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 1-6 of 6 results.

A111344 Pierpont 4-almost primes: numbers with exactly 4 prime divisors, not necessarily distinct, of the form 2^K*3^L + 1.

Original entry on oeis.org

513, 13825, 32769, 59050, 110593, 157465, 177148, 186625, 262145, 279937, 497665, 1259713, 1327105, 2097153, 2125765, 2519425, 4718593, 4782970, 5668705, 6718465, 17915905, 18874369, 22674817, 33554433, 38263753, 56623105
Offset: 1

Views

Author

Jonathan Vos Post, Nov 08 2005

Keywords

Examples

			a(1) = 513 = (2^9)*(3^0)+1 = 3 * 3 * 3 * 19.
a(2) = 13825 = (2^9)*(3^3)+1 = 5 * 5 * 7 * 79.
a(3) = 32769 = (2^15)*(3^0)+1 = 3 * 3 * 11 * 331.
a(4) = 59050 = (2^0)*(3^10)+1 = 2 * 5 * 5 * 1181.
a(10) = 279937 = (2^7)*(3^7)+1 = 7 * 7 * 29 * 197 (lots of sevens).
a(24) = 33554433 = (2^25)*(3^0) = 3 * 11 * 251 * 4051.
a(60) = 31381059610 = (2^0)*(3^22)+1 = 2 * 5 * 5501 * 570461.
a(168) = 16677181699666570 = (2^0)*(3^34)+1 = 2 * 5 * 956353 * 1743831169.

Links

Crossrefs

Intersection of A014613 and A055600.
A005109 gives the Pierpont primes, which are primes of the form (2^K)*(3^L)+1.
A113432 gives the Pierpont semiprimes, 2-almost primes of the form (2^K)*(3^L)+1.
A112797 gives the Pierpont 3-almost primes, of the form (2^K)*(3^L)+1.
A111345 gives the Pierpont 5-almost primes, of the form (2^K)*(3^L)+1.
A111346 gives the Pierpont 6-almost primes, of the form (2^K)*(3^L)+1.
A113739 gives the Pierpont 7-almost primes, of the form (2^K)*(3^L)+1.
A113740 gives the Pierpont 8-almost primes, of the form (2^K)*(3^L)+1.
A113741 gives the Pierpont 9-almost primes, of the form (2^K)*(3^L)+1.

Programs

  • PARI
    is(n)=bigomega(n)==4 && n-1 == 2^valuation(n-1,2)*3^valuation(n-1,3) \\ Charles R Greathouse IV, Feb 01 2017
  • PARI
    list(lim)=my(v=List(),L=lim\1-1); for(e=0,logint(L,3), my(t=3^e); while(t<=L, if(bigomega(t+1)==4, listput(v, t+1)); t*=2)); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

Extensions

Extended by Ray Chandler, Nov 08 2005
Name edited by Charles R Greathouse IV, Feb 01 2017

A113739 Pierpont 7-almost primes. 7-almost primes of form (2^K)*(3^L)+1.

Original entry on oeis.org

339738625, 10460353204, 83682825625, 669462604993, 2641807540225, 3761479876609, 7625597484988, 18075490334785, 35184372088833, 481469424205825, 488038239039169, 570630428688385, 1125899906842625
Offset: 1

Views

Author

Jonathan Vos Post, Nov 08 2005

Keywords

Examples

			a(1) = 339738625 = (2^22)*(3^4)+1 = 5 * 5 * 5 * 17 * 29 * 37 * 149.
a(2) = 10460353204 = (2^0)*(3^21)+1 = 2 * 2 * 7 * 7 * 43 * 547 * 2269.
a(3) = 83682825625 = (2^3)*(3^21)+1 = 5 * 5 * 5 * 5 * 7 * 631 * 30313.
a(4) = 669462604993 = (2^6)*(3^21)+1 = 7 * 13 * 19 * 31 * 67 * 277 * 673.
a(7) = 7625597484988 = (2^0)*(3^27)+1 = 2 * 2 * 7 * 19 * 37 * 19441 * 19927.
a(9) = 35184372088833 = (2^45)*(3^0)+1 = 3 * 3 * 3 * 11 * 19 * 331 * 18837001.
a(13) = 1125899906842625 = (2^50)*(3^0)+1 = 5 * 5 * 5 * 41 * 101 * 8101 * 268501.
a(16) = 5559060566555524 = (2^0)*(3^33)+1 = 2 * 2 * 7 * 67 * 661 * 25411 * 176419.
a(28) = 9223372036854775809 = (2^63)*(3^0)+1 = 3 * 3 * 3 * 19 * 43 * 5419 * 77158673929.

