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 16 results. Next

A068443 Triangular numbers which are the product of two primes.

Original entry on oeis.org

6, 10, 15, 21, 55, 91, 253, 703, 1081, 1711, 1891, 2701, 3403, 5671, 12403, 13861, 15931, 18721, 25651, 34453, 38503, 49141, 60031, 64261, 73153, 79003, 88831, 104653, 108811, 114481, 126253, 146611, 158203, 171991, 188191, 218791, 226801, 258121, 269011
Offset: 1

Views

Author

Stephan Wagler (stephanwagler(AT)aol.com), Mar 09 2002

Keywords

Comments

These triangular numbers are equal to p * (2p +- 1).
All terms belong to A006987. For n>2 all terms are odd and belong to A095147. - Alexander Adamchuk, Oct 31 2006
A156592 is a subsequence. - Reinhard Zumkeller, Feb 10 2009
Triangular numbers with exactly 4 divisors. - Jon E. Schoenfield, Sep 05 2018

Examples

			Triangular numbers begin 0, 1, 3, 6, 10, ...; 6=2*3, and 2 and 3 are two distinct primes; 10=2*5, and 2 and 5 are two distinct primes, etc. - _Vladimir Joseph Stephan Orlovsky_, Feb 27 2009
a(11) = 1891 and 1891 = 31 * 61.

Links

Crossrefs

Cf. A000217, A005382, A005384, A006987, A095147, A001358, A005385, A006881, A007304, A066179, A111206, A157342, A157344-A157347, A157352-A157357, A164977.
Cf. A075875.

Programs

  • Maple
    q:= n-> is(numtheory[bigomega](n)=2):
select(q, [i*(i+1)/2$i=0..1000])[];  # Alois P. Heinz, Mar 27 2024
  • Mathematica
    Select[ Table[ n(n + 1)/2, {n, 1000}], Apply[Plus, Transpose[ FactorInteger[ # ]] [[2]]] == 2 &]
Select[Accumulate[Range[1000]],PrimeOmega[#]==2&] (* Harvey P. Dale, Apr 03 2016 *)
  • PARI
    list(lim)=my(v=List());forprime(p=2,(sqrtint(lim\1*8+1)+1)\4, if(isprime(2*p-1),listput(v,2*p^2-p)); if(isprime(2*p+1), listput(v,2*p^2+p))); Vec(v) \\ Charles R Greathouse IV, Jun 13 2013

Formula

A010054(a(n))*A064911(a(n)) = 1. - Reinhard Zumkeller, Dec 03 2009
a(n) = A000217(A164977(n)). - Zak Seidov, Feb 16 2015

Extensions

Edited by Robert G. Wilson v, Jul 08 2002
Definition corrected by Zak Seidov, Mar 09 2008

A157342 Semiprimes that are the product of two non-Sophie Germain primes.

Original entry on oeis.org

49, 91, 119, 133, 169, 217, 221, 247, 259, 289, 301, 323, 329, 361, 403, 413, 427, 469, 481, 497, 511, 527, 553, 559, 589, 611, 629, 679, 703, 707, 721, 731, 749, 763, 767, 793, 799, 817, 871, 889, 893, 923, 949, 959, 961, 973, 1003, 1027, 1037, 1043, 1057
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

Keywords

Examples

			49 = 7*7, 2*7 + 1 = 15 (not prime);
91 = 7*13, 2*7 + 1 = 15 (not prime), 2*13 + 1 = 27 (not prime); ...

Links

Crossrefs

Cf. A001358, A005384, A111206.

Programs

  • Mathematica
    lst={};Do[If[Plus@@Last/@FactorInteger[n]==2,a=First/@FactorInteger[n];b=a[[1]];k=0;If[Length[a]==2,c=a[[2]];If[PrimeQ[2*c+1],k=1]];If[ !PrimeQ[2*b+1]&&k==0,AppendTo[lst,n]]],{n,7!}];lst
With[{nn=60},Take[Times@@@Tuples[Select[Prime[Range[nn]],!PrimeQ[ 2#+1]&], 2] // Union,nn]] (* Harvey P. Dale, Feb 15 2017 *)

A157344 Semiprimes that are the product of two distinct Sophie Germain primes.

Original entry on oeis.org

6, 10, 15, 22, 33, 46, 55, 58, 69, 82, 87, 106, 115, 123, 145, 159, 166, 178, 205, 226, 249, 253, 262, 265, 267, 319, 339, 346, 358, 382, 393, 415, 445, 451, 466, 478, 502, 519, 537, 562, 565, 573, 583, 586, 655, 667, 699, 717, 718, 753, 838, 843, 862, 865
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

Keywords

Comments

6=2*3; 2 and 3 are Sophie Germain primes, 10=2*5; 2 and 5 are Sophie Germain primes, 15=3*5; 3 and 5 are Sophie Germain primes, ...

