A001358 -id:A001358 - cp's OEIS frontend

A330478 Semiprimes A001358(k) = pq such that pq+p+q and rs+r+s are consecutive primes, where A001358(k+1)=rs.

33, 1718, 4174, 7971, 8434, 11114, 13011, 14005, 16645, 17571, 29787, 30574, 43647, 58414, 63177, 65006, 69694, 71794, 87218, 95314, 97827, 104485, 125738, 126394, 150334, 193594, 196341, 198694, 200378, 201094, 212631, 212847, 227554, 239314, 243591, 254427, 276085, 277594, 288818, 291514

Offset: 1

J. M. Bergot and Robert Israel, Dec 15 2019

nonn

			a(3)=4174=2*2087, the next semiprime is 4178=2*2089, and 4174+2+2087=6263 and 4178+2+2089=6269 are consecutive primes.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A001358.

g:= proc(n) local F;
  F:= ifactors(n)[2];
  if nops(F)=2 then n+F[1][1]+F[2][1] else n+2*F[1][1] fi
end proc:
SP:= select(t -> numtheory:-bigomega(t)=2, [seq(i,i=4..3*10^5)]):
nSP:= nops(SP):
P1:= map(g, SP):
SP[select(t -> isprime(P1[t]) and nextprime(P1[t])=P1[t+1], [$1..nSP-1])];

Select[Partition[Union@ Apply[Join, Table[Flatten@ {p #, Sort[{p, #}]} & /@ Prime@ Range@ PrimePi@ Floor[Max[#]/p], {p, #}]] &@ Prime@ Range[3*10^4], 2, 1], And[AllTrue[{#1, #2}, PrimeQ], #2 == NextPrime@ #1] & @@ {Total@ #1, Total@ #2} & @@ # &][[All, 1, 1]] (* Michael De Vlieger, Dec 15 2019 *)

A339314 a(n) is the least semiprime k > n-th semiprime s = A001358(n) such that k-s and k+s are both semiprimes.

Original entry on oeis.org

10, 15, 86, 25, 35, 106, 25, 55, 94, 51, 58, 85, 39, 77, 94, 95, 74, 55, 106, 178, 143, 155, 69, 118, 95, 142, 121, 118, 119, 91, 146, 142, 115, 206, 115, 115, 206, 169, 134, 146, 143, 178, 133, 158, 155, 262, 177, 158, 178, 155, 159, 183, 254, 194, 205, 202, 226, 187, 298, 206, 226, 209

Offset: 1

Zak Seidov, Dec 17 2020

nonn

Differences k - s: 6, 9, 77, 15, 21, 91, 4, 33, 69, 25, ... with minimal value 4.
What about the maximal value of k - s?
k-s is unbounded, because the gaps between semiprimes are unbounded. In fact, given any n distinct primes, by the Chinese Remainder Theorem there exist n consecutive positive integers that are each divisible by the cube of one of these primes (and thus not semiprimes). - Robert Israel, Dec 27 2020

			s=4, k=10, 6 and 14 are all semiprimes,
s=6, k=15, 9 and 21 are all semiprimes,
s=9, k=86, 77 and 95 are all semiprimes.

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

Cf. A001358.

N:= 10^3:
SP:= select(t -> numtheory:-bigomega(t)=2, [$4..N]):
f:= proc(n) local i,s;
  s:= SP[n];
  for i from n+1 do
    if numtheory:-bigomega(SP[i]-s)=2 and numtheory:-bigomega(SP[i]+s)=2 then return SP[i] fi
  od;
end proc:
map(f, [$1..100]); # Robert Israel, Dec 27 2020

issemip(n) = bigomega(n)==2;
lista(nn) = {my(v = select(issemip, [1..nn])); for (n=1, #v, my(ik=n+1, s=v[n]); while (!(issemip(v[ik]+s) && issemip(v[ik]-s)), ik++; if (ik>#v, return)); print1(v[ik], ", "););} \\ Michel Marcus, Dec 19 2020

A365939 Gilbreath transform of semiprimes (A001358).

Original entry on oeis.org

4, 2, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1

Offset: 1

N. J. A. Sloane, Sep 25 2023

nonn

Leading terms of rows of the array in A131749.

Cf. A001358, A196251, A361897, A362451.

