A111206 -id:A111206 - cp's OEIS frontend

A157357 Products of 3 distinct triple-safe primes.

777239, 1555559, 3112199, 4409399, 10635959, 12192599, 23348519, 23796743, 30612839, 47610023, 48628127, 55778519, 67454423, 91581239, 95286263, 97290047, 99883319, 102996599, 104812679, 135002663, 137841647, 148398599, 162707543, 170450999, 172007639, 186520823

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

			777239=23*47*719; 23, 47, and 719 are triple-safe prime numbers.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A157358, A001358, A005384, A005385, A006881, A007304, A066179, A111206, A157342, A157344, A157345, A157346, A157347, A157352, A157353, A157354, A157355, A157356

lst={};Do[If[Plus@@Last/@FactorInteger[n]==3,a=Length[First/@FactorInteger[n]];If[a==3,b=First/@FactorInteger[n];c=b[[1]];d=b[[2]];e=b[[3]];If[PrimeQ[cx=(c-1)/2]&&PrimeQ[cy=(cx-1)/2]&&PrimeQ[(cy-1)/2]&&PrimeQ[dx=(d-1)/2]&&PrimeQ[dy=(dx-1)/2]&&PrimeQ[(dy-1)/2]&&PrimeQ[ex=(e-1)/2]&&PrimeQ[ey=(ex-1)/2]&&PrimeQ[(ey-1)/2],AppendTo[lst,n]]]],{n,9!,11!}];lst

list(lim)=my(v=List(), P=select(p->isprime(p\2) && isprime(p\4) && isprime(p\8), primes([11, sqrtint(lim\11+1)-1])), p, q, t); for(i=1, #P, p=P[i]; if(p^3>=lim, break); for(j=i+1, #P, q=P[j]; t=p*q; forprime(r=q+4, lim\t, if(isprime(r\2) && isprime(r\4) && isprime(r\8), listput(v, r*t))))); Set(v); \\ Charles R Greathouse IV, Oct 14 2021

a(5)-a(26) from Charles R Greathouse IV, Oct 14 2021

A113433 Semi-Pierpont semiprimes: products of exactly two Pierpont primes A005109.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 21, 25, 26, 34, 35, 38, 39, 49, 51, 57, 65, 74, 85, 91, 95, 111, 119, 133, 146, 169, 185, 194, 218, 219, 221, 247, 259, 289, 291, 323, 326, 327, 361, 365, 386, 481, 485, 489, 511, 514, 545, 579, 629, 679, 703, 763, 771, 815, 866, 949, 965

Offset: 1

Jonathan Vos Post, Nov 01 2005

Semiprime both of whose prime factors are Pierpont primes (A005109), which are primes of the form (2^K)*(3^L)+1. Not to be confused with A113432: Pierpont semiprimes [Semiprimes of the form (2^K)*(3^L)+1]. This terminology itself is by analogy to what Tomaszewski used for the Sophie Germain counterparts A111153 and A111206.

			a(1) = 4 = 2^2 = [(2^0)*(3^0)+1]*[(2^1)*(3^0)+1] = A005109(1)*A005109(1).
a(2) = 6 = 2*3 = [(2^0)*(3^0)+1]*[(2^1)*(3^0)+1] = A005109(1)*A005109(2).
a(3) = 9 = 3^2 = [(2^1)*(3^0)+1]*[(2^1)*(3^0)+1] = A005109(2)*A005109(2).
a(4) = 10 = 2*5 = [(2^0)*(3^0)+1]*[(2^2)*(3^0)+1] = A005109(1)*A005109(3).
a(5) = 14 = 2*7 = [(2^0)*(3^0)+1]*[(2^1)*(3^1)+1] = A005109(1)*A005109(4).
a(6) = 15 = 3*5 = [(2^1)*(3^0)+1]*[(2^2)*(3^0)+1] = A005109(2)*A005109(3).

