A007700 -id:A007700 - cp's OEIS frontend

A059762 Initial primes of Cunningham chains of first type with length exactly 3. Primes in A059453 that survive as primes just two "2p+1 iterations", forming chains of exactly 3 terms.

41, 1031, 1451, 1481, 1511, 1811, 1889, 1901, 1931, 3449, 3491, 3821, 3911, 5081, 5441, 5849, 6101, 6131, 7151, 7349, 7901, 8969, 9221, 10691, 10709, 11171, 11471, 11801, 12101, 12821, 12959, 13229, 14009, 14249, 14321, 14669, 14741, 15161

Offset: 1

Labos Elemer, Feb 20 2001

nonn

Primes p such that {(p-1)/2, p, 2p+1, 4p+3, 8p+7} = {composite, prime, prime, prime, composite}.

			41 is a term because 20 and 325 are composites, and 41, 83, and 167 are primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris Caldwell's Prime Glossary, Cunningham chains.
Warut Roonguthai, Yves Gallot's Proth.exe and Cunningham Chains.
Eric Weisstein's World of Mathematics, Cunningham Chain.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059453, A059454, A059455, A007700.

ipccQ[n_]:=Module[{c=(n-1)/2},PrimeQ[NestList[2#+1&,c,4]]=={False, True, True, True, False}]; Select[Prime[Range[2000]],ipccQ] (* Harvey P. Dale, Nov 10 2014 *)

Definition corrected by Alexandre Wajnberg, Aug 31 2005

Offset corrected by Amiram Eldar, Jul 15 2024

A059456 Unsafe primes: primes not in A005385.

Original entry on oeis.org

2, 3, 13, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 71, 73, 79, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 173, 181, 191, 193, 197, 199, 211, 223, 229, 233, 239, 241, 251, 257, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337

Offset: 1

Labos Elemer, Feb 02 2001

nonn

A010051(a(n))*(1-A156659(a(n))) = 1; subsequence of A156657. - Reinhard Zumkeller, Feb 18 2009
Also, primes p such that p-1 is a non-semiprime. - Juri-Stepan Gerasimov, Apr 28 2010
Conjecture: From the sequence of prime numbers, let 2 and remove the first data iteration of 2*p+1; leave 3 and remove the prime data by the iteration 2*p+1 and we get the sequence. Example for p=2, remove(5,11,23,47); p=3, remove(7); p=13, p=17, p=19, p=23, remove(47); and so on. - Vincenzo Librandi, Aug 07 2010

			31 is here because (31-1)/2=15 is not prime. 2 and 3 are here because 1/2 and 1 are not prime numbers.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A005384, A005385, A053176, A059452-A059456, A007700, A005602, A023272, A023302, A023330.

Initial terms for groups in A075712.

Complement[Prime@ Range@ PrimePi@ Max@ #, #] &@ Select[Prime@ Range@ 90, PrimeQ[(# - 1)/2] &] (* Michael De Vlieger, May 01 2016 *)
Select[Prime[Range[100]],PrimeOmega[#-1]!=2&] (* Harvey P. Dale, May 13 2018 *)

is(n)=isprime(n) && !isprime(n\2) \\ Charles R Greathouse IV, May 02 2016

a(n) ~ n log n. - Charles R Greathouse IV, Dec 29 2024

A059761 Initial primes of Cunningham chains of first type with length exactly 2. Primes in A059453 that survive as primes only one "2p-1 iteration", forming chains of exactly 2 terms.

Original entry on oeis.org

3, 29, 53, 113, 131, 173, 191, 233, 239, 251, 281, 293, 419, 431, 443, 491, 593, 641, 653, 659, 683, 743, 761, 809, 911, 953, 1013, 1049, 1103, 1223, 1289, 1499, 1559, 1583, 1601, 1733, 1973, 2003, 2069, 2129, 2141, 2273, 2339, 2351, 2393, 2399, 2543

Offset: 1

Labos Elemer, Feb 20 2001

nonn

Primes p such that {(p-1)/2, p, 2p+1, 4p+3} = {composite, prime, prime, composite}.

			53 is a term because 26 and 215 are composites, and 53 and 107 are primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Chris Caldwell's Prime Glossary, Cunningham chains.
Warut Roonguthai, Yves Gallot's Proth.exe and Cunningham Chains. [Wayback Machine link]
Eric Weisstein's World of Mathematics, Cunningham Chain.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059453, A059454, A059455, A007700.

ccftQ[p_]:=Boole[PrimeQ[{(p-1)/2,p,2 p+1,4 p+3}]]=={0,1,1,0}; Select[ Prime[ Range[400]],ccftQ] (* Harvey P. Dale, Jun 19 2021 *)

A099645 Number of iterations until n reaches a number in A039943 under "x goes to sum of squares of digits of x" map.

