Tom Mueller - cp's OEIS frontend

A272894 a(n) is the largest natural number k such that the composite number (2n+1) 2^k+1 has a nontrivial divisor of the form h2^s+1 (h odd) with s>k. If such a natural number does not exist, we set a(n)=0.

0, 0, 0, 1, 0, 2, 1, 0, 1, 3, 2, 0, 1, 1, 2, 2, 0, 4, 3, 0, 2, 1, 1, 2, 1, 2, 3, 3, 1, 1, 2, 0, 2, 5, 4, 2, 3, 0, 1, 1, 2, 2, 1, 3, 2, 3, 1, 0, 1, 0, 4, 4, 0, 2, 3, 2, 0, 1, 1, 2, 3, 2, 0, 3, 1, 6, 5, 0, 4, 1, 2, 1, 3, 1, 1, 2, 3, 3, 2, 2, 2, 3, 0, 4, 3, 4, 2, 1, 1, 0, 3, 0, 2, 1, 3, 2, 1, 1, 5, 5, 2

Offset: 0

Tom Mueller, May 09 2016

			We always have 2^k + 1 < h2^s + 1 if k < s. Thus a(1)=0.

Tom Müller, On the Exponents of Non-Trivial Divisors of Odd Numbers and a Generalization of Proth's Primality Theorem, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.7.

Cf. A272895.

a:= proc(n)
H:=2*n+1:
smax:=floor(evalf(log[2](H))):
R:=0:
for r from 1 to smax-1 do;
for s from r+1 to smax do;
kmax:=floor(evalf(H/2^s)):
for k from 1 to kmax by 2 do;
h:=(H-2^(s-r)*k)/(2^s*k+1):
if h<1 then break fi;
if type(h,integer) and R

A272895 a(n) is the largest natural number k such that the composite number (2n+1) 2^k+1 has a nontrivial divisor of the form h2^s+1 (h odd) with 2s>k. If such a natural number does not exist, we set a(n)=0.

Original entry on oeis.org

0, 3, 4, 5, 5, 2, 6, 7, 6, 4, 3, 4, 7, 1, 8, 9, 8, 6, 3, 4, 6, 5, 1, 6, 8, 2, 6, 3, 9, 4, 10, 11, 10, 8, 4, 6, 5, 5, 5, 9, 6, 7, 2, 3, 8, 7, 3, 8, 9, 4, 8, 4, 4, 6, 4, 9, 10, 6, 7, 4, 11, 3, 12, 13, 12, 10, 5, 8, 8, 6, 6, 6, 5, 6, 6, 7, 3, 5, 10, 4, 8, 9, 0, 4, 8, 4, 5, 11, 8, 8, 3, 6, 10, 9, 4, 10

Offset: 0

Tom Mueller, May 09 2016

Tom Müller, On the Exponents of Non-Trivial Divisors of Odd Numbers and a Generalization of Proth's Primality Theorem, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.7.

Cf. A272894.

H:=2n+1:
smax:=floor(evalf(log[2](H))):
R:=A272894(n)
for m from 0 to smax do;
for s from m+1 to smax+1 do;
hmax:=floor(evalf(H/2^m)):
for h from 1 to hmax by 2 do;
k:=(2^(s-m)*H-h)/(2^s*h+1);
if k

A133187 Prime numbers formed by the concatenation of q and p, where q > p are also primes.

Original entry on oeis.org

53, 73, 113, 137, 173, 193, 197, 233, 293, 313, 317, 373, 433, 593, 613, 617, 673, 677, 733, 797, 977, 1013, 1033, 1093, 1097, 1277, 1373, 1493, 1637, 1733, 1913, 1933, 1973, 1993, 1997, 2113, 2237, 2273, 2293, 2297, 2311, 2333, 2393, 2417, 2633, 2693, 2713

Offset: 1

Tom Mueller (muel4503(AT)uni-trier.de), Dec 17 2007

These numbers are called Caesar primes because the birth date of Julius Caesar (July 13th) provides one example of such a number, i.e. p=7 and q=13 give the prime 137.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A019549, A105184.

