A084126 -id:A084126 - cp's OEIS frontend

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

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

Juri-Stepan Gerasimov, Apr 18 2010

nonn

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

			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.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A084126, A084127.

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)

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 *)

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

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

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

Metin Sariyar, Table of n, a(n) for n = 1..10000
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, Journal of Integer Sequences, Vol. 14 (2011), Article 11.9.8. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
Caldwell and Honaker, Prime Curio for 58.

Cf. A001358, A087062, A084126, A084127.

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 *)

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

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

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

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

			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.

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

Cf. A001358, A084126, A084127.

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);

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 *)

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

Juri-Stepan Gerasimov, Aug 12 2010

nonn

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.

Cf. A084126, A084127.

A176550 Numbers k such that (k-th odd semiprime)/(j-th prime) is prime and ((k+1)-th odd semiprime)/((j+1)-th prime) is prime for some j.

Original entry on oeis.org

1, 2, 3, 5, 9, 11, 18, 20, 21, 22, 24, 29, 34, 35, 42, 43, 57, 61, 74, 79, 81, 95, 101, 102, 111, 112, 118, 120, 123, 128, 136, 151, 153, 154, 163, 166, 167, 170, 173, 177, 190, 194, 195, 198, 199, 203, 205, 208, 212, 213, 239, 242, 245, 263, 267, 271, 278, 283

Offset: 1

Juri-Stepan Gerasimov, Apr 20 2010

nonn

Cf. A046315, A084126, A084127, A176484.

A046315 := proc(n) option remember; if n = 1 then 9; else for a from procname(n-1)+2 by 2 do if numtheory[bigomega](a) = 2 then return a; end if; end do: end if; end proc:
A020639 := proc(n) numtheory[factorset](n) ; min(op(%)) ; end proc:
isA176550 := proc(n) os := A046315(n) ; p := A020639(os) ; q := os/p ; ( A046315(n+1) mod nextprime(p) ) = 0 or (A046315(n+1) mod nextprime(q) = 0 ) ; end proc:
for n from 1 to 300 do if isA176550(n) then printf("%d,",n) ; end if; end do: # R. J. Mathar, May 30 2010

Corrected (39 removed, 43 inserted) and extended by R. J. Mathar, May 30 2010

A177193 The twin primes p such that m and k are also twin primes where m*k=semiprime(p).

Original entry on oeis.org

3, 7, 11, 13, 17, 19, 31, 43, 59, 61, 71, 101, 103, 139, 179, 311, 313, 347, 349, 421, 463, 523, 569, 571, 599, 601, 617, 659, 823, 1019, 1021, 1031, 1061, 1093, 1153, 1319, 1321, 1429, 1619, 1667, 1697, 1721, 1787, 1789

Offset: 1

Juri-Stepan Gerasimov, May 04 2010

nonn

			a(1)=3 because 3*3=semiprime(3) and 3=twin prime, a(2)=7 because 3*7=semiprime(7) and 3,7 are twin primes.

Cf. A001097, A001358, A084126, A084127.

Entries checked and sequence extended beyond 139 - R. J. Mathar, May 06 2010

cp's OEIS Frontend

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

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A321788 Product of semiprime factors using lunar arithmetic.

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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).

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

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A176550 Numbers k such that (k-th odd semiprime)/(j-th prime) is prime and ((k+1)-th odd semiprime)/((j+1)-th prime) is prime for some j.

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A177193 The twin primes p such that m and k are also twin primes where m*k=semiprime(p).

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Extensions

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