Vincenzo Librandi, Table of n, a(n) for n = 1..498
Chris Caldwell, "Pierpont primes." primeform posting, Oct 25, 2005.
Chris Caldwell, "Pierpont primes." primeform posting, Oct 25, 2005. [Cached copy]
Eric Weisstein's World of Mathematics, Pierpont Prime
Eric Weisstein's World of Mathematics, Semiprime

Cf. A001358, A003586, A005109, A055600, A111153, A111206, A113432, A113434.

Select[Range[10^3], Plus @@ Last /@ FactorInteger[ # ] == 2 && And @@ (Max @@ First /@ FactorInteger[ # - 1] < 5 &) /@ First /@ FactorInteger[ # ] &] (* Ray Chandler, Jan 24 2006 *)

{a(n)} = Semiprimes A001358 both of whose factors are of the form (2^K)*(3^L)+1. {a(n)} = {A005109(i)*A005109(j) for integers i and j not necessarily distinct}.

A113434 Semi-Pierpont semiprimes which are also Pierpont semiprimes.

Original entry on oeis.org

4, 9, 10, 25, 49, 65, 289

Offset: 1

Jonathan Vos Post, Nov 01 2005

Semiprimes both of whose prime factors are Pierpont primes (A005109), which are primes of the form (2^K)*(3^L)+1 and where the semiprime is itself of the form (2^K)*(3^L)+1.
No more under 10^50; what is the next element of this sequence?
No more terms <= 10^100. - Robert Israel, Mar 10 2017
This sequence is complete, see Links. - Charlie Neder, Feb 04 2019

			a(1) = 4 = 2^2 = [(2^0)*(3^0)+1]*[(2^0)*(3^0)+1] = (2^0)*(3^1)+1.
a(2) = 9 = 3^2 = [(2^1)*(3^0)+1]*[(2^1)*(3^0)+1] = (2^3)*(3^0)+1.
a(3) = 10 = 2*5 = [(2^0)*(3^0)+1]*[(2^2)*(3^0)+1] = (2^0)*(3^2)+1.
a(4) = 25 = 5^2 = [(2^2)*(3^0)+1]*[(2^2)*(3^0)+1] = (2^3)*(3^1)+1.
a(5) = 49 = 7^2 = [(2^1)*(3^1)+1]*[(2^1)*(3^1)+1] = (2^4)*(3^1)+1.
a(6) = 65 = 5*13 = [(2^2)*(3^0)+1]*[(2^2)*(3^1)+1] = (2^6)*(3^0)+1.
a(7) = 289 = 17^2 = [(2^4)*(3^0)+1]*[(2^4)*(3^0)+1] = (2^5)*(3^2)+1.

Chris Caldwell, "Pierpont primes." primeform posting, Oct 25, 2005.
Chris Caldwell, "Pierpont primes." primeform posting, Oct 25, 2005. [Cached copy]
Charlie Neder, Proof of the completeness of this sequence
Eric Weisstein's World of Mathematics, Pierpont Prime
Eric Weisstein's World of Mathematics, Semiprime

Cf. A001358, A003586, A005109, A055600, A111153, A111206, A113432, A113433.

N:= 10^100: # to get all terms <= N
PP:= select(isprime, {seq(seq(1+2^i*3^j, i=0..ilog2((N-1)/3^j)),j=0..floor(log[3](N-1)))}):
SP:= select(t -> t <= N and t = 1+2^padic:-ordp(t-1,2)*3^padic:-ordp(t-1,3), [seq(seq(PP[i]*PP[j], j=1..i),i=1..nops(PP))]):
sort(convert(SP,list)); # Robert Israel, Mar 10 2017

{a(n)} = intersection of A113432 and A113433. {a(n)} = Semiprimes A001358 of the form (2^K)*(3^L)+1 both of whose factors are of the form (2^K)*(3^L)+1. {a(n)} = {integers P such that, for nonnegative integers I, J, K, L, m, n there is a solution to (2^I)*(3^J)+1 = [(2^K)*(3^L)+1]*[(2^m)*(3^n)+1] where both [(2^K)*(3^L)+1] and [(2^m)*(3^n)+1] are prime}.

A111207 Numbers that are both Sophie Germain semiprimes and semi-Sophie Germain semiprimes.