lim=2700;plim=Max[FromDigits[Rest[IntegerDigits[lim]]],FromDigits[Drop[IntegerDigits[lim],-1]]];f2p[{p_,q_}]:=FromDigits[Join[IntegerDigits[q],IntegerDigits[p]]];p=Prime[Range[PrimePi[plim]]];p2=Subsets[p,{2}];Union[Select[f2p/@p2,PrimeQ[#]&&#<=lim&]] (* James C. McMahon, Mar 12 2025 *)

from sympy import isprime
def ok(n):
    if not isprime(n): return False
    s = str(n)
    return any(s[i]!='0' and (q:=int(s[:i])) > (p:=int(s[i:])) and isprime(q) and isprime(p) for i in range(1, len(s)))
print([k for k in range(2800) if ok(k)]) # Michael S. Branicky, Apr 05 2025

a(27)-a(47) from James C. McMahon, Mar 12 2025

A128852 Anti-elite primes: a prime number p is called anti-elite if only a finite number of Fermat numbers 2^(2^n)+1 are quadratic non-residues mod p.

Original entry on oeis.org

2, 13, 17, 97, 193, 241, 257, 641, 673, 769, 2689, 5953, 8929, 12289, 40961, 49921, 61681, 65537, 101377, 114689, 274177, 286721, 319489, 414721, 417793, 550801, 786433, 974849, 1130641, 1376257, 1489153, 1810433, 2424833, 3602561, 6700417

Offset: 1

Tom Mueller, Apr 16 2007

nonn

There are infinitely many anti-elite primes.

			Let F_r:=2^(2^r)+1 = r-th Fermat number. Then a(2)=13 because for all r>1 we have F_r == 4 (mod 13) if r is even, resp. F_r == 10 (mod 13) if r is odd. Notice that 4 and 10 are quadratic residues modulo 13.

Alexander Aigner; Über Primzahlen, nach denen (fast) alle Fermatzahlen quadratische Nichtreste sind. Monatsh. Math. 101 (1986), pp. 85-93

Dennis Martin, Table of n, a(n) for n = 1..101
M. Krizek, F. Luca, I. E. Shparlinski, L. Somer, On the complexity of testing elite primes, J. Int. Seq. 14 (2011) # 11.1.2
Dennis Martin, Anti-Elite Prime Search
Dennis Martin, Anti-Elite Prime Search [Cached copy, with permission of author]
Tom Müller, On Anti-Elite Prime Numbers, J. Integer Sequences, Vol. 10 (2007), Article 07.9.4.
Tom Müller, On the Fermat Periods of Natural Numbers, J. Int. Seq. 13 (2010) # 10.9.5.
Tom Müller, On the Exponents of Non-Trivial Divisors of Odd Numbers and a Generalization of Proth's Primality Theorem, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.7.

Cf. A102742.

Contains all Fermat prime factors of Fermat numbers (A023394) that are greater than 5.

isAntiElite(n) = if(isprime(n) && n > 2, my(d = znorder(Mod(2,n)), StartPoint = valuation(d,2), LengthTest = znorder(Mod(2, d >> StartPoint))); for(i = StartPoint, StartPoint + LengthTest - 1, if(!issquare(Mod(2,n)^2^i + 1), return(0))); 1, n == 2) \\ Jianing Song, May 15 2024

A102739 Numbers k such that 6*10^k-11 is prime.

Original entry on oeis.org

9, 15, 21, 44, 58, 64, 96, 108, 138, 160, 222, 534, 2060, 2446, 3884, 8794, 9879, 18303

Offset: 1

Tom Mueller (muel4503(AT)uni-trier.de), Feb 08 2005

Some of the larger entries may only correspond to probable primes.

Numbers corresponding to terms <= 534 are certified primes. - Klaus Brockhaus, Feb 16 2005

is(n)=ispseudoprime(6*10^n-11) \\ Charles R Greathouse IV, Jun 13 2017

a(12)-a(14) from Klaus Brockhaus, Feb 16 2005

a(15)-a(18) from Michael S. Branicky, Jul 12 2023

A102737 Numbers k such that 3*10^k - 11 is prime.

Original entry on oeis.org

1, 4, 7, 11, 14, 16, 22, 29, 36, 40, 65, 139, 149, 204, 842, 1031, 1331, 1345, 1505, 1894, 3386, 3526, 11092, 23836, 37836, 138811, 182614

Offset: 1

Tom Mueller (muel4503(AT)uni-trier.de), Feb 08 2005

Some of the larger entries may only correspond to probable primes.
Numbers corresponding to terms <= 842 are certified primes. - Klaus Brockhaus, Feb 16 2005
Next term > 12500. - Ryan Propper, Jul 21 2006
For k > 1, numbers k such that the digit 2 followed by k-2 occurrences of the digit 9 followed by the digits 89 is prime. - Robert Price, Nov 25 2017
a(28) > 2*10^5. - Robert Price, Jul 04 2018

Makoto Kamada, Factorization of near-repdigit-related numbers.
Makoto Kamada, Search for 29w89.

