A006556 -id:A006556 - cp's OEIS frontend

A076517 Partial sums of A060370 (cf. A006556).

1, 6, 8, 9, 10, 11, 12, 14, 26, 34, 36, 37, 41, 42, 43, 45, 47, 56, 62, 64, 66, 67, 92, 95, 97, 98, 99, 102, 103, 120, 123, 124, 126, 128, 130, 131, 135, 136, 137, 139, 140, 142, 144, 151, 152, 154, 155, 156, 190, 198, 203, 204, 205, 206, 260, 264, 274, 276, 278

Offset: 1

Benoit Cloitre, Nov 09 2002

nonn

a(n) seems to be asymptotic to c*n*log(n) with 0.5 < c < 1

A334904 a(n) is the least integer b such that the fractions (b^0)/p, (b^1)/p, ..., (b^(r-1))/p where p is the n-th prime, produce the A006556(n) distinct cycles.

Original entry on oeis.org

1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 3, 2, 3, 2, 1, 2, 1, 1, 2, 7, 2, 3, 2, 3, 1, 2, 2, 2, 1, 1, 3, 1, 2, 2, 1, 3, 2, 2, 1, 2, 1, 1, 7, 1, 2, 3, 2, 1, 2, 1, 1, 7, 7, 3, 1, 1, 1, 6, 2, 3, 2, 2, 2, 11, 1, 2, 2, 1, 2, 2, 2, 7, 1, 2, 1, 1, 1, 2, 3, 7, 1, 2, 7, 1, 3, 2, 3, 3, 1, 2, 2, 13

Offset: 1

George Plousos, May 15 2020

With the exception of the prime numbers 2 and 5, the values of r mentioned above form the sequence A006556.

Detection of all different cycles of digits in the decimal expansions of 1/p, 2/p, ..., (p-1)/p where p=n-th prime. If for the n-th prime p the number of different cycles of digits is equal to r, then there will be the smallest integer b in the interval 0 < b < p with the following property: The fractions (b^0)/p, (b^1)/p, ..., (b^(r-1))/p will produce r different cycles of digits. In this case the term a(n) of the sequence becomes equal to b.

			For n=13, prime(13)=41, there are A006556(13)=8 cycles.
With b=3, we get (normally, these fractions should be in the form (b^k mod p)/p):
  frac(3^0 / 41) = 0.02439 (1)
  frac(3^1 / 41) = 0.07317 (2)
  frac(3^2 / 41) = 0.21951 (3)
  frac(3^3 / 41) = 0.65853 (4)
  frac(3^4 / 41) = 0.97560 (5)
  frac(3^5 / 41) = 0.92682 (6)
  frac(3^6 / 41) = 0.78048 (7)
  frac(3^7 / 41) = 0.34146 (8=r)
So a(13) = 3.

Cf. A006556.

\\ default(realprecision, 1000)
nbc(p) = (p-1)/znorder(Mod(10, p));
len(p) = znorder(Mod(10, p));
pad(x, sz) = {while(#digits(x) != sz, x*=10); x;}
cmpc(x,y) = {if (x==y, return (0)); my(dx=digits(x), dy=digits(y), v=dx); for (k=1, #dx, v=vector(#v, k, if (k==#v, v[1], v[k+1])); if (v == dy, return (0));); return (1);}
decimals(x, sz) = pad(floor(1.0*10^sz*x), sz);
a(n) = {my(p=prime(n)); if ((p==2), return (1)); if ((p==5), return (2)); my(sz=len(p), nb=nbc(p), m=1); while (#vecsort(vector(f(p), k, decimals((m^(k-1) % p)/p, sz)),cmpc,8) != nb, m++); m;} \\ Michel Marcus, May 29 2020

A005384 Sophie Germain primes p: 2p+1 is also prime.