Original entry on oeis.org

4, 10, 25, 46, 55, 106, 123, 145, 159, 205, 226, 267, 339, 358, 415, 466, 529, 573, 583, 718, 753, 843, 865, 979, 1077, 1195, 1243, 1257, 1293, 1366, 1405, 1465, 1473, 1486, 2098, 2157, 2206, 2427, 2455, 2545, 2563, 2581, 2599, 2629, 2809, 2818, 2998, 3057

Offset: 1

Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Oct 24 2005

nonn

This is the intersection of the sequence of Sophie Germain semiprimes (A111153) and semi-Sophie Germain semiprimes (A111206).

			a(4)=46 because 46 is the 4th semiprime such that 2*46+1=93 is a semiprime and both of its factors are Sophie Germain primes: 2*2+1=5 and 2*23+1=47.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A111153, A001358, A005384, A111206.

seqQ[n_] := AllTrue[{n, 2*n + 1}, PrimeOmega[#] == 2 &] && AllTrue[First /@ FactorInteger[n], PrimeQ[2*# + 1] &]; Select[Range[3000], seqQ] (* Amiram Eldar, May 10 2020 *)

Extended by Ray Chandler, Oct 31 2005

A157355 Products of 3 distinct not safe primes.

Original entry on oeis.org

78, 102, 114, 174, 186, 222, 246, 258, 318, 366, 402, 426, 438, 442, 474, 494, 534, 582, 606, 618, 646, 654, 663, 678, 741, 754, 762, 786, 806, 822, 834, 894, 906, 942, 962, 969, 978, 986, 1038, 1054, 1066, 1086, 1102, 1118, 1131, 1146, 1158, 1178, 1182

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

nonn

78=2*3*13; 2,3 and 13 are not safe prime numbers,...

Cf. A001358, A005384, A005385, A006881, A007304, A111206, A157342, A157344, A157345, A157346, A157347, A157352, A157353, A157354

lst={};Do[If[Plus@@Last/@FactorInteger[n]==3,a=Length[First/@FactorInteger[n]];If[a==3,b=First/@FactorInteger[n];c=b[[1]];d=b[[2]];e=b[[3]];If[ !PrimeQ[(c-1)/2]&&!PrimeQ[(d-1)/2]&&!PrimeQ[(e-1)/2],AppendTo[lst,n]]]],{n,7!}];lst

A157356 Products (semiprimes) of two distinct double-safe primes.

Original entry on oeis.org

253, 517, 1081, 1837, 3841, 3949, 7849, 7909, 8257, 15829, 16537, 16873, 22429, 31669, 33097, 33793, 44869, 45397, 46897, 54109, 59953, 62029, 63877, 65197, 66217, 66517, 67633, 79717, 83149, 83677, 84997, 93817, 94921, 95833, 108229

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

nonn

253=11*23; 11 and 23 are double safe prime numbers; (11-1)/2=5; (5-1)/2=2(prime); (23-1)/2=11; (11-1)/2=5(prime), ...

Cf. A001358, A005384, A005385, A006881, A007304, A066179, A111206, A157342, A157344, A157345, A157346, A157347, A157352, A157353, A157354, A157355

lst={};Do[If[Plus@@Last/@FactorInteger[n]==2,a=Length[First/@FactorInteger[n]];If[a==2,b=First/@FactorInteger[n];c=b[[1]];d=b[[2]];If[PrimeQ[cx=(c-1)/2]&&PrimeQ[(cx-1)/2]&&PrimeQ[dx=(d-1)/2]&&PrimeQ[(dx-1)/2],AppendTo[lst,n]]]],{n,9!}];lst

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A157357 Products of 3 distinct triple-safe primes.

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A113433 Semi-Pierpont semiprimes: products of exactly two Pierpont primes A005109.

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A113434 Semi-Pierpont semiprimes which are also Pierpont semiprimes.

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A111207 Numbers that are both Sophie Germain semiprimes and semi-Sophie Germain semiprimes.

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A157355 Products of 3 distinct not safe primes.

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A157356 Products (semiprimes) of two distinct double-safe primes.

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