Select[Range[1,500],PrimeQ[3*10^# - 11]&] (* Julien Kluge, Sep 19 2016 *)

is(n)=ispseudoprime(3*10^n - 11) \\ Charles R Greathouse IV, Jun 13 2017

a(15)-a(19) from Klaus Brockhaus, Feb 16 2005
a(20)-a(23) from Ryan Propper, Jul 21 2006
a(24)-a(25) from Robert Price, Nov 25 2017
a(26) from Robert Price, Jul 04 2018
a(27) from Robert Price, Jul 25 2018

A102738 Numbers k such that 4*10^k - 11 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 15, 35, 61, 256, 357, 628, 767, 1064, 1096, 6608, 14821, 15341, 18795, 22648, 24199, 31919, 38519, 44279

Offset: 1

Tom Mueller (muel4503(AT)uni-trier.de), Feb 08 2005

Numbers corresponding to terms <= 767 are certified primes. - Klaus Brockhaus, Feb 16 2005

The next term is larger than 2500. - Stefan Steinerberger, Feb 18 2006

For[n=1, n<2500,n++,If[PrimeQ[4*10^n-11], Print[n]]] (* Stefan Steinerberger, Feb 18 2006 *)

is(n)=ispseudoprime(4*10^n-11) \\ Charles R Greathouse IV, Jun 13 2017

a(9)-a(14) from Klaus Brockhaus, Feb 16 2005
a(15) from Ryan Propper, Jul 21 2006
a(16)-a(17) from Michael S. Branicky, May 01 2023
a(18)-a(23) from Kamada data by Tyler Busby, May 03 2024

A102742 Elite primes: a prime number p is called elite if only a finite number of Fermat numbers 2^(2^n)+1 are quadratic residues mod p.

Original entry on oeis.org

3, 5, 7, 41, 15361, 23041, 26881, 61441, 87041, 163841, 544001, 604801, 6684673, 14172161, 159318017, 446960641, 1151139841, 3208642561, 38126223361, 108905103361, 171727482881, 318093312001, 443069456129, 912680550401, 1295536619521, 1825696645121, 2061584302081

Offset: 1

Tom Mueller, Feb 08 2005; extended Jun 16 2005

nonn

Křížek, Luca, Shparlinski, & Somer show that a(n) >> n log^2 n. - Charles R Greathouse IV, Jan 25 2017

Let d = 2^r*d' be the multiplicative order of 2 modulo p. Note that 2^2^s == 2^d == 1 (mod p), so p divides none of.

Dennis Martin, Table of n, a(n) for n = 1..29
Alexander Aigner, Über Primzahlen, nach denen (fast) alle Fermatzahlen quadratische Nichtreste sind, Monatsh. Math., Vol. 101 (1986), pp. 85-93; alternative link.
Alain Chaumont and Tom Mueller, All Elite Primes Up to 250 Billion, J. Integer Sequences, Vol. 9 (2006), Article 06.3.8.
Matthew Just, On upper bounds for the count of elite primes, arXiv:2102.00906 [math.NT], 2021.
Michal Křížek, Florian Luca, Igor E. Shparlinski, and Lawrence Somer, On the complexity of testing elite primes, Journal of Integer Sequences, Vol. 14 (2011), Article 11.1.2, 5 pp.
Xiaoquin Li, Verifying Two Conjectures on Generalized Elite Primes, JIS 12 (2009) 09.4.7.
Dennis Martin, Elite Prime Search. [Broken link]
Dennis Martin, Elite Prime Search. [Cached copy, with permission of author]
Dennis Martin, Elite and Anti-Elite Prime Search Methodology [Cached copy, with permission of author]
Tom Müller, Searching for large elite primes, Experimental Mathematics, Vol. 15, Nol. 2 (2006), pp. 183-186.
Tom Muller and A. Reinhart, On generalized Elite Primes, Journal of Integer Sequences, Vol. 11 (2008), Article 08.3.1.
Tom Müller, On the Fermat Periods of Natural Numbers, Journal of Integer Sequences, Vol. 13 (2010), Article 10.9.5.
Tom Müller, On the Exponents of Non-Trivial Divisors of Odd Numbers and a Generalization of Proth's Primality Theorem, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.7.

