47, 31, 23, 41, 29, 43, 53, 31, 47, 37, 59, 29, 61, 53, 41, 37, 31, 43, 67, 59, 41, 71, 61, 47, 73, 43, 29, 37, 79, 67, 47, 31, 53, 83
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 22 2014

Keywords

Comments

a(n) = A006530(A078972(n)) = A078972(n) / A239585(n).
A055642(a(n)) = A055642(A239585(n)).

Examples

			See A239585.

Links

Crossrefs

Subsequence of A084127.
Cf. A006530, A055642, A078972, A239585.

Programs

  • Haskell
    a239586 n = a078972 n `div` a239585 n
  • Mathematica
    Table[With[{f = FactorInteger[k]}, If[Total[f[[All, 2]]] == 2 && Length[Union[IntegerLength[f[[All, 1]]]]] == 1, f[[-1, 1]], Nothing]], {k, 1000}] (* Paolo Xausa, Oct 02 2024 *)
dlist2[d_] := Union[Times @@@ Tuples[Prime[Range[PrimePi[10^(d-1)] + 1, PrimePi[10^d]]], 2]]; (* Generates terms with d-digits prime factors -- faster but memory intensive *)
Map[FactorInteger[#][[-1,1]]&,Flatten[Array[dlist2,2]]] (* Paolo Xausa, Oct 08 2024 *)

A176347 n-th semiprime minus sum of its prime factors.

Original entry on oeis.org

0, 1, 3, 3, 5, 7, 11, 9, 15, 11, 19, 15, 23, 17, 23, 21, 35, 31, 39, 35, 27, 29, 47, 43, 35, 59, 39, 63, 41, 55, 71, 59, 45, 71, 51, 71, 87, 57, 95, 99, 59, 79, 83, 107, 65, 91, 69, 119, 111, 71, 119, 77, 103, 131, 81, 143, 115, 87, 119, 143, 159, 95, 131, 99, 167, 159, 101
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Apr 15 2010

Keywords

Programs

  • Maple
    From R. J. Mathar, Apr 20 2010: (Start)
isA001358 := proc(n) numtheory[bigomega](n) = 2 ; end proc:
A001358 := proc(n) option remember ; if n = 1 then return 4 ; else for a from procname(n-1)+1 do if isA001358(a) then return a; end if; end do; end if; end proc:
A084126 := proc(n) min(op(numtheory[factorset](A001358(n)))) ; end proc:
A084127 := proc(n) max(op(numtheory[factorset](A001358(n)))) ; end proc:
A068318 := proc(n) A084126(n)+A084127(n) ; end proc:
A176347 := proc(n) A001358(n)-A068318(n) ; end proc: seq(A176347(n),n=1..80) ; (End)

Formula

a(n) = A001358(n) - A068318(n).

Extensions

Entries checked by R. J. Mathar, Apr 20 2010

A176486 Numbers n such that semiprime(n)/prime(k)=prime and semiprime(n+1)/prime(k+1)=prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 14, 18, 21, 29, 35, 36, 39, 41, 42, 45, 52, 58, 59, 62, 71, 73, 87, 91, 96, 97, 104, 116, 120, 127, 137, 141, 142, 156, 168, 169, 170, 178, 179, 181, 185, 188, 204, 211, 227, 245, 246, 249, 250, 254, 255, 261, 263, 279, 281, 285, 290, 297, 305
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Apr 18 2010

Keywords

Comments

Indices n such that the (n+1)st semiprime has a prime factor which is the next prime after one of the prime factors of the n-th semiprime. - R. J. Mathar, Apr 20 2010

Examples

			a(1)=1 because semiprime(1)/prime(1)=2 and semiprime(2)/prime(2)=2;
a(2)=2 because semiprime(2)/prime(1)=3 and semiprime(3)/prime(2)=3;
a(3)=3 because semiprime(3)/prime(2)=3 and semiprime(4)/prime(3)=2.

Links

Crossrefs

Cf. A084126, A084127.

Programs

  • Maple
    From R. J. Mathar, Apr 20 2010: (Start)
isA001358 := proc(n) numtheory[bigomega](n) = 2 ; end proc:
A001358 := proc(n) option remember ; if n = 1 then return 4 ; else for a from procname(n-1)+1 do if isA001358(a) then return a; end if; end do; end if; end proc:
A084126 := proc(n) min(op(numtheory[factorset](A001358(n)))) ; end proc:
A084127 := proc(n) max(op(numtheory[factorset](A001358(n)))) ; end proc:
A176486 := proc(n) if n = 1 then 1; else for a from procname(n-1)+1 do spl := A084126(a) ; sph := A084127(a) ; sp2l := A084126(a+1) ; sp2h := A084127(a+1) ; if sp2l = nextprime(spl) or sp2h = nextprime(spl) or sp2l = nextprime(sph) or sp2h = nextprime(sph) then return a; end if; end do: end if; end proc:
seq(A176486(n),n=1..80) ; (End)
  • Mathematica
    sppQ[{a_,b_}]:=Module[{t=NextPrime[Transpose[FactorInteger[a]][[1]]],c,d}, c=t[[1]];d=If[Length[t]>1,t[[2]],t[[1]]];Divisible[b,c]|| Divisible[ b,d]]; Flatten[ Position[Partition[Select[Range[1500],PrimeOmega[#] == 2&],2,1],?sppQ]] (* _Harvey P. Dale, Mar 16 2015 *)

Extensions

Corrected (59, 137, 142 inserted, 147 removed) and extended by R. J. Mathar, Apr 20 2010

A321788 Product of semiprime factors using lunar arithmetic.

