A177215 -id:A177215 - cp's OEIS frontend

A177216 Numbers k that are the products of two distinct primes such that 2k-1, 4k-3, 8k-7, 16k-15, 32k-31, 64k-63 and 128*k-127 are also products of two distinct primes.

11293, 12139, 25399, 31261, 36199, 44869, 49471, 62521, 72397, 83086, 89737, 91705, 98941, 124846, 125041, 134023, 138994, 144793, 164041, 166171, 170431, 173311, 182527, 199543, 224962, 244294, 258169, 259891, 263086, 275281, 277987

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 04 2010

nonn

			11293 is a term because 11293 = 23*491, 2*11293 - 1 = 22585 = 5*4517, 4*11293 - 1 = 45169 = 17*2657, 8*11293 - 1 = 90337 = 13*6949, 16*11293 - 1 = 180673 = 79*2287, 32*11293 - 1 = 361345 = 5*72269, 64*11293 - 1 = 722689 = 11*65699, and 128*11293 - 1 = 1445377 = 193*7489.

Cf. A006881, A177210, A177211, A177212, A177213, A177214, A177215.

f[n_]:=Last/@FactorInteger[n]=={1,1}; lst={};Do[If[f[n]&&f[2*n-1]&&f[4*n-3]&&f[8*n-7]&&f[16*n-15]&&f[32*n-31]&&f[64*n-63]&&f[128*n-127],AppendTo[lst,n]],{n,11293,4*9!}];lst
tdpQ[n_]:=Module[{f=Table[n*2^i-(2^i-1),{i,0,7}]},And@@(Transpose[ FactorInteger[ #]][[2]]=={1,1}&/@f)]; Select[Range[300000],tdpQ] (* Harvey P. Dale, Apr 02 2015 *)

Example moved from Comments field to Example field by Harvey P. Dale, Apr 02 2015

A177221 Numbers k that are the products of two distinct primes such that 2*k + 1 is also the product of two distinct primes.

Original entry on oeis.org

10, 34, 38, 46, 55, 57, 77, 91, 93, 106, 118, 123, 129, 133, 143, 145, 159, 161, 177, 185, 201, 203, 205, 206, 213, 218, 226, 235, 259, 267, 291, 295, 298, 305, 314, 327, 334, 335, 339, 358, 365, 377, 381, 394, 395, 403, 407, 415, 417, 446, 447, 458, 466, 469

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 05 2010

nonn

			10 is in the sequence because 10 = 2*5 and 2*10+1 = 21 = 3*7.

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

Cf. A006881, A111153, A177210, A177211, A177212, A177213, A177214, A177215, A177216, A177217, A177220.

isA006881:= proc(n) local F;
  F:= ifactors(n)[2];
  nops(F)=2 and F[1, 2]+F[2, 2]=2
end proc:
filter:= n -> andmap(isA006881, [n, 2*n+1]); select(filter, [$1..1000]); # Robert Israel, Nov 09 2017

f[n_]:=Last/@FactorInteger[n]=={1,1}; lst={};Do[If[f[n]&&f[2*n+1],AppendTo[lst,n]],{n,0,3*6!}];lst
Select[Range[500],PrimeNu[#]==PrimeOmega[#]==PrimeNu[2#+1] == PrimeOmega[ 2#+1] == 2&] (* Harvey P. Dale, Feb 22 2018 *)

A177217 The products k of two distinct primes such that 2k-1, 4k-3, 8k-7, 16k-15, 32k-31, 64k-63, 128k-127 and 256k-255 are also products of two distinct primes.

Original entry on oeis.org

31261, 36199, 44869, 49471, 62521, 72397, 83086, 138994, 173311, 182527, 224962, 259891, 277987, 346621, 370126, 415423, 443746, 464245, 480331, 519781, 544006, 563326, 599245, 693241, 784681, 880561, 928489, 980743, 991237, 1032937

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 04 2010

nonn

			31261 is a term because 31261 = 43*727, 2*31261 - 1 = 62521 = 103*607, ..., 256*31261 - 1 = 8002561 = 7*1143223, ...

Cf. A006881, A177210, A177211, A177212, A177213, A177214, A177215, A177216.

f[n_]:=Last/@FactorInteger[n]=={1,1}; lst={};Do[If[f[n]&&f[2*n-1]&&f[4*n-3]&&f[8*n-7]&&f[16*n-15]&&f[32*n-31]&&f[64*n-63]&&f[128*n-127]&&f[256*n-255],AppendTo[lst,n]],{n,31261,5*9!}];lst

A177220 Smallest numbers with properties: products of two distinct primes of the form a(k)=2^n*m-(2^n-1), n:0->k.

Original entry on oeis.org

6, 26, 33, 247, 247, 634, 694, 11293, 31261, 31261, 173311, 173311, 2212801

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 04 2010

nonn

6=2*3, 26=2*13;2*26-1=51=3*17, 33=3*11;2*33-1=65=5*13;2*65-1=4*33-3=129=3*43, 247=13*19;2*247-1=493=17*29;4*247-3=985=5*197;8*247-1=1969=11*179, 247=13*19;2*247-1=493=17*29;4*247-3=985=5*197;8*247-1=1969=11*179;16*247-15=3937=31*127, 634=2*317;2*634=1267=7*181;4*634-3=2533=17*149;8*634-7=5065=5*1013;16*634=10129=7*1447;32*634=20257=47*431, ..

