keyword:hard - cp's OEIS frontend

A007850 Giuga numbers: composite numbers n such that p divides n/p - 1 for every prime divisor p of n.

30, 858, 1722, 66198, 2214408306, 24423128562, 432749205173838, 14737133470010574, 550843391309130318, 244197000982499715087866346, 554079914617070801288578559178, 1910667181420507984555759916338506

Offset: 1

D. Borwein, J. M. Borwein, P. B. Borwein and R. Girgensohn

There are no other Giuga numbers with 8 or fewer prime factors. I did an exhaustive search using a PARI script which implemented Borwein and Girgensohn's method for finding n factor solutions given n - 2 factors. - Fred Schneider, Jul 04 2006
One further Giuga number is known with 10 prime factors, namely:
420001794970774706203871150967065663240419575375163060922876441614\
2557211582098432545190323474818 =
2 * 3 * 11 * 23 * 31 * 47059 * 2217342227 * 1729101023519 * 8491659218261819498490029296021 * 58254480569119734123541298976556403 but this may not be the next term. (See the Butske et al. paper.)
Conjecture: Giuga numbers are the solution of the differential equation n' = n + 1, where n' is the arithmetic derivative of n. - Paolo P. Lava, Nov 16 2009
n is a Giuga number if and only if n' = a*n + 1 for some integer a > 0 (see our preprint in arXiv:1103.2298). - José María Grau Ribas, Mar 19 2011
A composite number n is a Giuga number if and only if Sum_{i = 1..n-1} i^phi(n) == -1 (mod n), where phi(n) = A000010(n). - Jonathan Sondow, Jan 03 2014
A composite number n is a Giuga number if and only if Sum_{prime p|n} 1/p = 1/n + an integer. (In fact, all known Giuga numbers n satisfy Sum_{prime p|n} 1/p = 1/n + 1.) - Jonathan Sondow, Jan 08 2014
The prime factors of a(n) are listed as n-th row of A236434. - M. F. Hasler, Jul 13 2015
Conjecture: let k = a(n) and k be the product of x(n) distinct prime factors where x(n) <= x(n+1). Then, for any even n, n/2 + 2 <= x(n) <= n/2 + 3 and, for any odd n, (n+1)/2 + 2 <= x(n) <= (n+1)/2 + 3. For any n > 1, there are y "old" distinct prime factors o(1)...o(y) such that o(1) = 2, o(2) = 3, and z "new" distinct prime factors n(1)...n(z) such that none of them - unlike the "old" ones - can be a divisor of a(q) while q < n; n(1) > o(y), y = x(n) - z >= 2, 2 <= z <= b where b is either 4, or 1/2*n. - Sergey Pavlov, Feb 24 2017
Conjecture: a composite n is a Giuga number if and only if Sum_{k=1..n-1} k^lambda(n) == -1 (mod n), where lambda(n) = A002322(n). - Thomas Ordowski and Giovanni Resta, Jul 25 2018
A composite number n is a Giuga number if and only if A326690(n) = 1. - Jonathan Sondow, Jul 19 2019
A composite n is a Giuga number if and only if n * A027641(phi(n)) == - A027642(phi(n)) (mod n^2). Note: Euler's phi function A000010 can be replaced by the Carmichael lambda function A002322. - Thomas Ordowski, Jun 07 2020
By von Staudt and Clausen theorem, a composite n is a Giuga number if and only if n * A027759(phi(n)) == A027760(phi(n)) (mod n^2). Note: Euler's phi function can be replaced by the Carmichael lambda function. - Thomas Ordowski, Aug 01 2020

			From _M. F. Hasler_, Jul 13 2015: (Start)
The prime divisors of 30 are {2, 3, 5}, and 2 divides 30/2-1 = 14, 3 divides 30/3-1 = 9, and 5 divides 30/5-1 = 5.
The prime divisors of 858 are {2, 3, 11, 13} and 858/2-1 = 428 is even, 858/3-1 = 285 is divisible by 3, 858/11-1 = 77 is a multiple of 11, and 858/13-1 = 65 = 13*5.
(End)

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 30, pp 11, Ellipses, Paris 2008.