Links

Crossrefs

Cf. A001358, A005384, A111206, A157342, A006881.

Programs

  • Mathematica
    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[2*c+1]&&PrimeQ[2*d+1],AppendTo[lst,n]]]],{n,7!}];lst
nn=100;With[{sgp=Select[Prime[Range[nn]],PrimeQ[2#+1]&]},Take[ Union[ Select[ Times @@@ Subsets[sgp,{2}],PrimeOmega[#]==2&]],nn]] (* Harvey P. Dale, Nov 22 2012 *)

A113432 Pierpont semiprimes: semiprimes of the form (2^K)*(3^L)+1.

Original entry on oeis.org

4, 9, 10, 25, 33, 49, 55, 65, 82, 129, 145, 217, 289, 649, 865, 973, 1537, 1945, 2049, 2305, 3073, 4097, 4609, 5833, 6145, 6913, 8193, 8749, 9217, 11665, 13123, 15553, 20737, 23329, 24577, 27649, 31105, 34993, 41473, 62209, 69985, 73729, 78733
Offset: 1

Views

Author

Jonathan Vos Post, Nov 01 2005

Keywords

Examples

			a(1) = 4 = (2^0)*(3^1)+1 = 2^2 hence the semiprime A001358(1).
a(2) = 9 = (2^3)*(3^0)+1 = 3^2 hence the semiprime A001358(3).
a(3) = 10 = (2^0)*(3^2)+1 = 2 * 5 hence the semiprime A001358(4).
a(4) = 25 = (2^3)*(3^1)+1 = 5^2 hence the semiprime A001358(9).
a(5) = 33 = (2^5)*(3^0)+1 = 3 * 11 hence the semiprime A001358(11).
a(6) = 49 = (2^4)*(3^1)+1 = 7^2 hence the semiprime A001358(17).
a(7) = 55 = (2^1)*(3^3)+1 = 5 * 11 hence the semiprime A001358(19).

Links

Crossrefs

Cf. A001358, A003586, A005109, A055600, A111153, A111206, A113433, A113434.

Programs

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

Formula

{a(n)} = Intersection of {(2^K)*(3^L)+1} A055600 and semiprimes A001358. a(n) is in this sequence iff there exist nonnegative integers K and L such that Omega((2^K)*(3^L)+1) = 2.

A157345 Semiprimes that are the product of two distinct non-Sophie Germain primes.

Original entry on oeis.org

91, 119, 133, 217, 221, 247, 259, 301, 323, 329, 403, 413, 427, 469, 481, 497, 511, 527, 553, 559, 589, 611, 629, 679, 703, 707, 721, 731, 749, 763, 767, 793, 799, 817, 871, 889, 893, 923, 949, 959, 973, 1003, 1027, 1037, 1043, 1057, 1099, 1121, 1139, 1141
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

Keywords

Comments

91 = 7*13; 7 and 13 are not Sophie Germain primes, ...

Links

Crossrefs

Cf. A001358, A005384, A111206, A157342, A006881, A157344.

Programs

  • Mathematica
    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[2*c+1]&&!PrimeQ[2*d+1],AppendTo[lst,n]]]],{n,7!}];lst
With[{nn=50},Take[Union[Times@@@Subsets[Select[Prime[Range[nn]], !PrimeQ[ 2#+1]&],{2}]],nn]] (* Harvey P. Dale, May 04 2015 *)

A157352 Products (semiprimes) of two distinct safe primes.

Original entry on oeis.org

35, 55, 77, 115, 161, 235, 253, 295, 329, 413, 415, 517, 535, 581, 649, 749, 835, 895, 913, 1081, 1135, 1169, 1177, 1253, 1315, 1357, 1589, 1735, 1795, 1837, 1841, 1909, 1915, 1969, 2335, 2395, 2429, 2461, 2497, 2513, 2515, 2681, 2773, 2815, 2893, 2935
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

Keywords

Comments

35=5*7; 5 and 7 are safe primes, 55=5*11; 5 and 11 are safe primes,...

Examples

			a(1) = 35 since 35 = 5 * 7, and (5 - 1)/2 = 2 and (7 - 1)/2 = 3 are both prime, thus 5 and 7 are distinct safe primes.