Links

Crossrefs

Intersection of A046308 and A055600.
A005109 gives the Pierpont primes, which are primes of the form (2^K)*(3^L)+1.
A113432 gives the Pierpont semiprimes, 2-almost primes of the form (2^K)*(3^L)+1.
A112797 gives the Pierpont 3-almost primes, of the form (2^K)*(3^L)+1.
A111344 gives the Pierpont 4-almost primes, of the form (2^K)*(3^L)+1.
A111345 gives the Pierpont 5-almost primes, of the form (2^K)*(3^L)+1.
A111346 gives the Pierpont 6-almost primes, of the form (2^K)*(3^L)+1.
A113740 gives the Pierpont 8-almost primes, of the form (2^K)*(3^L)+1.
A113741 gives the Pierpont 9-almost primes, of the form (2^K)*(3^L)+1.

Programs

  • PARI
    list(lim)=my(v=List(), L=lim\1-1); for(e=0, logint(L, 3), my(t=3^e); while(t<=L, if(bigomega(t+1)==7, listput(v, t+1)); t*=2)); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

Formula

a(n) is in this sequence iff there exist nonnegative integers K and L such that Omega((2^K)*(3^L)+1) = 7.

Extensions

Extended by Ray Chandler, Nov 08 2005

A113741 Pierpont 9-almost primes. 9-almost primes of form (2^K)*(3^L)+1.

Original entry on oeis.org

1601009443167990625, 1897492673384285185, 39346408075296537575425, 46005119909369701466113, 221073919720733357899777, 2153693963075557766310748, 3925770232266214525108225
Offset: 1

Views

Author

Jonathan Vos Post, Nov 08 2005

Keywords

Examples

			a(1) = 1601009443167990625 = (2^5)*(3^35)+1 = 5 * 5 * 5 * 5 * 5 * 7 * 11 * 241 * 27608073601.
a(2) = 1897492673384285185 = (2^10)*(3^32)+1 = 5 * 13 * 13 * 13 * 41 * 41 * 373 * 2357 * 116881.

Links

Crossrefs

Intersection of A046312 and A055600.
A005109 gives the Pierpont primes, which are primes of the form (2^K)*(3^L)+1.
A113432 gives the Pierpont semiprimes, 2-almost primes of the form (2^K)*(3^L)+1.
A112797 gives the Pierpont 3-almost primes, of the form (2^K)*(3^L)+1.
A111344 gives the Pierpont 4-almost primes, of the form (2^K)*(3^L)+1.
A111345 gives the Pierpont 5-almost primes, of the form (2^K)*(3^L)+1.
A111346 gives the Pierpont 6-almost primes, of the form (2^K)*(3^L)+1.
A113739 gives the Pierpont 7-almost primes, of the form (2^K)*(3^L)+1.
A113740 gives the Pierpont 8-almost primes, of the form (2^K)*(3^L)+1.

Programs

  • PARI
    list(lim)=my(v=List(), L=lim\1-1); for(e=0, logint(L, 3), my(t=3^e); while(t<=L, if(bigomega(t+1)==9, listput(v, t+1)); t*=2)); Set(v) \\ Charles R Greathouse IV, Feb 02 2017

Formula

a(n) is in this sequence iff there exist nonnegative integers K and L such that Omega((2^K)*(3^L)+1) = 9.

Extensions

Extended by Ray Chandler, Nov 08 2005

A111345 Pierpont 5-almost primes. 5-almost primes of form (2^K)*(3^L)+1.