Original entry on oeis.org

2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, 1013, 1019, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1439, 1451, 1481, 1499, 1511, 1559

Offset: 1

N. J. A. Sloane

Then 2p+1 is called a safe prime: see A005385.
Primes p such that the equation phi(x) = 2p has solutions, where phi is the totient function. See A087634 for another such collection of primes. - T. D. Noe, Oct 24 2003
Subsequence of A117360. - Reinhard Zumkeller, Mar 10 2006
Let q = 2n+1. For these n (and q), the difference of two cyclotomic polynomials can be written as a cyclotomic polynomial in x^2: Phi(q,x) - Phi(2q,x) = 2x Phi(n,x^2). - T. D. Noe, Jan 04 2008
A Sophie Germain prime p is 2, 3 or of the form 6k-1, k >= 1, i.e., p = 5 (mod 6). A prime p of the form 6k+1, k >= 1, i.e., p = 1 (mod 6), cannot be a Sophie Germain prime since 2p+1 is divisible by 3. - Daniel Forgues, Jul 31 2009
Also solutions to the equation: floor(4/A000005(2*n^2+n)) = 1. - Enrique Pérez Herrero, May 03 2012
In the spirit of the conjecture related to A217788, we conjecture that for any integers n >= m > 0 there are infinitely many integers b > a(n) such that the number Sum_{k=m..n} a(k)*b^(n-k) is prime. - Zhi-Wei Sun, Mar 26 2013
If k is the product of a Sophie Germain prime p and its corresponding safe prime 2p+1, then a(n) = (k-phi(k))/3, where phi is Euler's totient function. - Wesley Ivan Hurt, Oct 03 2013
Giovanni Resta found the first Sophie Germain prime which is also a Brazilian number (A125134), 28792661 = 1 + 73 + 73^2 + 73^3 + 73^4 = (11111)73. - _Bernard Schott, Mar 07 2019
For all Sophie Germain primes p >= 5, 2*p + 1 = min(A, B) where A is the smallest prime factor of 2^p - 1 and B the smallest prime factor of (2^p + 1) / 3. - Alain Rocchelli, Feb 01 2023
Consider a pair of numbers (p, 2*p+1), with p >= 3. Then p is a Sophie Germain prime iff (p-1)!^2 + 6*p == 1 (mod p*(2*p+1)). - Davide Rotondo, May 02 2024

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
A. Peretti, The quantity of Sophie Germain primes less than x, Bull. Number Theory Related Topics, Vol. 11, No. 1-3 (1987), pp. 81-92.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 76, 227-230.
Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 83.
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 114.

