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.

Showing 1-10 of 10 results.

A177212 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.

Original entry on oeis.org

247, 249, 295, 395, 422, 478, 493, 502, 519, 589, 634, 694, 721, 755, 955, 1255, 1267, 1294, 1306, 1351, 1387, 1441, 1522, 1546, 1727, 1762, 1942, 2031, 2119, 2155, 2323, 2374, 2449, 2491, 2509, 2533, 2587, 2623, 2661, 2733, 2773, 3005, 3039, 3091, 3334
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 04 2010

Keywords

Examples

			247 is a term because 247 = 13*19, 2*247 - 1 = 493 = 17*29, 4*247-3 = 985 = 5*197, and 8*247 - 1 = 1969 = 11*179.

Links

Crossrefs

Cf. A006881, A177210, A177211.

Programs

  • Maple
    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-1,4*n-3,8*n-7]);
select(filter, [$1..10000]); # Robert Israel, Jul 11 2017
  • Mathematica
    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,0,3*7!}];lst
p2dpQ[n_]:=Transpose[FactorInteger[n]][[2]]=={1,1}; With[{s=Select[Range[ 3500], p2dpQ]},Select[s,AllTrue[{2#-1,4#-3,8#-7},p2dpQ]&]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 27 2015 *)

A177213 Numbers k that are the products of two distinct primes such that 2*k-1, 4*k-3, 8*k-7 and 16*k-15 are also products of two distinct primes.

Original entry on oeis.org

247, 295, 478, 634, 694, 721, 1255, 1267, 1294, 1387, 1546, 1762, 1942, 2323, 2374, 2773, 3005, 3334, 3403, 3883, 3949, 4126, 4714, 4741, 4777, 5062, 5269, 5287, 5353, 5422, 5617, 6583, 6805, 7273, 7495, 8587, 8767, 9017, 9406, 9427, 9847, 10018
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 04 2010

Keywords

Examples

			247 is a term because 247 = 13*19, 2*247 - 1 = 493 = 17*29, 4*247 - 3 = 985 = 5*197, 8*247 - 1 = 1969 = 11*179, and 16*247 - 15 = 3937 = 31*127.

Links

Crossrefs

Cf. A006881, A177210, A177211, A177212.

Programs

  • Mathematica
    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],AppendTo[lst,n]],{n,0,8!}];lst
ptdpQ[n_]:=PrimeNu[n]==PrimeOmega[n]==2; Select[Range[11000],AllTrue[ {#,2#-1,4#-3,8#-7,16#-15},ptdpQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 30 2016 *)

A177214 Numbers k that are the products of two distinct primes such that 2*k-1, 4*k-3, 8*k-7, 16*k-15 and 32*k-31 are also products of two distinct primes.

Original entry on oeis.org

634, 694, 1387, 1942, 3403, 4714, 5062, 5269, 5353, 5617, 6805, 7495, 8587, 9427, 9847, 10018, 10123, 10705, 10942, 11293, 12139, 13162, 13798, 14191, 14989, 15406, 17197, 19735, 20866, 21439, 22114, 22585, 24277, 25009, 25351, 25399, 26734
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 04 2010

Keywords

Examples

			634 is a term since 634 = 2*317, 2*634 - 1 = 1267 = 7*181, 4*634 - 3 = 2533 = 17*149, 8*634 - 7 = 5065 = 5*1013, 16*634 = 10129 = 7*1447, and 32*634 = 20257 = 47*431.

Crossrefs

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

Programs

  • Mathematica
    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],AppendTo[lst,n]],{n,0,9!}];lst

A177215 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 and 64*k-63 are also products of two distinct primes.

Original entry on oeis.org

694, 3403, 4714, 5062, 5353, 7495, 11293, 12139, 13798, 14191, 19735, 21439, 22585, 24277, 25009, 25399, 26734, 26899, 31261, 32959, 35299, 36199, 44869, 48949, 49471, 50797, 58003, 60181, 62521, 70759, 72397, 73909, 75631, 79021, 83086
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 04 2010

Keywords

Examples

			694 is a term because 694 = 2*347, 2*694 - 1 = 1387 = 19*73, 4*694 - 1 = 2773 = 47*59, 8*694 - 1 = 5545 = 5*1109, 16*694 - 1 = 11089 = 13*853, 32*694 - 1 = 22177 = 67*331, and 64*694 - 1 = 44353 = 17*3609.

Crossrefs

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

Programs

  • Mathematica
    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],AppendTo[lst,n]],{n,0,9!}];lst

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.

Original entry on oeis.org

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

Views

Author

Vladimir Joseph Stephan Orlovsky, May 04 2010

Keywords

Examples

			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.

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

Vladimir Joseph Stephan Orlovsky, May 05 2010

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 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.

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

Views

Author

Vladimir Joseph Stephan Orlovsky, May 04 2010

Keywords

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Vladimir Joseph Stephan Orlovsky, May 04 2010

Keywords

Comments

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, ..

Crossrefs

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

Programs

  • Mathematica
    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 2*k + 1 and 4*k + 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

Views

Author

Vladimir Joseph Stephan Orlovsky, May 05 2010

Keywords

Examples

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

Crossrefs

Cf. A006881, A111153, A177210, A177211, A177212, A177213, A177214, A177215, A177216, A177217, A177220, A177221.
Cf. A133123 (allows squares also).

Programs

  • Mathematica
    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 2*k+1, 4*k+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

Views

Author

Vladimir Joseph Stephan Orlovsky, May 05 2010

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
Showing 1-10 of 10 results.

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