Cf. A128852, A344785. Subsequence of A129802.

list_upto(N)={forprime(p=3,N,r=2^valuation(p-1,2); a=Mod(3,p); v=List(); k=0; while(1,listput(v,a); a=(a-1)^2+1; for(j=1,#v,if(v[j]==a,k=j;break(2)))); for(i=k,#v,znorder(v[i]) % r != 0 && next(2)); print1(p,", "))} \\ Slow, only for illustration, Jeppe Stig Nielsen, Jan 28 2020

isElite(n) = if(isprime(n) && n > 2, my(d = znorder(Mod(2,n)), StartPoint = valuation(d,2), LengthTest = znorder(Mod(2, d >> StartPoint))); for(i = StartPoint, StartPoint + LengthTest - 1, if(issquare(Mod(2,n)^2^i + 1), return(0))); 1, 0) \\ Jianing Song, May 15 2024

Sum_{n>=1} 1/a(n) = A344785. - Amiram Eldar, May 30 2021

a(17) from Tom Mueller, Jul 20 2005
a(18)-a(21) from Tom Mueller, Apr 18 2006
6 further terms from Tom Mueller, Apr 16 2007

A102740 Numbers k such that 7*10^k - 11 is prime.

Original entry on oeis.org

1, 6, 7, 9, 10, 11, 16, 42, 53, 78, 321, 699, 1858, 3425, 4899, 5734, 11081, 11675, 12136, 14056, 16074, 77969, 158465

Offset: 1

Tom Mueller (muel4503(AT)uni-trier.de), Feb 08 2005

a(24) > 2*10^5.
Numbers corresponding to terms <= 699 are certified primes. - Klaus Brockhaus, Feb 15 2005
For k > 1, numbers k such that the digit 6 followed by k-2 occurrences of the digit 9 followed by the digits 89 is prime (see Example section).

			Initial terms and associated primes:
a(1) = 1, 59;
a(2) = 6, 6999989;
a(3) = 7, 69999989;
a(4) = 9, 6999999989;
a(5) = 10, 69999999989; etc.

Makoto Kamada, Factorization of near-repdigit-related numbers.
Makoto Kamada, Search for 69w89.

Cf. A056654, A268448, A269303, A270339, A270613, A270831, A270890, A270929, A271269.

Select[Range[1, 100000], PrimeQ[7*10^# - 11] &]

is(n)=ispseudoprime(7*10^n - 11) \\ Charles R Greathouse IV, Jun 13 2017

a(11)-a(13) from Klaus Brockhaus, Feb 15 2005
a(14)-a(22) from Robert Price, Oct 29 2017
a(23) from Robert Price, Jun 21 2019

A070213 Numbers k such that A056542(k) is prime.

Original entry on oeis.org

4, 8, 18, 20, 70

Offset: 1

Tom Mueller (muel4503(AT)uni-trier.de), May 07 2002

The 6th term (if it exists) is larger than 1019.
No additional primes for n < 14000. - T. D. Noe, Jul 07 2005
a(6) > 25000. - Michael S. Branicky, Apr 16 2025

Tom Mueller, Prime and Composite Terms in Sloane's Sequence A056542, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.3.

Cf. A056542.

a(5) from Tom Mueller (muel4503(AT)uni-trier.de), Oct 30 2004

Name clarified by Sean A. Irvine, Jun 02 2024

cp's OEIS Frontend

User: Tom Mueller

A272894 a(n) is the largest natural number k such that the composite number (2n+1) 2^k+1 has a nontrivial divisor of the form h2^s+1 (h odd) with s>k. If such a natural number does not exist, we set a(n)=0.

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A272895 a(n) is the largest natural number k such that the composite number (2n+1) 2^k+1 has a nontrivial divisor of the form h2^s+1 (h odd) with 2s>k. If such a natural number does not exist, we set a(n)=0.

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A133187 Prime numbers formed by the concatenation of q and p, where q > p are also primes.

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A128852 Anti-elite primes: a prime number p is called anti-elite if only a finite number of Fermat numbers 2^(2^n)+1 are quadratic non-residues mod p.

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A102739 Numbers k such that 6*10^k-11 is prime.

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A102737 Numbers k such that 3*10^k - 11 is prime.

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A102738 Numbers k such that 4*10^k - 11 is prime.

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A102742 Elite primes: a prime number p is called elite if only a finite number of Fermat numbers 2^(2^n)+1 are quadratic residues mod p.

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