Original entry on oeis.org

0, 1, 5, 0, 4, 9, 5, 5, 4, 1, 2, 5, 2, 6, 3, 0, 5, 3, 4, 0, 5, 6, 3, 1, 3, 2, 6, 3, 2, 5, 2, 3, 4, 4, 5, 8, 0, 2, 5, 1, 6, 0, 4, 4, 7, 4, 3, 6, 4, 4, 3, 3, 5, 7, 5, 2, 4, 0, 2, 9, 1, 2, 8, 4, 2, 7, 2, 2, 5, 5, 5, 6, 1, 3, 4, 2, 2, 4, 3, 5, 3, 3, 2, 6, 1, 2, 4, 7, 0, 4, 4, 2, 5, 4, 2, 5, 3, 1, 8, 1, 2, 5, 2, 6, 3

Offset: 1

Labos Elemer, Nov 08 2004

Length of transient when the f[n]=Sum[digit^2 of n] function is iterated.

In A031176 including cycle lengths[=c] of this iteration only c=1 and c=8 occur. A007770 lists cases of c=1, the happy numbers.

			n=99999999999: iteration-list={99999999999,891,146,53,34,25,29,85,89,145,42,20,[4,16,37,58,89,145,42,20],4,...}. Lengths of transient=12, of cycle=8.

Hugo Steinhaus: "Sto zadan" (1958), "One Hundred Problems in Elementary Mathematics" (1964), problem 2. - M. F. Hasler, May 24 2009

T. D. Noe, Table of n, a(n) for n = 1..1000
Arthur Porges, A set of eight numbers, Amer. Math. Monthly 52 (1945), 379-382. - _M. F. Hasler_, May 24 2009

Cf. A007700, A031176.
Cf. A000216, A003132, A003621
Cf. A039943, A031176, A007770, A000216 (orbit of 2), A000218 (orbit of 3), A080709 (orbit of 4), A000221 (orbit of 5), A008460 (orbit of 6), A008462 (orbit of 8), A008463 (orbit of 9), A139566 (orbit of 15), A122065 (orbit of 74169). - M. F. Hasler, May 24 2009

a099645 = length . takeWhile (`notElem` a039943_list) . iterate a003132
a099645_list = map a099645 [1..]
-- Reinhard Zumkeller, Aug 24 2011

fu[x_] :=Apply[Plus, IntegerDigits[x]^2];hs=20; (* transient lengths are obtained by: *) a[n_] :=-1+Min[Flatten[Position[NestList[fu, n, Length[Union[NestList[fu, n, hs]]]] -Last[NestList[fu, n, Length[Union[NestList[fu, n, hs]]]]], 0]]]; Table[a[n], {n, 1, 256}]

A099645(n)={ local( c=0, S=Set([1,4,16,37,58,89,145,42,20])); while( !setsearch(S,n), n=A003132(n); c++); c} \\ M. F. Hasler, May 24 2009

Terms checked using the given PARI code. However, according to the domain of A003132 and the definition of A039943 (which both include 0), an initial a(0)=0 should be added here, too. - M. F. Hasler, May 24 2009

A023213 Primes p such that 4p + 3 is prime.

Original entry on oeis.org

2, 5, 7, 11, 17, 19, 31, 37, 41, 47, 59, 67, 89, 107, 109, 149, 151, 157, 179, 181, 227, 229, 241, 257, 271, 307, 331, 349, 359, 367, 389, 401, 439, 457, 461, 467, 487, 499, 509, 521, 571, 577, 587, 599, 647, 661, 677, 691, 719, 769, 797, 829, 839, 877, 881, 907, 929, 941

Offset: 1

David W. Wilson

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A007700.

[ n: n in PrimesUpTo(1000) | IsPrime(4*n+3) ]; // Vincenzo Librandi, Nov 20 2010

Select[Prime@Range@500, PrimeQ[4 # + 3] &] (* Vincenzo Librandi, May 19 2014 *)

select(p->isprime(4*p+3),primes(100)) \\ Charles R Greathouse IV, Mar 21 2013

A059763 Primes starting a Cunningham chain of the first kind of length 4.