M. A. Alekseyev, J. M. Grau, A. M. Oller-Marcen. Computing solutions to the congruence 1^n + 2^n + ... + n^n == p (mod n). Discrete Applied Mathematics, 2018. doi:10.1016/j.dam.2018.05.022 arXiv:1602.02407 [math.NT], 2016.
D. Borwein, J. M. Borwein, P. B. Borwein and R. Girgensohn, Giuga's Conjecture on Primality, Amer. Math. Monthly 103, No. 1, 40-50 (1996).
J. M. Borwein and E. Wong, A Survey of Results Relating to Giuga's Conjecture on Primality, Vinet, Luc (ed.): Advances in Mathematical Sciences: CRM's 25 Years. Providence, RI: American Mathematical Society. CRM Proc. Lect. Notes. 11, 13-27 (1997).
William Butske, Lynda M. Jaje, and Daniel R. Mayernik, On the equation Sum_{p | N} 1/p + (1/N)=1, pseudoperfect numbers and perfectly weighted graphs, Math. Comp. 69 (2000), no. 229, 407-420.
José María Grau and Antonio M. Oller-Marcén, Giuga Numbers and the arithmetic derivative., arXiv:1103.2298 [math.NT], 2011; J. Int. Seq. 15 (2012) 12.4.1
José María Grau and Antonio M. Oller-Marcén, Generalizing Giuga's conjecture, arXiv:1103.3483 [math.NT], 2011.
J. M. Grau and A. M. Oller-Marcén, On the congruence sum_{j=1}^{n-1} j^{k(n-1)} == -1 (mod n); k-strong Giuga and k-Carmichael numbers, arXiv preprint arXiv:1311.3522 [math.NT], 2013.
J. M. Grau, A. M. Oller-Marcén, and D. Sadornil, On µ-Sondow Numbers, arXiv:2111.14211 [math.NT], 2021.
John Machacek, Egyptian Fractions and Prime Power Divisors, arXiv:1706.01008 [math.NT], 2017.
Mersenne Forum, Giuga numbers
Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867 [math.NT], May 04 2013.
R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
J. Sondow and K. MacMillan, Primary pseudoperfect numbers, arithmetic progressions, and the Erdős-Moser equation, Amer. Math. Monthly, 124 (2017) 232-240; arXiv:math/1812.06566 [math.NT], 2018.
J. Sondow and E. Tsukerman, The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers, arXiv:1401.0322 [math.NT], 2014; see section 4.
Eric Weisstein's World of Mathematics, Giuga Number.
Wikipedia, Agoh-Giuga conjecture
Wikipedia, Giuga number

Cf. A054377, A216823, A216824, A235137, A235138, A235140, A235363, A236434, A326690.

fQ[n_] := AllTrue[First /@ FactorInteger@ n, Divisible[n/# - 1, #] &]; Select[Range@ 100000, CompositeQ@ # && fQ@ # &] (* Michael De Vlieger, Oct 05 2015 *)

is(n)=if(isprime(n), return(0)); my(f=factor(n)[,1]); for(i=1,#f, if((n/f[i])%f[i]!=1, return(0))); n>1 \\ Charles R Greathouse IV, Apr 28 2015

from itertools import count, islice
from sympy import isprime, primefactors
def A007850_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda x: not isprime(x) and all((x//p-1) % p == 0 for p in primefactors(x)), count(max(startvalue,2)))
A007850_list = list(islice(A007850_gen(),4)) # Chai Wah Wu, Feb 19 2022

Sum_{i = 1..a(n)-1} i^phi(a(n)) == -1 (mod a(n)). - Jonathan Sondow, Jan 03 2014

a(12) from Fred Schneider, Jul 04 2006
Further references from Fred Schneider, Aug 19 2006
Definition corrected by Jonathan Sondow, Sep 16 2012

A003022 Length of shortest (or optimal) Golomb ruler with n marks.

Original entry on oeis.org

1, 3, 6, 11, 17, 25, 34, 44, 55, 72, 85, 106, 127, 151, 177, 199, 216, 246, 283, 333, 356, 372, 425, 480, 492, 553, 585