J. S. Cheema, Table of n, a(n) for n = 1..100000. [This replaces an earlier b-file computed by T. D. Noe]
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
R. P. Boas & N. J. A. Sloane, Correspondence, 1974
P. Bruillard, S.-H. Ng, E. Rowell and Z. Wang, On modular categories, arXiv preprint arXiv:1310.7050 [math.QA], 2013.
Chris K. Caldwell, The Prime Glossary, Sophie Germain Prime
Andrea Del Centina, Letters of Sophie Germain preserved in Florence Historia Mathematica, Vol. 32 (2005), pp. 60-75.
Harvey Dubner, Large Sophie Germain Primes, Math. Comp., Vol. 65, No. 213 (1996), pp. 393-396.
Luis H. Gallardo and Olivier Rahavandrainy, There are finitely many even perfect polynomials over F_p with p+1 irreducible divisors, Acta Mathematica Universitatis Comenianae, Vol. 83, No. 2, 2016, 261-275.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Reikku Kulon, Sublinear arbitrary precision generator of Sophie Germain and safe primes in C (public domain).
Henri Lifchitz, A new and simpler primality test for Sophie-Germain numbers(q=2*p+1).
Victor Meally, Letter to N. J. A. Sloane, no date.
Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867v1 [math.NT], May 04 2013.
Frans Oort, Prime numbers, 2013.
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 31, 114.
Larry Riddle, Sophie Germain and Fermat's Last Theorem, Agnes Scott College, Math. Dept., Jul, 1999.
Carlos Rivera, Puzzle 1122. OEIS A005385, The Prime Puzzles & Problems Connection.
Carlos Rivera, Puzzle 1140. Test for Sophie Germain primes, The Prime Puzzles & Problems Connection.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Anthony G. Shannon, Peter J.-S. Shiue, Tian-Xiao He, and Christopher Saito, On the special cases of Carmichael's totient conjecture, Notes Num. Theor. Disc. Math. (2025) Vol. 31, No. 3, 504-534.
Rosemary Sullivan and Neil Watling, Independent divisibility pairs on the set of integers from 1 to n, INTEGERS, Vol. 13 (2013), Article A65.
Agoh Takashi, On Sophie Germain primes, Number theory (Liptovský Ján, 1999), Tatra Mt. Math. Publ., Vol. 20 (2000), pp. 65-73.
Terence Tao, Obstructions to uniformity and arithmetic patterns in the primes, arXiv:math/0505402 [math.NT], 2005.
Apoloniusz Tyszka, On sets X subset of N for which we know an algorithm that computes a threshold number t(X) in N such that X is infinite if and only if X contains an element greater than t(X), 2019.
Vmoraru, PlanetMath.org, Sophie Germain prime.
Samuel S. Wagstaff, Jr., Sum of Reciprocals of Germain Primes, Journal of Integer Sequences, Vol. 24, No. 2 (2021), Article 21.9.5.
Eric Weisstein's World of Mathematics, Sophie Germain Prime.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.
Wikipedia, Sophie Germain prime.
Samuel Yates, Sophie Germain primes, in "The mathematical heritage of C. F. Gauss," World Sci. Publ., River Edge, NJ, 1991, pp. 882-886.

Cf. A005385, A057331, A005602, A087634.
Cf. also A000355, A156541, A156542, A156592, A161896, A156660, A156874, A092816, A023212, A007528 (primes of the form 6k-1).
For primes p that remains prime through k iterations of the function f(x) = 2x + 1: this sequence (k=1), A007700 (k=2), A023272 (k=3), A023302 (k=4), A023330 (k=5), A278932 (k=6), A138025 (k=7), A138030 (k=8).

Filtered([1..1600],p->IsPrime(p) and IsPrime(2*p+1)); # Muniru A Asiru, Mar 06 2019

[ p: p in PrimesUpTo(1560) | IsPrime(2*p+1) ]; // Klaus Brockhaus, Jan 01 2009

A:={}: for n from 1 to 246 do if isprime(2*ithprime(n)+1) then A:=A union {ithprime(n)} fi od: A:=A; # Emeric Deutsch, Dec 09 2004

Select[Prime[Range[1000]],PrimeQ[2#+1]&]
lst = {}; Do[If[PrimeQ[n + 1] && PrimeOmega[n] == 2, AppendTo[lst, n/2]], {n, 2, 10^4}]; lst (* Hilko Koning, Aug 17 2021 *)

select(p->isprime(2*p+1), primes(1000)) \\ In old PARI versions <= 2.4.2, use select(primes(1000), p->isprime(2*p+1)).

forprime(n=2, 10^3, if(ispseudoprime(2*n+1), print1(n, ", "))) \\ Felix Fröhlich, Jun 15 2014

is_A005384=(p->isprime(2*p+1)&&isprime(p));
  {A005384_vec(N=100,p=1)=vector(N,i,until(isprime(2*p+1),p=nextprime(p+1));p)} \\ M. F. Hasler, Mar 03 2020

from sympy import isprime, nextprime
def ok(p): return isprime(2*p+1)
def aupto(limit): # only test primes
  alst, p = [], 2
  while p <= limit:
    if ok(p): alst.append(p)
    p = nextprime(p)
  return alst
print(aupto(1559)) # Michael S. Branicky, Feb 03 2021