Original entry on oeis.org

4375, 19684, 7077889, 7962625, 34012225, 100663297, 129140164, 452984833, 459165025, 544195585, 644972545, 918330049, 5159780353, 7346640385, 8589934593, 13947137605, 14495514625, 23219011585, 27518828545, 28991029249
Offset: 1

Views

Author

Jonathan Vos Post, Nov 08 2005

Keywords

Examples

			a(1) = 4375 = (2^1)*(3^7)+1 = 5 * 5 * 5 * 5 * 7.
a(2) = 19684 = (2^0)*(3^9)+1 = 2 * 2 * 7 * 19 * 37.
a(3) = 7077889 = (2^18)*(3^3)+1 = 7 * 13 * 13 * 31 * 193 (prime factors each have all odd digits).
a(4) = 7962625 = (2^15)*(3^5)+1 = 5 * 5 * 5 * 11 * 5791 (again, coincidentally, prime factors each have all odd
digits).
a(7) = 129140164 = (2^0)*(3^17)+1 = 2 * 2 * 103 * 307 * 1021.
a(15) = 8589934593 = (2^33)*(3^0)+1 = 3 * 3 * 67 * 683 * 20857.
a(21) = 34359738369 = (2^35)*(3^0)+1 = 3 * 11 * 43 * 281 * 86171.
a(30) = 793437161473 = (2^11)*(3^18)+1 = 11 * 11 * 11 * 43 * 13863281.
a(32) = 847288609444 = (2^0)*(3^25)+1 = 2 * 2 * 61 * 151 * 22996651.
a(47) = 68630377364884 = (2^0)*(3^29)+1 = 2 * 2 * 523 * 6091 * 5385997.
a(48) = 70368744177665 = (2^46)*(3^0)+1 = 5 * 277 * 1013 * 1657 * 30269.
a(81) = 50031545098999708 = (2^0)*(3^35)+1 = 2 * 2 * 61 * 547 * 374857981681.
a(89) = 144115188075855873 = (2^57)*(3^0)+1 = 3 * 3 * 571 * 174763 * 160465489.
a(99) = 450283905890997364 = (2^0)*(3^37)+1 = 2 * 2 * 18427 * 107671 * 56737873.
a(113) = 4611686018427387905 = (2^62)*(3^0)+1 = 5 * 5581 * 8681 * 49477 * 384773.

Links

Crossrefs

Intersection of A014614 and A055600.
A005109 gives the Pierpont primes, which are primes of the form (2^K)*(3^L)+1.
A113432 gives the Pierpont semiprimes, 2-almost primes of the form (2^K)*(3^L)+1.
A112797 gives the Pierpont 3-almost primes, of the form (2^K)*(3^L)+1.
A111344 gives the Pierpont 4-almost primes, of the form (2^K)*(3^L)+1.
A111346 gives the Pierpont 6-almost primes, of the form (2^K)*(3^L)+1.
A113739 gives the Pierpont 7-almost primes, of the form (2^K)*(3^L)+1.
A113740 gives the Pierpont 8-almost primes, of the form (2^K)*(3^L)+1.
A113741 gives the Pierpont 9-almost primes, of the form (2^K)*(3^L)+1.

Programs

  • PARI
    list(lim)=my(v=List(), L=lim\1-1); for(e=0, logint(L, 3), my(t=3^e); while(t<=L, if(bigomega(t+1)==5, listput(v, t+1)); t*=2)); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

Formula

a(n) is in this sequence iff there exist nonnegative integers K and L such that Omega((2^K)*(3^L)+1) = 5.

Extensions

Extended by Ray Chandler, Nov 08 2005

A111346 Pierpont 6-almost primes. 6-almost primes of form (2^K)*(3^L)+1.

Original entry on oeis.org

14348908, 134217729, 1073741825, 139314069505, 231928233985, 264479053825, 282429536482, 618475290625, 705277476865, 3570467226625, 4398046511105, 8349416423425, 21134460321793, 35664401793025, 91507169819845
Offset: 1

Views

Author

Jonathan Vos Post, Nov 08 2005

Keywords

Examples

			a(1) = 14348908 = (2^0)*(3^15)+1 = 2 * 2 * 7 * 31 * 61 * 271.
a(2) = 134217729 = (2^27)*(3^0)+1 = 3 * 3 * 3 * 3 * 19 * 87211.
a(3) = 1073741825 = (2^30)*(3^0)+1 = 5 * 5 * 13 * 41 * 61 * 1321.
a(4) = 139314069505 = (2^18)*(3^12)+1 = 5 * 13 * 17 * 61 * 337 * 6133.
a(100) = 151115727451828646838273 = (2^77)*(3^0)+1 = 3 * 43 * 617 * 683 * 78233 * 35532364099.
a(127) = 9671406556917033397649409 = (2^83)*(3^0)+1 = 3 * 499 * 1163 * 2657 * 155377 * 13455809771.
a(153) = 523347633027360537213511522 = (2^0)*(3^56)+1 = 2 * 17 * 113 * 193 * 19489 * 36214795668330833.
a(169) = 2475880078570760549798248449 = (2^91)*(3^0)+1 = 3 * 43 * 2731 * 224771 * 1210483 * 25829691707.