Offset: 2

N. J. A. Sloane

a(n) is the least integer such that there is an n-element set of integers between 0 and a(n), the sums of pairs (of not necessarily distinct elements) of which are distinct.
From David W. Wilson, Aug 17 2007: (Start)
An n-mark Golomb ruler has a unique integer distance between any pair of marks and thus measures n(n-1)/2 distinct integer distances.
An optimal n-mark Golomb ruler has the smallest possible length (distance between the two end marks) for an n-mark ruler.
A perfect n-mark Golomb ruler has length exactly n(n-1)/2 and measures each distance from 1 to n(n-1)/2. (End)
Positions where A143824 increases (see also A227590). - N. J. A. Sloane, Apr 08 2016
From Gus Wiseman, May 17 2019: (Start)
Also the smallest m such that there exists a length-n composition of m for which every restriction to a subinterval has a different sum. Representatives of compositions for the first few terms are:
   0: ()
   1: (1)
   3: (2,1)
   6: (2,3,1)
  11: (3,1,5,2)
  17: (4,2,3,7,1)
Representatives of corresponding Golomb rulers are:
  {0}
  {0,1}
  {0,2,3}
  {0,2,5,6}
  {0,3,4,9,11}
  {0,4,6,9,16,17}
(End)

			a(5)=11 because 0-1-4-9-11 (0-2-7-10-11) resp. 0-3-4-9-11 (0-2-7-8-11) are shortest: there is no b0-b1-b2-b3-b4 with different distances |bi-bj| and max. |bi-bj| < 11.

CRC Handbook of Combinatorial Designs, 1996, p. 315.
A. K. Dewdney, Computer Recreations, Scientific Amer. 253 (No. 6, Jun), 1985, pp. 16ff; 254 (No. 3, March), 1986, pp. 20ff.
S. W. Golomb, How to number a graph, pp. 23-37 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.
Richard K. Guy, Unsolved Problems in Number Theory (2nd edition), Springer-Verlag (1994), Section C10.
A. Kotzig and P. J. Laufer, Sum triangles of natural numbers having minimum top, Ars. Combin. 21 (1986), 5-13.
Miller, J. C. P., Difference bases. Three problems in additive number theory. Computers in number theory (Proc. Sci. Res. Council Atlas Sympos. No. 2, Oxford, 1969), pp. 299--322. Academic Press, London,1971. MR0316269 (47 #4817)
Rhys Price Jones, Gracelessness, Proc. 10th S.-E. Conf. Combin., Graph Theory and Computing, 1979, pp. 547-552.
Ana Salagean, David Gardner and Raphael Phan, Index Tables of Finite Fields and Modular Golomb Rulers, in Sequences and Their Applications - SETA 2012, Lecture Notes in Computer Science. Volume 7280, 2012, pp. 136-147.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Anonymous, In Search Of The Optimal 20, 21 and 22 Mark Golomb Rulers
Thomas Bloom, Problem 30, Problem 43, Problem 155, and Problem 861, Erdős Problems.
A. K. Dewdney, Computer Recreations, Scientific Amer. 253 (No. 6, Jun), 1985, pp. 16ff; 254 (No. 3, March), 1986, pp. 20ff. [Annotated scanned copy]
Distributed.Net, Project OGR
Kent Freeman, Unpublished notes. [Scanned copy]
Michael Geißer, Theresa Körner, Sascha Kurz, and Anne Zahn, Squares with three digits, arXiv:2112.00444 [math.NT], 2021.
S. W. Golomb, Letter to N. J. A. Sloane, 1972.
A. Kotzig and P. J. Laufer, Sum triangles of natural numbers having minimum top, Ars. Combin. 21 (1986), 5-13. [Annotated scanned copy]
Joseph Malkevitch, Weird Rulers.
Greg Martin and Kevin O'Bryant, Constructions of generalized Sidon sets, arXiv:math/0408081 [math.NT], 2004-2005.
Lloyd Miller, Golomb Rulers
Kevin O'Bryant, Sets of Natural Numbers with Proscribed Subsets, J. Int. Seq. 18 (2015) # 15.7.7
Ed Pegg, Jr., Math Games: Rulers, Arrays, and Gracefulness
Bill Rankin, Golomb Ruler Calculations
Walter Schneider, Golomb Rulers
James B. Shearer, Golomb ruler table
James B. Shearer, Table of Known Optimal Golomb Rulers
James B. Shearer, Difference Triangle Sets: Known optimal solutions.
James B. Shearer, Difference Triangle Sets: Discoverers
David Singmaster, David Fielker, and N. J. A. Sloane, Correspondence, August 1979
N. J. A. Sloane, First few optimal Golomb rulers
Terence Tao, Erdős problem database, see nos. 30, 43, 155, 861.
D. Vanderschel et al., In Search Of The Optimal 20, 21 and 22 Mark Golomb Rulers
Eric Weisstein's World of Mathematics, Golomb Ruler.
Wikipedia, Golomb ruler
Index entries for sequences related to Golomb rulers

See A106683 for triangle of marks.
Cf. A008404, A036501, A039953, A078106, A030873.
0-1-4-9-11 corresponds to 1-3-5-2 in A039953: 0+1+3+5+2=11
A row or column of array in A234943.
Adding 1 to these terms gives A227590. Cf. A143824.
For first differences see A270813.
Cf. A103295, A108917, A143823, A169942.
Cf. A325466, A325545, A325676, A325677, A325678, A325683.