a(n) mod 10 <> 7. - Reinhard Zumkeller, Feb 12 2009
A156660(a(n)) = 1; A156874 gives numbers of Sophie Germain primes <= n. - Reinhard Zumkeller, Feb 18 2009
tau(4*a(n) + 2) = tau(4*a(n)) - 2, for n > 1. - Arkadiusz Wesolowski, Aug 25 2012
eulerphi(4*a(n) + 2) = eulerphi(4*a(n)) + 2, for n > 1. - Arkadiusz Wesolowski, Aug 26 2012
A005097 INTERSECT A000040. - R. J. Mathar, Mar 23 2017
Sum_{n>=1} 1/a(n) is in the interval (1.533944198, 1.8026367) (Wagstaff, 2021). - Amiram Eldar, Nov 04 2021
a(n) >> n log^2 n. - Charles R Greathouse IV, Jul 25 2024

A005231 Odd abundant numbers (odd numbers m whose sum of divisors exceeds 2m).

Original entry on oeis.org

945, 1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, 6615, 6825, 7245, 7425, 7875, 8085, 8415, 8505, 8925, 9135, 9555, 9765, 10395, 11025, 11655, 12285, 12705, 12915, 13545, 14175, 14805, 15015, 15435, 16065, 16695, 17325, 17955

Offset: 1

N. J. A. Sloane

nonn

While the first even abundant number is 12 = 2^2*3, the first odd abundant is 945 = 3^3*5*7, the 232nd abundant number.
Schiffman notes that 945+630k is in this sequence for all k < 52. Most of the first initial terms are of the form. Among the 1996 terms below 10^6, 1164 terms are of that form, and only 26 terms are not divisible by 5, cf. A064001. - M. F. Hasler, Jul 16 2016
From M. F. Hasler, Jul 28 2016: (Start)
Any multiple of an abundant number is again abundant, see A006038 for primitive terms, i.e., those which are not a multiple of an earlier term.
An odd abundant number must have at least 3 distinct prime factors, and 5 prime factors when counted with multiplicity (A001222), whence a(1) = 3^3*5*7. To see this, write the relative abundancy A(N) = sigma(N)/N = sigma[-1](N) as A(Product p_i^e_i) = Product (p_i-1/p_i^e_i)/(p_i-1) < Product p_i/(p_i-1).
See A115414 for terms not divisible by 3, A064001 for terms not divisible by 5, A112640 for terms coprime to 5*7, and A047802 for other generalizations.
As of today, we don't know of a difference between this set S of odd abundant numbers and the set S' of odd semiperfect numbers: Elements of S' \ S would be perfect (A000396), and elements of S \ S' would be weird (A006037), but no odd weird or perfect number is known. (End)
For any term m in this sequence, A064989(m) is also an abundant number (in A005101), and for any term x in A115414, A064989(x) is in this sequence. If there are no odd perfect numbers, then applying A064989 to these terms and sorting into ascending order gives A337386. - Antti Karttunen, Aug 28 2020
There exist infinitely many terms m such that 2*m+1 is also a term. An example of such a term is given by m = 985571808130707987847768908867571007187. - Max Alekseyev, Nov 16 2023

W. Dunham, Euler: The Master of Us All, The Mathematical Association of America Inc., Washington, D.C., 1999, p. 13.
R. K. Guy, Unsolved Problems in Number Theory, B2.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 128.

Metin Sariyar, Table of n, a(n) for n = 1..32000 (terms 1..1000 from T. D. Noe)
Jill Britton, Perfect Number Analyzer.
L. E. Dickson, Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, American Journal of Mathematics 35 (1913), pp. 413-422.
Mitsuo Kobayashi, Paul Pollack and Carl Pomerance, On the distribution of sociable numbers, Journal of Number Theory, Vol. 129, No. 8 (2009), pp. 1990-2009. See Theorem 10 on p. 2007.
Victor Meally, Letter to N. J. A. Sloane, no date.
Walter Nissen, Abundancy : Some Resources.
Tyler Ross, A Perfect Number Generalization and Some Euclid-Euler Type Results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.7.5. See p. 10.
Jay L. Schiffman, Odd Abundant Numbers, Mathematical Spectrum, Volume 37, Number 2 (January 2005), pp 73-75.
Jay L. Schiffman and Christopher S. Simons, More Odd Abundant Sequences, Volume 38, Number 1 (September 2005), pp. 7-8.