Original entry on oeis.org

509, 1229, 1409, 2699, 3539, 6449, 10589, 11549, 11909, 12119, 17159, 19709, 19889, 22349, 26189, 27479, 30389, 43649, 55229, 57839, 60149, 71399, 74699, 75329, 82499, 87539, 98369, 101399, 104369, 112919, 122099, 139439, 148829, 166739

Offset: 1

Labos Elemer, Feb 20 2001

Initial (unsafe) primes of Cunningham chains of first type with length exactly 4. Primes in A059453 that survive as primes just three "2p+1 iterations", forming chains of exactly 4 terms.
The definition indicates each chain is exactly 4 primes long (i.e., the chain cannot be a subchain of a longer one). That is why this sequence is different from A023272, which also gives primes included in longer chains ("starting" them or not).
Prime p such that {(p-1)/2, p, 2p+1, 4p+3, 8p+7, 16p+15} = {composite, prime, prime, prime, prime, composite}.

			1229 is a term because, through 2p+1, 1229 -> 2459 -> 4919 -> 9839 and the chain ends here since 2*9839 + 1 = 11*1789 is composite.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059455, A007700.

Cf. A059759, A059760, A059761, A059762, A059763, A059764, A059765, A038397, A104349, A091314, A069362, A016093, A014937, A057326.

isA059763 := proc(p) local pitr,itr ; if isprime(p) then if isprime( (p-1)/2 ) then RETURN(false) ; else pitr := p ; for itr from 1 to 3 do pitr := 2*pitr+1 ; if not isprime(pitr) then RETURN(false) ; fi ; od: pitr := 2*pitr+1 ; if isprime(pitr) then RETURN(false) ; else RETURN(true) ; fi ; fi ; else RETURN(false) ; fi ; end: for i from 2 to 100000 do p := ithprime(i) ; if isA059763(p) then printf("%d,",p) ; fi ; od: # R. J. Mathar, Jul 23 2008

Edited and extended by R. J. Mathar, Jul 23 2008, Aug 18 2008

A059764 Initial (unsafe) primes of Cunningham chains of first type with length exactly 5. Primes in A059453 that survive as primes just four "2p+1 iterations", forming chains of exactly 5 terms.

Original entry on oeis.org

2, 53639, 53849, 61409, 66749, 143609, 167729, 186149, 206369, 268049, 296099, 340919, 422069, 446609, 539009, 594449, 607319, 658349, 671249, 725009, 775949, 812849, 819509, 926669, 1008209, 1092089, 1132949, 1271849

Offset: 1

Labos Elemer, Feb 20 2001

nonn

Primes p such that {(p-1)/2, p, 2p+1, 4p+3, 8p+7, 16p+15, 32p+31} = {nonprime, prime, prime, prime, prime, prime, composite}.

			2 is here because (2-1)/2 = 1/2 and 32*2+31 = 95 are not primes, while 2, 5, 11, 23, and 47 give a first-kind Cunningham chain of 5 primes which cannot be continued.
53639 is here because through <2p+1>, 53639 -> 107279 -> 214559 -> 429119 -> 858239 and the chain ends here (with this operator).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A023272, A023302, A023330, A005384, A005385, A059452-A059455, A007700, A059759, A059760, A059761, A059762, A059763, A059764, A059765, A038397, A104349, A091314, A069362, A016093, A014937, A057326.

l5Q[n_]:=Module[{a=PrimeQ[(n-1)/2],b=PrimeQ[ NestList[2#+1&,n,5]]}, Join[{a},b]=={False,True,True,True,True,True,False}]; Select[Range[ 1300000],l5Q] (* Harvey P. Dale, Oct 14 2012 *)

Definition corrected by Alexandre Wajnberg, Aug 31 2005

Entry revised by N. J. A. Sloane, Apr 01 2006

A059453 Sophie Germain primes (A005384) that are not safe primes (A005385).

Original entry on oeis.org

2, 3, 29, 41, 53, 89, 113, 131, 173, 191, 233, 239, 251, 281, 293, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 743, 761, 809, 911, 953, 1013, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1451, 1481, 1499, 1511, 1559, 1583, 1601, 1733, 1811, 1889, 1901

Offset: 1

Labos Elemer, Feb 02 2001

nonn

Except for 2 and 3 these primes are congruent to 5 or 11 modulo 12.

Introducing terms of Cunningham chains of first kind.

			89 is a term because (89-1)/2 = 44 is not prime, but 2*89 + 1 = 179 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris K. Caldwell, Cunningham Chains.

Intersection of A005384 and A059456.