Min@@Total/@#&/@GatherBy[Select[Join@@Permutations/@Join@@Table[IntegerPartitions[i],{i,0,15}],UnsameQ@@ReplaceList[#,{_,s__,_}:>Plus[s]]&],Length] (* Gus Wiseman, May 17 2019 *)

from itertools import combinations, combinations_with_replacement, count
def a(n):
    for k in count(n-1):
        for c in combinations(range(k), n-1):
            c = c + (k, )
            ss = set()
            for s in combinations_with_replacement(c, 2):
                if sum(s) in ss: break
                else: ss.add(sum(s))
            if len(ss) == n*(n+1)//2: return k # Jianing Song, Feb 14 2025, adapted from the python program of A345731

a(n) >= n(n-1)/2, with strict inequality for n >= 5 (Golomb). - David W. Wilson, Aug 18 2007

425 sent by Ed Pegg Jr, Nov 15 2004
a(25), a(26) proved by OGR-25 and OGR-26 projects, added by Max Alekseyev, Sep 29 2010
a(27) proved by OGR-27, added by David Consiglio, Jr., Jun 09 2014
a(28) proved by OGR-28, added by David Consiglio, Jr., Jan 19 2023

A006879 Number of primes with n digits.

Original entry on oeis.org

0, 4, 21, 143, 1061, 8363, 68906, 586081, 5096876, 45086079, 404204977, 3663002302, 33489857205, 308457624821, 2858876213963, 26639628671867, 249393770611256, 2344318816620308, 22116397130086627, 209317712988603747, 1986761935284574233, 18906449883457813088, 180340017203297174362

Offset: 0

N. J. A. Sloane, Simon Plouffe

The number of primes between 10^(n-1) and 10^n. - Cino Hilliard, May 31 2008 [Corrected by Jon E. Schoenfield, Nov 29 2008]

			As 2, 3, 5, and 7 are the only primes less than 10, a(1) = 4.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 21, pp 8, Ellipses, Paris 2008.
C. T. Long, Elementary Introduction to Number Theory. Prentice-Hall, Englewood Cliffs, NJ, 1987, p. 77.
P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 179.
D. Shanks, Solved and Unsolved Problems in Number Theory. Chelsea, NY, 2nd edition, 1978, p. 15.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 113.

Jianing Song, Table of n, a(n) for n = 0..29 (terms 0..24 by Charles R Greathouse IV, a(25) by Vladimir Pletser, a(26)-a(28) from David Baugh, a(29) based on A006880)
C. K. Caldwell, How Many Primes Are There?
Vladimir Pletser, Global Generalized Mersenne Numbers: Definition, Decomposition, and Generalized Theorems, Preprints.org, 2024. See p. 20.
Index entries for sequences related to numbers of primes in various ranges.

First differences of A006880.

Cf. A309329.

Differences[PrimePi[10^Range[-1, 25]]] (* Paolo Xausa, Apr 16 2024 *)

a(n)=primepi(10^n)-primepi(10^(n-1)) \\ Charles R Greathouse IV, May 03 2012

a(n) = pi(10^n)-pi(10^(n-1)) where pi(10^(-1)) := 0 (cf. A000720 and A006880).

Limit_{n->oo} a(n)/a(n-1) = 10. - Stefano Spezia, Aug 31 2025

a(11) and a(12) corrected by Jud McCranie and Enoch Haga
a(19) corrected and a(20) added by Paul Zimmermann
a(21)-a(22) from Vladeta Jovovic, Nov 07 2001

A123910 Numbers k such that k!!-4 is prime.