Cf. A000203, A005835, A006038, A115414, A064001, A112640, A122036, A136446, A005101, A173490, A039725, A064989, A337386.

A005231 := proc(n) option remember ; local a ; if n = 1 then 945 ; else for a from procname(n-1)+2 by 2 do if numtheory[sigma](a) > 2*a then return a; end if; end do: end if; end proc: # R. J. Mathar, Mar 20 2011

fQ[n_] := DivisorSigma[1, n] > 2n; Select[1 + 2Range@ 9000, fQ] (* Robert G. Wilson v, Mar 20 2011 *)

je=[]; forstep(n=1,15000,2, if(sigma(n)>2*n, je=concat(je,n))); je

is_A005231(n)={bittest(n,0)&&sigma(n)>2*n} \\ M. F. Hasler, Jul 28 2016

list(lim)=my(v=List()); forfactored(n=945,lim\1, if(n[2][1,1]>2 && sigma(n,-1)>2, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Apr 21 2022

a(n) ~ k*n for some constant k (perhaps around 500). - Charles R Greathouse IV, Apr 21 2022

482.8 < k < 489.8 (based on density bounds by Kobayashi et al., 2009). - Amiram Eldar, Jul 17 2022

More terms from James Sellers

A005114 Untouchable numbers, also called nonaliquot numbers: impossible values for the sum of aliquot parts function (A001065).

Original entry on oeis.org

2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 206, 210, 216, 238, 246, 248, 262, 268, 276, 288, 290, 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, 516, 518, 520, 530, 540, 552, 556, 562, 576, 584, 612, 624, 626, 628, 658

Offset: 1

N. J. A. Sloane

Complement of A078923. - Lekraj Beedassy, Jul 19 2005
Chen & Zhao show that the lower density of this sequence is at least 0.06, improving on te Riele. - Charles R Greathouse IV, Dec 28 2013
Numbers k such that A048138(k) = 0. A048138(k) measures how "touchable" k is. - Jeppe Stig Nielsen, Jan 12 2020
From Amiram Eldar, Feb 13 2021: (Start)
The term "untouchable number" was coined by Alanen (1972). He found the 570 terms below 5000.
Erdős (1973) proved that the lower asymptotic density of untouchable numbers is positive, te Riele (1976) proved that it is > 0.0324, and Banks and Luca (2004, 2005) proved that it is > 1/48.
Pollack and Pomerance (2016) conjectured that the asymptotic density is ~ 0.17. (End)
The upper asymptotic density is less than 1/2 by the 'almost all' binary Goldbach conjecture, independently proved by Nikolai Chudakov, Johannes van der Corput, and Theodor Estermann. (In this context, this shows that the density of the odd numbers of this form is 0 (consider A001065(p*q) for prime p, q); full Goldbach would prove that 5 is the only odd number in this sequence.) - Charles R Greathouse IV, Dec 05 2022

Richard K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, section B10, pp. 100-101.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 65.
József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 93.
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 147.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 125.