A365939list[upto_]:=If[upto<4,{},Module[{d=Select[Range[upto],PrimeOmega[#]==2&]},Join[{4},Table[First[d=Abs[Differences[d]]],Length[d]-1]]]];A365939list[500] (* Paolo Xausa, Sep 25 2023 *)

More terms from Pontus von Brömssen, Sep 25 2023

A105934 Positive integers n such that n^22 + 1 is semiprime (A001358).

Original entry on oeis.org

116, 176, 184, 300, 444, 470, 584, 690, 696, 950

Offset: 1

Jonathan Vos Post, Apr 26 2005

We have the polynomial factorization: n^22 + 1 = (n^2 + 1) * (n^20 - n^18 + n^16 - n^14 + n^12 - n^10 + n^8 - n^6 + n^4 - n^2 + 1). Hence after the initial n=1 prime, the binomial can never be prime. It can be semiprime iff n^2+1 is prime and (n^20 - n^18 + n^16 - n^14 + n^12 - n^10 + n^8 - n^6 + n^4 - n^2 + 1) is prime.

			116^22 + 1 = 2618639792014920380336685706161496723088736257 = 13457 * 194593133091693570657404005808240820620401,
300^22 + 1 = 3138105960900000000000000000000000000000000000000000001 = 90001 * 34867456593815624270841435095165609271008099910001,
950^22 + 1 = 323533544973709366507562922501564025878906250000000000000000000001 = 902501 * 358485525194663902319845543109164450653136395416736380347501.

Cf. A000040, A001358, A006313, A103854, A104238, A104335, A105041, A105066, A105078, A105122, A105142, A105237, A104479, A104494, A104657, A105282.

IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [2..1000] | IsSemiprime(n^22+1)]; // Vincenzo Librandi, Dec 21 2010

Select[Range[1000], PrimeOmega[#^22 + 1]==2&] (* Vincenzo Librandi, May 24 2014 *)

a(n)^22 + 1 is in A001358. a(n)^2+1 is in A000040 and (a(n)^20 - a(n)^18 + a(n)^16 - a(n)^14 + a(n)^12 - a(n)^10 + a(n)^8 - a(n)^6 + a(n)^4 - a(n)^2 + 1) is in A000040.

A115698 Semiprimes (A001358) whose digit reversal is the product of 2 palindromes greater than 1.

Original entry on oeis.org

4, 6, 9, 21, 22, 33, 46, 51, 55, 65, 77, 82, 94, 121, 202, 203, 253, 262, 303, 393, 407, 427, 445, 446, 451, 485, 505, 559, 583, 626, 667, 669, 671, 687, 707, 789, 803, 849, 869, 889, 939, 1042, 1111, 1441, 1622, 1661, 1991, 2031, 2123, 2157, 2173, 2321

Offset: 1

Giovanni Resta, Jan 31 2006

			669=3*223 is semiprime and 966=6*161.

Cf. A001358, A115683.

A116026 phi(n) plus n gives a semiprime (A001358).

Original entry on oeis.org

4, 5, 9, 10, 11, 13, 17, 21, 26, 29, 30, 43, 45, 47, 49, 55, 57, 58, 61, 66, 67, 70, 71, 73, 75, 82, 87, 89, 99, 101, 102, 103, 106, 107, 109, 111, 115, 119, 123, 127, 129, 130, 146, 151, 153, 154, 175, 181, 182, 183, 185, 190, 191, 195, 197, 202, 203, 205, 207

Offset: 1

Giovanni Resta, Feb 13 2006

nonn

			phi(101)+101=201=3*67.

Cf. A001358, A116015.

A116691 Continued fraction expansion of concatenation of semiprimes (A001358).

Original entry on oeis.org

0, 2, 7, 1, 1, 2, 4, 26, 1, 4, 1, 7, 2, 2, 1, 1, 6, 1, 3, 1, 8, 1, 7, 6, 1, 1, 1, 22, 3, 18, 2, 1, 24, 11, 1, 2, 1, 5, 7, 1, 2, 1, 3, 20, 1, 1, 5, 2, 70, 1, 1, 2, 2, 1, 1, 1, 1, 2, 5, 1, 2, 1, 1, 6, 2, 153, 1, 2, 1, 10, 13, 4, 1, 4, 1, 3, 1, 1, 3, 2, 2, 6, 3, 1, 4, 1, 7, 1, 1, 6, 1, 1, 14, 2, 1, 2, 2, 2, 2, 2

Offset: 0

Jonathan Vos Post, Mar 15 2006

This is the semiprime analog of A030168: continued fraction expansion of Copeland-Erdős constant (concatenated primes). Decimal expansion of real number formed from concatenation of first 101 digits of A001358, to first 50 terms of continued fraction expansion: 0.469101415212225... = 0 + 1/2+ 1/7+ 1/1+ 1/1+ 1/2+ 1/4+ 1/26+ 1/1+ 1/4+ 1/1+... It seems likely that the real number itself is transcendental.