Original entry on oeis.org

5, 7, 9, 11, 13, 15, 19, 25, 35, 79, 81, 105, 171, 243, 271, 295, 355, 523, 591, 1211, 3073, 11157, 12887, 19825

Offset: 1

Alexander Adamchuk, Oct 28 2006

Corresponding primes of the form k!!-4 = a(n)!!-4 are {11, 101, 941, 10391, 135131, 2027021, 654729071, 7905853580621, 221643095476699771871, ...}.

a(25) > 50000. - Robert Price, May 08 2015

Joe McLean, Interesting Sources of Probable Primes

Cf. A007749, A094144.

Do[f=n!!-4;If[PrimeQ[f],Print[{n,f}]],{n,1,355}]

More terms from Farideh Firoozbakht, Nov 19 2006
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 03 2008
a(23)-a(24) from Robert Price, May 08 2015

A156695 Odd numbers that are not of the form p + 2^a + 2^b, a, b > 0, p prime.

Original entry on oeis.org

1, 3, 5, 6495105, 848629545, 1117175145, 2544265305, 3147056235, 3366991695, 3472109835, 3621922845, 3861518805, 4447794915, 4848148485, 5415281745, 5693877405, 6804302445, 7525056375, 7602256605, 9055691835, 9217432215

Offset: 1

Charles R Greathouse IV, Feb 13 2009

Crocker shows that this sequence is infinite.
All members above 5 found so far (up to 2.5 * 10^11) are divisible by 255 = 3 * 5 * 17, and many are divisible by 257. I conjecture that all members of this sequence greater than 5 are divisible by 255. This implies that all odd numbers (greater than 7) are the sum of a prime and at most three positive powers of two.
Pan shows that, for every c > 1, a(n) << x^c. More specifically, there are constants C,D > 0 such that there are at least Dx/exp(C log x log log log log x/log log log x) members of this sequence up to x. - Charles R Greathouse IV, Apr 11 2016
All terms > 5 are numbers k > 3 such that k - 2^n is a de Polignac number (A006285) for every n > 0 with 2^n < k. Are there numbers K such that |K - 2^n| is a Riesel number (A101036) for every n > 0? If so, ||K - 2^n| - 2^m| is composite for every pair m,n > 0, by the dual Riesel conjecture. - Thomas Ordowski, Jan 06 2024
In keeping with the example's connection to A000215, the lowest ki for ki * Product_{i=0..11} (F(i)) to belong to A156695 are 1, 433007, 25471, 17047, 1291, 7, 101, 807, 83, 347, 9, 179. So for example, 433007*(3*5) is a term. This implies a variant of the first commented conjecture accordingly. - Bill McEachen, Apr 17 2025

			Prime factorization of terms:
F_0 = 3, F_1 = 5, F_2 = 17, F_3 = 257 are Fermat numbers (cf. A000215)
6495105    = 3   * 5   * 17               * 25471
848629545  = 3   * 5   * 17               * 461      * 7219
1117175145 = 3   * 5   * 17         * 257 * 17047
2544265305 = 3^2 * 5   * 17         * 257 * 12941
3147056235 = 3^2 * 5   * 17         * 257 * 16007
3366991695 = 3   * 5   * 17   * 83  * 257 * 619
3472109835 = 3   * 5   * 17         * 257 * 52981
3621922845 = 3   * 5   * 17^2       * 257 * 3251
3861518805 = 3^3 * 5   * 17         * 257 * 6547
4447794915 = 3^3 * 5   * 17         * 257 * 7541
4848148485 = 3^4 * 5   * 17               * 704161
5415281745 = 3   * 5   * 17               * 21236399
5693877405 = 3^2 * 5   * 17         * 257 * 28961
6804302445 = 3^2 * 5   * 17   * 53  * 257 * 653
7525056375 = 3^2 * 5^3 * 17         * 257 * 1531
7602256605 = 3   * 5   * 17         * 257 * 311      * 373
9055691835 = 3   * 5   * 17         * 257 * 138181
9217432215 = 3^2 * 5   * 17   * 173 * 257 * 271

Giovanni Resta, Table of n, a(n) for n = 1..233 (terms < 10^12)
Roger Crocker, "On the sum of a prime and of two powers of two", Pacific Journal of Mathematics 36:1 (1971), pp. 103-107.
Roger Crocker, Some counter-examples in the additive theory of numbers, Master's thesis (Ohio State University), 1962.
Hao Pan, On the integers not of the form p + 2^a + 2^b. arXiv:0905.3809 [math.NT], 2009.
Zhi-Wei Sun, Mixed sums of primes and other terms (2009-2010).