Cf. A005385, A053176, A059452, A059453, A059454, A059455, A007700, A005602, A023272, A023302, A023330.

lst={};Do[p=Prime[n];If[ !PrimeQ[(p-1)/2],If[PrimeQ[2*p+1],AppendTo[lst,p]]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 24 2009 *)
Select[Prime[Range[300]],PrimeQ[2#+1]&&!PrimeQ[(#-1)/2]&] (* Harvey P. Dale, Nov 10 2017 *)

is(p) = isprime(p) && isprime(2*p+1) && if(p > 2, !isprime((p-1)/2), 1); \\ Amiram Eldar, Jul 15 2024

from itertools import count, islice
from sympy import isprime, prime
def A059453_gen(): # generator of terms
    return filter(lambda p:not isprime(p>>1) and isprime(p<<1|1),(prime(i) for i in count(1)))
A059453_list = list(islice(A059453_gen(),10)) # Chai Wah Wu, Jul 12 2022

A156660(a(n))*(1-A156659(a(n))) = 1. - Reinhard Zumkeller, Feb 18 2009

A062737 Primes p such that 4p-1 is also prime.

Original entry on oeis.org

2, 3, 5, 11, 17, 41, 53, 71, 83, 131, 137, 173, 197, 227, 263, 281, 293, 383, 431, 467, 503, 521, 563, 587, 593, 617, 677, 683, 701, 743, 797, 827, 887, 911, 977, 983, 1013, 1061, 1091, 1151, 1163, 1181, 1277, 1307, 1361, 1481, 1511, 1523, 1553, 1607, 1613

Offset: 1

Reiner Martin, Jul 12 2001

For n>1, 4*a(n) is a solution to the equation phi(x-1) - phi(x) = x/2. - Farideh Firoozbakht, Dec 17 2014

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A005384, A007700, A005382.

select(n -> isprime(n) and isprime(4*n-1), [2,seq(2*k+1,k=1..1000)]); # Robert Israel, Dec 17 2014

{ n=0; forprime (p=2, 10^5, if (isprime(4*p - 1), write("b062737.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 10 2009

A059500 Primes p such that both q=(p-1)/2 and 2p + 1 = 4q + 3 are composite numbers. Intersection of A059456 and A053176.

Original entry on oeis.org

13, 17, 19, 31, 37, 43, 61, 67, 71, 73, 79, 97, 101, 103, 109, 127, 137, 139, 149, 151, 157, 163, 181, 193, 197, 199, 211, 223, 229, 241, 257, 269, 271, 277, 283, 307, 311, 313, 317, 331, 337, 349, 353, 367, 373, 379, 389, 397, 401, 409, 421, 433, 439, 449

Offset: 1

Labos Elemer, Feb 05 2001

nonn

Primes which are neither safe nor of Sophie Germain type.
Primes not in Cunningham chains of the first kind. - Alonso del Arte, Jun 30 2005
A010051(a(n))*(1-A156660(a(n)))*(1-A156659(a(n))) = 1; A156878 gives numbers of these numbers <= n. - Reinhard Zumkeller, Feb 18 2009

			Prime p=17 is here because both 35 and 8 are composite numbers. Such primes fall "out of" any Cunningham chain of first kind (or generate Cunningham chains of 0-length).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from _Reinhard Zumkeller)
C. K. Caldwell, Cunningham Chains

Cf. A005384, A005385, A053176, A059452-A059456, A007700, A005602, A023272, A023302, A023330, A156658.

Complement[Prime[Range[100]], Select[Prime[Range[100]], PrimeQ[2# + 1] &], Select[Prime[Range[100]], PrimeQ[(# - 1)/2] &]] (Delarte)
Select[Prime[Range[100]],!PrimeQ[q=2#+1]&&!PrimeQ[(#-1)/2]&] (* Zak Seidov, Mar 09 2013 *)

is(n)=isprime(n)&&!isprime(n\2)&&!isprime(2*n+1) \\ Charles R Greathouse IV, Jan 16 2013

a(n) ~ n log n. - Charles R Greathouse IV, Jan 16 2013

cp's OEIS Frontend

A059762 Initial primes of Cunningham chains of first type with length exactly 3. Primes in A059453 that survive as primes just two "2p+1 iterations", forming chains of exactly 3 terms.

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A059456 Unsafe primes: primes not in A005385.

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A059761 Initial primes of Cunningham chains of first type with length exactly 2. Primes in A059453 that survive as primes only one "2p-1 iteration", forming chains of exactly 2 terms.

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A099645 Number of iterations until n reaches a number in A039943 under "x goes to sum of squares of digits of x" map.

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A023213 Primes p such that 4p + 3 is prime.

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Magma

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A059763 Primes starting a Cunningham chain of the first kind of length 4.

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A059764 Initial (unsafe) primes of Cunningham chains of first type with length exactly 5. Primes in A059453 that survive as primes just four "2p+1 iterations", forming chains of exactly 5 terms.

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