A066179 -id:A066179 - cp's OEIS frontend

A068443 Triangular numbers which are the product of two primes.

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

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

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

			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.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

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

Cf. A075875.

q:= n-> is(numtheory[bigomega](n)=2):
select(q, [i*(i+1)/2$i=0..1000])[];  # Alois P. Heinz, Mar 27 2024

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 *)

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

A010054(a(n))*A064911(a(n)) = 1. - Reinhard Zumkeller, Dec 03 2009

a(n) = A000217(A164977(n)). - Zak Seidov, Feb 16 2015

Edited by Robert G. Wilson v, Jul 08 2002

Definition corrected by Zak Seidov, Mar 09 2008

A065966 Numbers k such that phi(k) / 2 is prime.

Original entry on oeis.org

5, 7, 8, 9, 10, 11, 12, 14, 18, 22, 23, 46, 47, 59, 83, 94, 107, 118, 166, 167, 179, 214, 227, 263, 334, 347, 358, 359, 383, 454, 467, 479, 503, 526, 563, 587, 694, 718, 719, 766, 839, 863, 887, 934, 958, 983, 1006, 1019, 1126, 1174, 1187, 1283, 1307, 1319

Offset: 1

Joseph L. Pe, Dec 08 2001

nonn

This is probably an infinite sequence, but a proof would be nice. Are there infinitely many consecutive terms of the sequence which are also consecutive integers? (For example, 7, 8 and 46, 47.)

Apart from 8, 9, 12 and 18, all the terms of the sequence are safe primes or twice safe primes. It is not known if there are infinitely many safe primes (cf. A005385, A005384). For consecutive terms of the sequence which are also consecutive integers see A066179. - Vladeta Jovovic, Dec 16 2001

			phi(46)/2 = 22/2 = 11, a prime.

Harry J. Smith, Table of n, a(n) for n = 1..1000
André Hernández-Espiet, Reyes M. Ortiz-Albino, On the Characterization of tau(n)-Atoms, arXiv:1905.02834 [math.NT], 2019. See Proposition 3.1.

Cf. A000010, A005385, A005384, A066179, A006530, A052126, A068211, A068212.

Select[Range[1400],PrimeQ[EulerPhi[#]/2]&] (* Harvey P. Dale, Feb 11 2020 *)

for(n=3,5000, if(isprime(eulerphi(n)/2),print1(n,",")))

{ n=0; for (m=3, 10^9, if (isprime(eulerphi(m)/2), write("b065966.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Nov 05 2009

Numbers k such that A068212(k) = 2.

More terms from Jason Earls, Dec 09 2001

Edited by Charles R Greathouse IV, Mar 18 2010

A292936 a(n) = the least k >= 0 such that floor(n/(2^k)) is a nonprime; a(n) is degree of the "safeness" of prime, 0 if n is not a prime, 1 for unsafe primes (A059456), and k >= 2 for primes that are (k-1)-safe but not k-safe.

Original entry on oeis.org

0, 1, 1, 0, 2, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 5, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Antti Karttunen, Sep 27 2017

nonn

Records occur at positions 1, 2, 5, 11, 23, 47, 2879, ... (A292937).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A007814, A078349, A292599, A292937.
Cf. A000040, A005385, A066179, A157358, A157359 (positions of terms that are > k, for k = 0..4).
Cf. A059456 (positions of ones).

A292936 := proc(n)
    for k from 0 do
        if not isprime(floor(n/2^k)) then
            return k;
        end if;
    end do:
end proc:
seq(A292936(n),n=1..100) ; # R. J. Mathar, Sep 28 2017

Table[SelectFirst[Range[0, 10], ! PrimeQ@ Floor[n/(2^#)] &], {n, 105}] (* Michael De Vlieger, Sep 29 2017 *)

A292936(n) = { my(k=0); while(isprime(n), n >>= 1; k++); k; };

(define (A292936 n) (A007814 (1+ (A292599 n))))

a(n) = A007814(1+A292599(n)).
For n >= 1, a(n) <= A078349(n).
For n > 47, a(n) <= A007814(1+n).

A157358 Triple-safe primes p: p, (p-1)/2, (p-3)/4, and (p-7)/8 are all prime.

Original entry on oeis.org

23, 47, 719, 1439, 2879, 4079, 9839, 11279, 21599, 28319, 51599, 84719, 92399, 95279, 96959, 137279, 157679, 159119, 178799, 209519, 219839, 243119, 349199, 429119, 430799, 441839, 462719, 481199, 491279, 507359, 533999, 571199, 597599

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

These occur in a proof of nonexistence of a certain type of permutation group for p (Theorem 8 by Ito). - R. J. Mathar, May 29 2011

			(23-1)/2=11, (11-1)/2=5, (5-1)/2=2(prime), ...

Noboru Ito, Transitive permutation groups of degree p=2q+1, p and q being prime numbers, Bull. AMS 69 (2) (1963), 165-192.

Cf. A005385, A066179, A157357.

lst={};Do[p=Prime[n];If[PrimeQ[a=(p-1)/2]&&PrimeQ[b=(a-1)/2]&&PrimeQ[(b-1)/2],AppendTo[lst,p]],{n,9!}];lst

is(n)=n%8==7 && isprime(n) && isprime(n\2) && isprime(n\4) && isprime(n\8) \\ Charles R Greathouse IV, Oct 14 2021

list(lim)=my(v=List()); forprimestep(p=23,lim\1,8, if(isprime(p\8) && isprime(p\4) && isprime(p\2), listput(v,p))); Vec(v); \\ Charles R Greathouse IV, Oct 14 2021

a(n) >> n log^(4) n. - 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

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

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

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

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

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

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

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

A157359 Quatro-safe primes.