Cf. A006285, A118955, A232565, A337487.

is(n)=if(n%2==0,return(0)); for(a=1,log(n)\log(2), for(b=1,a, if(isprime(n-2^a-2^b),return(0)))); 1 \\ Charles R Greathouse IV, Nov 27 2013

from itertools import count, islice
from sympy import isprime
def A156695_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue+(startvalue&1^1),1),2):
        l = n.bit_length()-1
        for a in range(l,0,-1):
            c = n-(1<A156695_list = list(islice(A156695_gen(),4)) # Chai Wah Wu, Nov 29 2023

Factorizations added by Daniel Forgues, Jan 20 2011

A003216 Number of Hamiltonian graphs with n nodes.

Original entry on oeis.org

1, 0, 1, 3, 8, 48, 383, 6196, 177083, 9305118, 883156024, 152522187830, 48322518340547

Offset: 1

N. J. A. Sloane

a(1) could also be taken to be 0, but I prefer a(1) = 1. - N. J. A. Sloane, Oct 15 2006

J. P. Dolch, Names of Hamiltonian graphs, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 259-271.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 219.
R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Asplund, N. Bradley Fox, and Arran Hamm, New Perspectives on Neighborhood-Prime Labelings of Graphs, arXiv:1804.02473 [math.CO], 2018.
J. P. Dolch, Names of Hamiltonian graphs, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 259-271. (Annotated scanned copy of 3 pages)
Scott Garrabrant and Igor Pak, Pattern Avoidance is Not P-Recursive, preprint, 2015.
Scott Garrabrant and Igor Pak, Pattern Avoidance is Not P-Recursive, arXiv:1505.06508 [math.CO], 2015.
Jan Goedgebeur, Barbara Meersman, and Carol T. Zamfirescu, Graphs with few Hamiltonian Cycles, arXiv:1812.05650 [math.CO], 2018-2019.
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Eric Weisstein's World of Mathematics, Hamiltonian Graph
Wikipedia, Hamiltonian path
Gus Wiseman, Non-isomorphic representatives of the a(5) = 8 simple graphs containing a Hamiltonian cycle.

Main diagonal of A325455 and of A325447 (for n>=3).
The labeled case is A326208.
The directed case is A326226 (with loops) or A326225 (without loops).
The case without loops is A326215.
Unlabeled simple graphs not containing a Hamiltonian cycle are A246446.
Unlabeled simple graphs containing a Hamiltonian path are A057864.
Cf. A000088, A006125, A283420.

A000088(n) = a(n) + A246446(n). - Gus Wiseman, Jun 17 2019

Extended to n=11 by Brendan McKay, Jul 15 1996
a(12) from Sean A. Irvine, Mar 17 2015
a(13) from A246446 added by Jan Goedgebeur, Sep 07 2019

A003459 Absolute primes (or permutable primes): every permutation of the digits is a prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, 1111111111111111111, 11111111111111111111111

Offset: 1

N. J. A. Sloane

From Bill Gosper, Jan 24 2003, in a posting to the Math Fun Mailing List: (Start)
Recall Sloane's old request for more terms of A003459 = (2 3 5 7 11 13 17 31 37 71 73 79 97 113 131 199 311 337 373 733 919 991 ...) and Richard C. Schroeppel's astonishing observation that the next term is 1111111111111111111. Absent Rich's analysis, trying to extend this sequence makes a great set of beginner's programming exercises. We may restrict the search to combinations of the four digits 1,3,7,9, only look at starting numbers with nondecreasing digits, generate only unique digit combinations, and only as needed. (We get the target sequence afterward by generating and merging the various permutations, and fudging the initial 2,3,5,7.)
To my amazement the (uncompiled, Macsyma) program printed 11,13,...,199,337, and after about a minute, 1111111111111111111!
And after a few more minutes, (10^23-1)/9! (End)
Boal and Bevis say that Johnson (1977) proves that if there is a term > 1000 with exactly two distinct digits then it must have more than nine billion digits. - N. J. A. Sloane, Jun 06 2015
Some authors require permutable or absolute primes to have at least two different digits. This produces the subsequence A129338. - M. F. Hasler, Mar 26 2008
See A039986 for a related problem with more sophisticated (PARI) code (iteration over only inequivalent digit permutations). - M. F. Hasler, Jul 10 2018

Richard C. Schroeppel, personal communication.
Wacław Sierpiński, Co wiemy, a czego nie wiemy o liczbach pierwszych. Warsaw: PZWS, 1961, pp. 20-21.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 113.