Cf. A033308 Decimal expansion of Copeland-Erdős constant: concatenate primes.

Extended and edited by Charles R Greathouse IV, Apr 25 2010

A131701 Decimal expansion of the continued fraction 4+6/(9+10/(14+15/21+...)) where the terms are the semiprimes: A001358.

Original entry on oeis.org

4, 6, 1, 9, 7, 7, 6, 8, 9, 0, 3, 3, 0, 3, 6, 8, 9, 3, 8, 0, 6, 8, 2, 1, 1, 9, 9, 6, 8, 4, 3, 7, 4, 2, 3, 8, 4, 0, 1, 7, 2, 8, 2, 5, 5, 6, 4, 4, 1, 2, 9, 2, 3, 3, 8, 9, 8, 4, 2, 0, 8, 2, 3, 2, 9, 6, 3, 7, 4, 4, 6, 3, 0, 8, 4, 4, 2, 5, 9, 6, 4, 4, 8, 6, 7, 2, 8, 7, 1, 8, 8, 0, 3, 1, 6, 6, 5, 0, 8, 2, 3, 6, 4, 2, 8

Offset: 1

Jonathan Vos Post & Robert G. Wilson v, Sep 08 2007

			4.6197768903303689380682119968437423840172825564412923389842082329...

Cf. A001358, A113011.

semiPrimeQ[n_] := Plus @@ Last /@ FactorInteger@n == 2; lst = Select[Range@ 400, semiPrimeQ@ # &]; First@ RealDigits@ N[ Fold[ Last@ #2 + First@ #2/#1 &, 4, Partition[Reverse@ lst, 2]], 111] (* Robert G. Wilson v *)

A137253 Semiprimes (A001358) which are not the sum of two semiprimes.

Original entry on oeis.org

4, 6, 9, 22, 33

Offset: 1

Zak Seidov, Mar 11 2008

Apparently the list is complete.

Cf. A001358.

A146490 Decimal expansion of Product_{q in A001358} (1-1/(q^3*(q-1))).

Original entry on oeis.org

9, 9, 3, 5, 2, 1, 5, 8, 9, 7, 1, 0, 5, 0, 5, 4, 6, 0, 6, 7, 5, 4, 0, 9, 2, 6, 9, 2, 4, 1, 4, 1, 6, 4, 2, 9, 4, 0, 1, 1, 1, 5, 0, 7, 8, 6, 7, 7, 8, 1, 5, 6, 6, 0, 1, 8, 8

Offset: 0

R. J. Mathar, Feb 13 2009

Semiprime analog of A065415.

			0.993521589710505460675409269.. = (1-1/192)*(1-1/1080)*(1-1/5832)*...

The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_2(s)/j at r=3, where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.

cp's OEIS Frontend

A330478 Semiprimes A001358(k) = p*q such that p*q+p+q and r*s+r+s are consecutive primes, where A001358(k+1)=r*s.

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Maple

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A339314 a(n) is the least semiprime k > n-th semiprime s = A001358(n) such that k-s and k+s are both semiprimes.

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PARI

A365939 Gilbreath transform of semiprimes (A001358).

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Mathematica

Extensions

A105934 Positive integers n such that n^22 + 1 is semiprime (A001358).

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Magma

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Formula

A115698 Semiprimes (A001358) whose digit reversal is the product of 2 palindromes greater than 1.

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A116026 phi(n) plus n gives a semiprime (A001358).

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Examples

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A116691 Continued fraction expansion of concatenation of semiprimes (A001358).

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Extensions

A131701 Decimal expansion of the continued fraction 4+6/(9+10/(14+15/21+...)) where the terms are the semiprimes: A001358.

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Examples

Crossrefs

Programs

Mathematica

A137253 Semiprimes (A001358) which are not the sum of two semiprimes.

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Author

Keywords

Comments

Crossrefs

A146490 Decimal expansion of Product_{q in A001358} (1-1/(q^3*(q-1))).

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Author

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Comments

A330478 Semiprimes A001358(k) = pq such that pq+p+q and rs+r+s are consecutive primes, where A001358(k+1)=rs.