Links

Crossrefs

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

Programs

  • Mathematica
    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[(c-1)/2]&&PrimeQ[(d-1)/2],AppendTo[lst,n]]]],{n,7!}];lst
Select[Select[Range@ 3000, PrimeNu@ # == 2 &], Times @@ Map[If[PrimeQ[(# - 1)/2], #, 0] &, Map[First, FactorInteger@ #]] == # &] (* Michael De Vlieger, Feb 28 2016 *)
Module[{upto=3000,sp},sp=Select[Prime[Range[PrimePi[upto/5]]],PrimeQ[(#-1)/2]&];Select[Union[Times@@@Subsets[sp,{2}]],#<+upto&]] (* Harvey P. Dale, Aug 25 2017 *)

Extensions

Example corrected by Harvey P. Dale, Aug 25 2017

A157346 Products of 3 distinct Sophie Germain primes.

Original entry on oeis.org

30, 66, 110, 138, 165, 174, 230, 246, 290, 318, 345, 410, 435, 498, 506, 530, 534, 615, 638, 678, 759, 786, 795, 830, 890, 902, 957, 1038, 1074, 1130, 1146, 1166, 1245, 1265, 1310, 1334, 1335, 1353, 1398, 1434, 1506, 1595, 1686, 1695, 1730, 1749, 1758, 1790
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

Keywords

Examples

			30 = 2*3*5; 2,3 and 5 are distinct Sophie Germain primes.
66 = 2*3*11; 2,3 and 11 are distinct Sophie Germain primes.

Links

Crossrefs

Cf. A001358, A005384, A111206, A157342, A006881, A157344, A157345, A007304.

Programs

  • Mathematica
    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[2*c+1]&&PrimeQ[2*d+1]&&PrimeQ[2*e+1],AppendTo[lst,n]]]],{n,7!}];lst
With[{sgps=Select[Prime[Range[100]],PrimeQ[2#+1]&]},Take[Union[ Times@@@ Subsets[sgps,{3}]],60]] (* Harvey P. Dale, Aug 10 2011 *)

A157347 Products of 3 distinct non-Sophie Germain primes.

Original entry on oeis.org

1547, 1729, 2261, 2821, 3367, 3689, 3913, 4123, 4199, 4277, 4403, 4921, 5117, 5369, 5551, 5593, 5719, 6097, 6251, 6461, 6643, 6851, 7021, 7189, 7259, 7657, 7847, 7973, 8029, 8113, 8177, 8449, 8687, 8827, 8911, 9139, 9191, 9331, 9373, 9401, 9443, 9503
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

Keywords

Examples

			1547 = 7*13*17 is a term: its prime factors 7, 13, and 17 are not Sophie Germain primes.

Links

Crossrefs

Cf. A001358, A005384, A111206, A157342, A006881, A157344, A157345, A007304, A157346.

Programs

  • Magma
    S:=[ p: p in PrimesUpTo(120) | not IsPrime(2*p+1) ]; T:=[ q: a, b, c in S | a lt b and b lt c and q lt 10000 where q is a*b*c ]; Sort(~T); T; // Klaus Brockhaus, Apr 11 2009
  • Mathematica
    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[2*c+1]&&!PrimeQ[2*d+1]&&!PrimeQ[2*e+1],AppendTo[lst,n]]]],{n,8!}];lst

Extensions

Entries verified by Klaus Brockhaus, Apr 11 2009

A157353 Products (semiprimes) of two distinct primes that are not safe primes.

Original entry on oeis.org

6, 26, 34, 38, 39, 51, 57, 58, 62, 74, 82, 86, 87, 93, 106, 111, 122, 123, 129, 134, 142, 146, 158, 159, 178, 183, 194, 201, 202, 206, 213, 218, 219, 221, 226, 237, 247, 254, 262, 267, 274, 278, 291, 298, 302, 303, 309, 314, 323, 326, 327, 339, 346, 362, 377
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

Keywords

Examples

			6=2*3; 2 and 3 are not safe primes.
26=2*13; 2 and 13 are not safe primes.

Crossrefs

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

Programs

  • Mathematica
    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[(c-1)/2]&&!PrimeQ[(d-1)/2],AppendTo[lst,n]]]],{n,7!}];lst

A157354 Products of 3 distinct safe primes.

Original entry on oeis.org

385, 805, 1265, 1645, 1771, 2065, 2585, 2905, 3245, 3619, 3745, 4543, 4565, 5405, 5845, 5885, 6265, 6391, 6785, 7567, 7945, 8239, 9185, 9205, 9499, 9545, 9845, 11891, 12145, 12305, 12485, 12565, 12859, 13363, 13405, 13783, 13865, 14465, 14927
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

Keywords

Examples

			385=5*7*11; 5,7 and 11 are safe primes.

Links

Crossrefs

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

Programs

  • Mathematica
    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
  • PARI
    list(lim)=my(v=List(),P=select(p->isprime(p\2), primes([5,sqrtint(lim\5+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), listput(v,r*t))))); Set(v); \\ Charles R Greathouse IV, Oct 14 2021
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