I. O. Angell and H. J. Godwin, On Truncatable Primes, Math. Comput. 31, 265-267, 1977. [Related paper, but primarily concerned with A023107 and A103443. - _N. J. A. Sloane_, Jun 06 2015]
T. N. Bhargava and P. H. Doyle, On the existence of absolute primes, Math. Mag., 47 (1974), 233.
J. L. Boal and J. H. Bevis, Permutable primes. Math. Mag., 55 (No. 1, 1982), 38-41.
C. Caldwell, The prime glossary: Permutable Prime.
J. P. Delahaye, Persistent Primes, Illustrating Permutable, Circular, Right & Left Truncatable Primes, Pour La Science no 256.
James Grime and Brady Haran, Absolute Primes, YouTube Numberphile video, 2024.
A. W. Johnson, Absolute primes, Mathematics Magazine, 1977, vol. 50, pp. 100-103.
R. Ondrejka, The Top Ten: a Catalogue of Primal Configurations.
W. Schneider, MATHEWS, Circular, Permutable, Truncatable and Deletable Primes.
A. Slinko, Absolute Primes Oct. 2000.
A. Slinko, Absolute Primes, Oct. 2000 [Cached copy, permission requested].
Wikipedia, Permutable prime.
Index entries for sequences related to truncatable primes.

Includes all of A004022 = A002275(A004023).
A258706 gives minimal representatives of the permutation classes.
Cf. A010051, A023107, A016114, A103443, A129338, A141263.
Cf. A039986.

import Data.List (permutations)
a003459 n = a003459_list !! (n-1)
a003459_list = filter isAbsPrime a000040_list where
   isAbsPrime = all (== 1) . map (a010051 . read) . permutations . show
-- Reinhard Zumkeller, Sep 15 2011

f[n_]:=Module[{b=Permutations[IntegerDigits[n]],q=1},Do[If[!PrimeQ[c=FromDigits[b[[m]]]],q=0;Break[]],{m,Length[b]}];q];Select[Range[1000],f[#]>0&] (* Vladimir Joseph Stephan Orlovsky, Feb 03 2011 *)
(* Linear complexity: can't reach R(19). See A258706. - Bill Gosper, Jan 06 2017 *)

for(n=1, oo, my(S=[],r=10^n\9); for(a=1, 9^(n>1), for(b=if(n>2, 1-a), 9-a, for(j=0, if(b, n-1), ispseudoprime(a*r+b*10^j)||next(2)); S=concat(S,vector(if(b,n,1),k,a*r+10^(k-1)*b))));apply(t->printf(t","),Set(S))) \\ M. F. Hasler, Jun 26 2018

Conjecture: for n >= 23, a(n) = A004022(n-21). - Max Alekseyev, Oct 08 2018

The next terms are a(25)=A002275(317), a(26)=A002275(1031), a(27)=A002275(49081).

A062589 Numbers k such that 23^k - 22^k is prime, or a strong pseudoprime.

Original entry on oeis.org

229, 241, 673, 5387, 47581

Offset: 1

Mike Oakes, May 18 2001, May 19 2001

Terms greater than 1000 often correspond to "unproven" strong pseudoprimes.

a(6) > 10^5. - Robert Price, Aug 22 2012

Cf. A000043, A057468, A059801, A059802, A062572-A062588, A214658, A214655, A062592-A062666.

a(5) from Robert Price, Aug 22 2012

Edited by M. F. Hasler, Sep 21 2013

A127821 a(n) = least k such that the remainder of 13^k divided by k is n.

Original entry on oeis.org

2, 11, 5, 51, 44, 7, 15, 371285, 10, 74853, 158, 13757837, 17, 5805311, 22, 2181, 38, 25, 30, 9667, 74, 87, 146, 23441, 88, 19629779, 35, 45, 70, 235433, 46, 55, 34, 309, 134

Offset: 1

Alexander Adamchuk, Jan 30 2007

a(36) > 10^16. - Max Alekseyev, Oct 25 2016

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..10000 with -1 for large entries where a(n) has not yet been found, Oct 31 2016 [With 207 new terms, this supersedes the earlier table from Robert G. Wilson v et al.]
Doug Strain, Table of n, a(n) for n = 1..10000 with -1 for large entries where a(n) has not yet been found, [With more terms, this supersedes the earlier table from Fausto A. C. Cariboni and Robert G. Wilson v et al.]
Robert G. Wilson v et al., Table of n, a(n) for n = 1..10000 with -1 for large entries where a(n) has not yet been found, Feb 06 2007; Nov 30 2010.