Original entry on oeis.org

47, 1439, 2879, 858239, 861599, 982559, 1014719, 1067999, 2029439, 2034239, 2297759, 2683679, 2978399, 3301919, 4068479, 4288799, 4737599, 5454719, 6484319, 6753119, 7145759, 8624159, 9511199, 9717119, 10533599, 10739999

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

nonn

(47-1)/2=23,(23-1)/2=11,(11-1)/2=5,(5-1)/2=2(prime),...

Cf. A005385, A066179, A157357, A157358

lst={};Do[p=Prime[n];If[PrimeQ[a=(p-1)/2]&&PrimeQ[b=(a-1)/2]&&PrimeQ[c=(b-1)/2]&&PrimeQ[(c-1)/2],AppendTo[lst,p]],{n,10!}];lst
Select[Prime[Range[711000]],AllTrue[Rest[NestList[(#-1)/2&,#,4]], PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 31 2018 *)

A162019 Double-safe primes which are also double-Sophie Germain primes.

Original entry on oeis.org

11, 359, 719, 214559, 215399, 245639, 253679, 266999, 507359, 508559, 574439, 670919, 744599, 825479, 1017119, 1072199, 1184399, 1363679, 1621079, 1688279, 1786439, 2156039, 2377799, 2429279, 2633399, 2684999, 2900039, 3103799

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 24 2009

nonn

The intersection of the primes in A066179 and those in A007700: they remain prime after each of two successive applications of the substitution p->(p-1)/2, and remain prime after each two successive applications of the substitution p->2p+1.

			a(1)=11 is double safe: (11-1)/2=5; (5-1)/2=2, and double Sophie-Germain: 2*11+1=23; 2*23+1=47.

Cf. A007700, A023302, A066179, A157359.

lst={};Do[p=Prime[n];If[PrimeQ[safe=(p-1)/2],If[PrimeQ[(safe-1)/2],If[PrimeQ[sophie=2*p+1],If[PrimeQ[2*sophie+1],AppendTo[lst,p]]]]],{n,3*9!}];lst

a(n) = 4*A023302(n) + 3 = (A157359(n)-3)/4. - R. J. Mathar, Jun 26 2009

Edited by R. J. Mathar, Jun 26 2009

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

Vladimir Joseph Stephan Orlovsky, Feb 27 2009

nonn

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), ...

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

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

A292937 a(0)=1, followed by highly safe primes: positions of records in A292936.

Original entry on oeis.org

1, 2, 5, 11, 23, 47, 2879, 71850239, 2444789759, 21981381119

Offset: 0

Antti Karttunen, Sep 28 2017

The starting offset is 0 to accommodate 1, which is only nonprime in this sequence, and also to align with the indexing used in A110056.

Sequence starts like A007505, and at least for terms a(5) .. a(9) is equal to A110056.

Cf. A007505, A066174, A110056, A292936.

Cf. A000040, A005385, A066179, A157358, A157359 (each starts with the term a(1) .. a(5) of this sequence).

With[{s = Table[SelectFirst[Range[0, 10], ! PrimeQ@ Floor[n/(2^#)] &], {n, 10^7}]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Sep 29 2017 *)

A328799 Primes p that are simultaneously of the forms 2q+1, 4r+3, 6s+5 and 8t+7 where q,r,s,t are primes.

Original entry on oeis.org

23, 47, 1439, 2879, 11279, 51599, 209519, 243119, 349199, 507359, 700319, 903359, 1190639, 1342079, 1650959, 1956719, 2978399, 3304079, 3376559, 3841679, 4858559, 5404319, 5454719, 6207599, 6486479, 7682399, 7825439, 8169599, 8826479, 8970959, 9546959

Offset: 1

J. M. Bergot and Robert Israel, Nov 18 2019

nonn

All terms == 23 (mod 24). All but the first == 47 (mod 48).

			a(3)=1439 is a term because 1439=2*719+1=4*359+3=6*239+5=8*179+7 and 1439, 719, 359, 239 and 179 are all primes.

Robert Israel, Table of n, a(n) for n = 1..2000

Cf. A005385, A066179, A157358.

map(t -> 24*t+23, select(k -> andmap(isprime, [3*k+2,4*k+3,6*k+5,12*k+11,24*k+23]), [0, seq(k,k=1..10^6,2)]));

Clarified definition. - N. J. A. Sloane, Nov 15 2021

cp's OEIS Frontend

A068443 Triangular numbers which are the product of two primes.

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A065966 Numbers k such that phi(k) / 2 is prime.

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A292936 a(n) = the least k >= 0 such that floor(n/(2^k)) is a nonprime; a(n) is degree of the "safeness" of prime, 0 if n is not a prime, 1 for unsafe primes (A059456), and k >= 2 for primes that are (k-1)-safe but not k-safe.

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A157358 Triple-safe primes p: p, (p-1)/2, (p-3)/4, and (p-7)/8 are all prime.

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A157357 Products of 3 distinct triple-safe primes.

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A157359 Quatro-safe primes.

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A162019 Double-safe primes which are also double-Sophie Germain primes.

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A328799 Primes p that are simultaneously of the forms 2q+1, 4r+3, 6s+5 and 8t+7 where q,r,s,t are primes.