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 1-8 of 8 results.

A269452 phi(A157352(n)), n >= 1, where phi is Euler's totient function A000010, and A157352 gives the products of two distinct safe primes.

Original entry on oeis.org

24, 40, 60, 88, 132, 184, 220, 232, 276, 348, 328, 460, 424, 492, 580, 636, 664, 712, 820, 1012, 904, 996, 1060, 1068, 1048, 1276, 1356, 1384, 1432, 1660, 1572, 1804, 1528, 1780, 1864, 1912, 2076, 2332, 2260, 2148, 2008, 2292, 2668, 2248, 2620, 2344, 2796, 2868, 3012, 2872, 3460, 3652, 3772, 3372
Offset: 1

Views

Author

Marina Ibrishimova, Feb 27 2016

Keywords

Comments

phi(p*q) = (p-1)(q-1) where p, q are distinct safe primes.
2^(a(n)/2) == 1 (mod A157352(n)). For the reference see a comment on A269454. - Wolfdieter Lang, Mar 31 2016

Links

Crossrefs

Cf. A000010, A157352, A269454.

Programs

  • Mathematica
    EulerPhi /@ Select[Select[Range@ 4000, PrimeNu@ # == 2 &], Times @@ Map[If[PrimeQ[(# - 1)/2], #, 0] &, Map[First, FactorInteger@ #]] == # &] (* Michael De Vlieger, Feb 28 2016 *)

Formula

a(n) = phi(A157352(n)), n >= 1.

Extensions

More terms from Michael De Vlieger, Feb 28 2016

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

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

A157357 Products of 3 distinct triple-safe primes.

Original entry on oeis.org

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

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

Keywords

Examples

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

Links

Crossrefs

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

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[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
  • PARI
    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

Extensions

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

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

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

Keywords

Comments

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

Crossrefs

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

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

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

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

Keywords

Comments

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), ...

Crossrefs

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

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[cx=(c-1)/2]&&PrimeQ[(cx-1)/2]&&PrimeQ[dx=(d-1)/2]&&PrimeQ[(dx-1)/2],AppendTo[lst,n]]]],{n,9!}];lst

A269453 The order of 2 mod m when m is the product of two distinct safe primes.

Original entry on oeis.org

12, 20, 30, 44, 33, 92, 110, 116, 69, 174, 164, 230, 212, 246, 290, 318, 332, 356, 410, 253, 452, 249, 530, 534, 524, 638, 678, 692, 716, 830, 393, 902, 764, 890, 932, 956, 1038, 1166, 1130, 537, 1004, 573, 1334, 1124, 1310, 1172, 1398, 717, 753, 1436, 1730, 913, 1886, 1686, 1790
Offset: 1

Views

Author

Marina Ibrishimova, Feb 27 2016

Keywords

Comments

The smallest positive integer k for which 2^k == 1 (mod m) where m = p*q with p, q distinct safe primes.

Crossrefs

Cf. A002326, A157352, A269452.

Programs

  • Mathematica
    MultiplicativeOrder[2, #] & /@ Select[Select[Range@ 4200, PrimeNu@ # == 2 &], Times @@ Map[If[PrimeQ[(# - 1)/2], #, 0] &, Map[First, FactorInteger@ #]] == # &] (* Michael De Vlieger, Feb 28 2016 *)

Formula

a(n) is the order of 2 modulo A157352(n). - Wolfdieter Lang, Mar 31 2016

Extensions

More terms from Michael De Vlieger, Feb 28 2016
Showing 1-8 of 8 results.

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