Cf. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820.

t = Table[0, {10000} ]; k = 1; While[ k < 3500000000, a = PowerMod[13, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t

More terms from Robert G. Wilson v, Feb 06 2007

a(264), a(798), a(884), a(896), a(976), a(980), a(152), a(171), a(296), a(464), a(824), a(870) from Daniel Morel, Jun 17, Nov 30 2010

A045882 Smallest term of first run of (at least) n consecutive integers which are not squarefree.

Original entry on oeis.org

4, 8, 48, 242, 844, 22020, 217070, 1092747, 8870024, 221167422, 221167422, 47255689915, 82462576220, 1043460553364, 79180770078548, 3215226335143218, 23742453640900972, 125781000834058568

Offset: 1

Erich Friedman

Solution for n=10 is same as for n=11.

This sequence is infinite and each term initiates a suitable arithmetic progression with large differences like squares of primorials or other suitable products of primes from prime factors being on power 2 in terms and in chains after. Proof includes solution of linear Diophantine equations and math. induction. See also A068781, A070258, A070284, A078144, A049535, A077640, A077647, A078143 of which first terms are recollected here. - Labos Elemer, Nov 25 2002

			a(3) = 48 as 48, 49 and 50 are divisible by squares.
n=5 -> {844=2^2*211; 845=5*13^2; 846=2*3^2*47; 847=7*11^2; 848=2^4*53}.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 242, p. 67, Ellipses, Paris 2008.

L. Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829 [math.NT], 2012. See also the author page.
"sikefield3", double-square (2019)
Eric Weisstein's World of Mathematics, Squarefree numbers
Eric Weisstein's World of Mathematics, Squareful

Cf. A013929, A053806, A049535, A077647, A078143. Also A069021 and A051681 are different versions.

cnt = 0; k = 0; Table[While[cnt < n, k++; If[! SquareFreeQ[k], cnt++, cnt = 0]]; k - n + 1, {n, 7}]

a(n)=my(s);for(k=1,9^99,if(issquarefree(k),s=0,if(s++==n,return(k-n+1)))) \\ Charles R Greathouse IV, May 29 2013

a(n) = 1 + A020754(n+1). - R. J. Mathar, Jun 25 2010
Correction from Jeppe Stig Nielsen, Mar 05 2022: (Start)
a(n) = 1 + A020754(n+1) for 1 <= n < 11.
a(n) = 1 + A020754(n) for 11 <= n < N where N is unknown.
Possibly a(n) = 1 + A020754(n-d) for some higher n, depending on how many repeated terms the sequence has. (End)
a(n) <= A061742(n) = A002110(n)^2 is the trivial bound obtained from the CRT. - Charles R Greathouse IV, Sep 06 2022

a(9)-a(11) from Patrick De Geest, Nov 15 1998, Jan 15 1999
a(12)-a(15) from Louis Marmet (louis(AT)marmet.org) and David Bernier (ezcos(AT)yahoo.com), Nov 15 1999
a(16) was obtained as a result of a team effort by Z. McGregor-Dorsey et al. [Louis Marmet (louis(AT)marmet.org), Jul 27 2000]
a(17) was obtained as a result of a team effort by E. Wong et al. [Louis Marmet (louis(AT)marmet.org), Jul 13 2001]
a(18) was obtained as a result of a team effort by L. Marmet et al.

cp's OEIS Frontend

A007850 Giuga numbers: composite numbers n such that p divides n/p - 1 for every prime divisor p of n.

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A003022 Length of shortest (or optimal) Golomb ruler with n marks.

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A006879 Number of primes with n digits.

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A123910 Numbers k such that k!!-4 is prime.

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A156695 Odd numbers that are not of the form p + 2^a + 2^b, a, b > 0, p prime.

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A003216 Number of Hamiltonian graphs with n nodes.

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A003459 Absolute primes (or permutable primes): every permutation of the digits is a prime.

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