Cf. A006881, A177210, A177211, A177212, A177213, A177214, A177215, A177216, A177217

f[n_]:=Last/@FactorInteger[n]=={1,1}; g[n_,m_]:=Module[{a=1,x,y,z},Do[x=2^k;y=x-1;z=x*n-y;If[ !f[z],a=0;Break[]],{k,0,m}];a]; q=0;e=5;lst={};Do[e++;If[g[e,q]==1,Print[e];AppendTo[lst,e];q++;e-- ],{n,0,4*10!}];lst

A177222 Numbers k that are the products of two distinct primes, such that 2k + 1 and 4k + 3 are also products of two distinct primes.

Original entry on oeis.org

38, 46, 106, 129, 133, 145, 201, 203, 235, 291, 298, 334, 335, 381, 407, 417, 458, 489, 497, 538, 579, 583, 597, 623, 626, 649, 685, 689, 694, 707, 758, 767, 781, 815, 898, 899, 921, 926, 959, 995, 1073, 1079, 1082, 1094, 1099, 1139, 1142, 1157, 1214, 1226

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 05 2010

nonn

			38 is a term because 38 = 2*19, 2*38 + 1 = 77 = 7*11, and 4*38 + 1 = 155 = 5*31.

Cf. A006881, A111153, A177210, A177211, A177212, A177213, A177214, A177215, A177216, A177217, A177220, A177221.

Cf. A133123 (allows squares also).

f[n_] := Last/@FactorInteger[n] == {1,1};  lst = {}; Do[If[f[n] && f[2*n+1] && f[4*n+3], AppendTo[lst, n]], {n, 1000}]; lst

A177223 Numbers k that are the products of two distinct primes such that 2k+1, 4k+3 and 8*k+7 are also products of two distinct primes.

Original entry on oeis.org

145, 203, 291, 298, 407, 497, 649, 707, 758, 815, 899, 926, 959, 995, 1079, 1094, 1139, 1142, 1157, 1313, 1403, 1415, 1461, 1497, 1538, 1639, 1658, 1691, 1857, 1934, 1945, 1991, 2123, 2159, 2217, 2234, 2315, 2603, 2629, 2807, 2991, 3215, 3254, 3279, 3305

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 05 2010

nonn

A number k is the product of two distinct primes iff k = p*q where p and q are distinct primes. - N. J. A. Sloane, Jan 11 2025

			145 is a term because 145 = 5*29, 2*145 + 1 = 291 = 3*97, 4*145 + 1 = 583 = 11*53, and 8*145 + 1 = 1167 = 3*389.

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

Cf. A006881, A111153, A177210, A177211, A177212, A177213, A177214, A177215, A177216, A177217, A177220, A177221, A177222.

f[n_]:=Last/@FactorInteger[n]=={1,1}; lst={};Do[If[f[n]&&f[2*n+1]&&f[4*n+3]&&f[8*n+7],AppendTo[lst,n]],{n,0,2*7!}];lst
tdpQ[n_]:=With[{c={n, 2n+1, 4n+3,8n+7}},PrimeNu[c]==PrimeOmega[c]=={2,2,2,2}]; Select[Range[3500],tdpQ] (* Harvey P. Dale, Jan 11 2025 *)

cp's OEIS Frontend

A177216 Numbers k that are the products of two distinct primes such that 2*k-1, 4*k-3, 8*k-7, 16*k-15, 32*k-31, 64*k-63 and 128*k-127 are also products of two distinct primes.

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A177221 Numbers k that are the products of two distinct primes such that 2*k + 1 is also the product of two distinct primes.

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A177217 The products k of two distinct primes such that 2*k-1, 4*k-3, 8*k-7, 16*k-15, 32*k-31, 64*k-63, 128*k-127 and 256*k-255 are also products of two distinct primes.

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A177220 Smallest numbers with properties: products of two distinct primes of the form a(k)=2^n*m-(2^n-1), n:0->k.

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A177222 Numbers k that are the products of two distinct primes, such that 2*k + 1 and 4*k + 3 are also products of two distinct primes.

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A177223 Numbers k that are the products of two distinct primes such that 2*k+1, 4*k+3 and 8*k+7 are also products of two distinct primes.

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A177216 Numbers k that are the products of two distinct primes such that 2k-1, 4k-3, 8k-7, 16k-15, 32k-31, 64k-63 and 128*k-127 are also products of two distinct primes.

A177217 The products k of two distinct primes such that 2k-1, 4k-3, 8k-7, 16k-15, 32k-31, 64k-63, 128k-127 and 256k-255 are also products of two distinct primes.

A177222 Numbers k that are the products of two distinct primes, such that 2k + 1 and 4k + 3 are also products of two distinct primes.

A177223 Numbers k that are the products of two distinct primes such that 2k+1, 4k+3 and 8*k+7 are also products of two distinct primes.