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 3, 11, 5, 12, 11, 12, 5, 12, 13, 22, 7, 13, 11, 13, 22, 21, 13, 23, 22, 11, 21, 15, 22, 23, 13, 31, 22, 15, 22, 33, 23, 22, 17, 111, 21, 31, 33, 17, 22, 33, 21, 111, 25, 22, 31, 22, 33, 23, 22, 113, 33, 22, 31, 35, 111, 22, 33, 101, 27, 41, 102, 111, 31, 102, 43, 31, 102, 33, 113, 112, 45
Offset: 1

Views

Author

G. L. Honaker, Jr., Nov 18 2018

Keywords

Examples

			a(16)=22 because the 16th semiprime is 46 = 2*23. In lunar arithmetic the product becomes 22.

Links

Crossrefs

Cf. A001358, A087062, A084126, A084127.

Programs

  • Mathematica
    ladd[x_, y_] := FromDigits[MapThread[Max, IntegerDigits[#, 10, Max@ IntegerLength [{x, y}]] & /@ {x, y}]]; lmult[x_, y_] := Fold[ladd, 0, Table[10^i, {i, IntegerLength[y] - 1, 0, -1}]*FromDigits /@ Transpose@Partition[Min[##] & @@@ Tuples[IntegerDigits[{x, y}]], IntegerLength[y]]]; s={}; Do[If[PrimeOmega[n]==2, f=FactorInteger[n]; x=f[[1,1]]; y=n/x; m=lmult[x,y]; AppendTo[s, m]],{n,1,300}]; s (* Amiram Eldar, Nov 19 2018 after Davin Park at A087062 *)

Formula

a(n) = A087062(A084126(n), A084127(n)). - Michel Marcus, Nov 20 2018

A339410 If the n-th semiprime is p*q with p<=q primes, a(n) is the area of the triangle with vertices (1,p), (p,q) and (q,p*q).

Original entry on oeis.org

1, 1, 6, 2, 9, 8, 6, 35, 40, 54, 10, 104, 54, 135, 24, 209, 126, 64, 70, 90, 350, 405, 72, 154, 594, 190, 740, 64, 819, 280, 216, 330, 989, 54, 1274, 504, 22, 1595, 256, 550, 1710, 640, 714, 270, 2079, 874, 2345, 648, 56, 2484, 90, 2925, 1144, 286, 3239, 936, 1450, 3740, 1560, 216, 832, 4464
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Dec 03 2020

Keywords

Examples

			For n = 5 the 5th semiprime is 14=2*7, and the area of the triangle with vertices (1,2), (2,7) and (7,14) is a(5)=9.

Links

Crossrefs

Cf. A001358, A084126, A084127.

Programs

  • Maple
    N:= 1000: # for semiprimes <= N
SP:= select(t -> numtheory:-bigomega(t)=2, [$4..N]):
f:= proc(n) local p,q;
  p,q:= (min,max)(numtheory:-factorset(n));
  (q-1)*abs(p^2-q)/2
end proc:
map(f, SP);
  • Mathematica
    ar[{a_,b_}]:=Abs[Det[{{1,a,b},{a,b,a b},{1,1,1}}]]/2; ar/@(If[Length[#]==1,Flatten[ {#,#}],#]&/@(FactorInteger[#][[;;,1]]&/@Select[Range[200],PrimeOmega[ #] == 2&])) (* Harvey P. Dale, Mar 05 2023 *)

Formula

a(n) = (q-1)*|p^2-q|/2 where p = A084126(n) and q = A084127(n).

A175711 Primes p such that p-th semiprime=r*k and r>=p>=k.

Original entry on oeis.org

2, 3, 5, 7, 11, 29, 41, 43, 47, 53, 59, 67, 73, 97, 107, 109, 113, 137, 149, 151, 167, 179, 191, 193, 211, 227, 229, 233, 241, 263, 269, 277, 281, 307, 311, 317, 359, 373, 379, 383, 389, 449, 457, 487, 491, 499, 521, 563, 571, 577, 587, 593, 607, 631, 661, 677
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Aug 12 2010

Keywords

Comments

a(1)=2 because 2nd semiprime=3*2 and 3>2=2, a(2)=3 because 3rd semiprime=3*3 and 3=3=3, a(3)=5 because 5th semiprime=7*2 and 7>5>2.

Crossrefs

Cf. A084126, A084127.
Previous Showing 31-40 of 42 results. Next

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