Links

Crossrefs

Intersection of A046306 and A055600.
A005109 gives the Pierpont primes, which are primes of the form (2^K)*(3^L)+1.
A113432 gives the Pierpont semiprimes, 2-almost primes of the form (2^K)*(3^L)+1.
A112797 gives the Pierpont 3-almost primes, of the form (2^K)*(3^L)+1.
A111344 gives the Pierpont 4-almost primes, of the form (2^K)*(3^L)+1.
A111345 gives the Pierpont 5-almost primes, of the form (2^K)*(3^L)+1.
A113739 gives the Pierpont 7-almost primes, of the form (2^K)*(3^L)+1.
A113740 gives the Pierpont 8-almost primes, of the form (2^K)*(3^L)+1.
A113741 gives the Pierpont 9-almost primes, of the form (2^K)*(3^L)+1.

Programs

  • PARI
    list(lim)=my(v=List(), L=lim\1-1); for(e=0, logint(L, 3), my(t=3^e); while(t<=L, if(bigomega(t+1)==6, listput(v, t+1)); t*=2)); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

Formula

a(n) is in this sequence iff there exist nonnegative integers K and L such that Omega((2^K)*(3^L)+1) = 6.

Extensions

Extended by Ray Chandler, Nov 08 2005

A112797 Pierpont 3-almost primes. 3-almost primes of form (2^K)*(3^L)+1.

Original entry on oeis.org

28, 244, 325, 385, 730, 1025, 1729, 2188, 5185, 6562, 7777, 16385, 26245, 36865, 46657, 49153, 55297, 82945, 93313, 221185, 354295, 419905, 531442, 559873, 589825, 663553, 708589, 884737, 1119745, 1572865, 1594324, 1889569, 2985985
Offset: 1

Views

Author

Jonathan Vos Post, Nov 08 2005

Keywords

Examples

			a(1) = 28 = (2^0)*(3^3)+1 = 2 * 2 * 7.
a(2) = 244 = (2^0)*(3^5)+1 = 2 * 2 * 61.
a(3) = 325 = (2^2)*(3^4)+1 = 5 * 5 * 13.
a(4) = 385 = (2^7)*(3^1)+1 = 5 * 7 * 11.
a(11) = 7777 = (2^5)*(3^5)+1 = 7 * 11 * 101.
a(115) = 94143178828 = (2^0)*(3^23)+1 = 2 * 2 * 23535794707.
a(119) = 137438953473 = (2^37)*(3^0)+1 = 3 * 1777 * 25781083.
a(196) = 281474976710657 = (2^48)*(3^0)+1 = 193 * 65537 * 22253377.

Links

Crossrefs

Intersection of A014612 and A055600.
A005109 gives the Pierpont primes, which are primes of the form (2^K)*(3^L)+1.
A113432 gives the Pierpont semiprimes, 2-almost primes of the form (2^K)*(3^L)+1.
A111344 gives the Pierpont 4-almost primes, of the form (2^K)*(3^L)+1.
A111345 gives the Pierpont 5-almost primes, of the form (2^K)*(3^L)+1.
A111346 gives the Pierpont 6-almost primes, of the form (2^K)*(3^L)+1.
A113739 gives the Pierpont 7-almost primes, of the form (2^K)*(3^L)+1.
A113740 gives the Pierpont 8-almost primes, of the form (2^K)*(3^L)+1.
A113741 gives the Pierpont 9-almost primes, of the form (2^K)*(3^L)+1.

Programs

  • Mathematica
    Take[Select[2^#[[1]] 3^#[[2]] + 1 & /@ Tuples[Range[0, 20], 2],
PrimeOmega[ #]  ==  3 &] // Union, 40] (* Harvey P. Dale, Jan 02 2021 *)
  • PARI
    list(lim)=my(v=List(), L=lim\1-1); for(e=0, logint(L, 3), my(t=3^e); while(t<=L, if(bigomega(t+1)==3, listput(v, t+1)); t*=2)); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

Formula

a(n) is in this sequence iff there exist nonnegative integers K and L such that Omega((2^K)*(3^L)+1) = 3.

Extensions

Extended by Ray Chandler, Nov 08 2005
Showing 1-6 of 6 results.

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