Giovanni Resta, Table of n, a(n) for n = 1..13863 (terms < 10^5; first 8153 terms from Klaus Brockhaus)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], p. 840.
Jack David Alanen, Empirical study of aliquot series, Ph.D Thesis, Yale University, 1972.
William D. Banks and Florian Luca, Noncototients and Nonaliquots, arXiv:math/0409231 [math.NT], 2004.
William D. Banks and Florian Luca, Nonaliquots and Robbins numbers, Colloq. Math., Vol. 103, No. 1 (2005), pp. 27-32.
Yong-Gao Chen and Qing-Qing Zhao, Nonaliquot numbers, Publ. Math. Debrecen, Vol. 78, No. 2 (2011), pp. 439-442.
K. Chum, R. K. Guy, M. J. Jacobson, Jr., and A. S. Mosunov, Numerical and statistical analysis of aliquot sequences, Experimental Mathematics (2018), pp. 1-12.
Paul Erdős, Über die Zahlen der Form sigma(n)-n und n-phi(n), Elemente der Math., Vol. 28 (1973), pp. 83-86; alternative link (in German).
Shyam Sunder Gupta, Beauty of Number 153, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 15, 399-410.
Victor Meally, Letter to N. J. A. Sloane, no date.
Paul Pollack, Not Always Buried Deep: A Second Course in Elementary Number Theory, AMS, 2009, p. 272.
Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B, Vol. 3 (2016), pp. 1-26; Errata.
Carl Pomerance and Hee-Sung Yang, On untouchable numbers and related problems, 2012.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Math. Comp., Vol. 83, No. 288 (2014), pp. 1903-1913; alternative link.
Giovanni Resta, Untouchable numbers (the 150232 terms up to 10^6).
H. J. J. te Riele, A theoretical and computational study of generalized aliquot sequences, Mathematisch Centrum, Amsterdam, 1976. See chapter 9.
Eric Weisstein's World of Mathematics, Untouchable Number.
Wikipedia, Untouchable number.
Robert G. Wilson v, Letter to N. J. A. Sloane, Jul. 1992.

Cf. A001065, A048138, A057709, A064000, A078923, A152454, A231964, A283152, A284147.

untouchableQ[n_] := Catch[ Do[ If[n == DivisorSigma[1, k]-k, Throw[True]], {k, 0, (n-1)^2}]] === Null; Reap[ Table[ If[ untouchableQ[n], Print[n]; Sow[n]], {n, 2, 700}]][[2, 1]] (* Jean-François Alcover, Jun 29 2012, after Benoit Cloitre *)

isA078923(n)=if(n==0 || n==1, return(1)); for(m=1,(n-1)^2, if( sigma(m)-m == n, return(1))); 0
isA005114(n)=!isA078923(n)
for(n=1,700, if (isA005114(n), print(n))) \\ R. J. Mathar, Aug 10 2006

is(n)=if(n%2 && n<4e18, return(n==5)); forfactored(m=1,(n-1)^2, if(sigma(m)-m[1]==n, return(0))); 1 \\ Charles R Greathouse IV, Dec 05 2022

from sympy import divisor_sigma as sigma
from functools import cache
@cache
def f(m): return sigma(m)-m
def okA005114(n):
    if n < 2: return 0
    return not any(f(m) == n for m in range(1, (n-1)**2+1))
print([k for k in range(289) if okA005114(k)]) # Michael S. Branicky, Nov 16 2024

# faster for intial segment of sequence
from itertools import count, islice
from sympy import divisor_sigma as sigma
def agen(): # generator of terms
    n, touchable, t = 2, {0, 1}, 1
    for m in count(2):
        touchable.add(sigma(m)-m)
        while m > t:
            if n not in touchable:
                yield n
            else:
                touchable.discard(n)
            n += 1
            t = (n-1)**2
print(list(islice(agen(), 20))) # Michael S. Branicky, Nov 16 2024

More terms from David W. Wilson

A006566 Dodecahedral numbers: a(n) = n(3n - 1)(3n - 2)/2.

Original entry on oeis.org

0, 1, 20, 84, 220, 455, 816, 1330, 2024, 2925, 4060, 5456, 7140, 9139, 11480, 14190, 17296, 20825, 24804, 29260, 34220, 39711, 45760, 52394, 59640, 67525, 76076, 85320, 95284, 105995, 117480, 129766, 142880, 156849, 171700, 187460, 204156

Offset: 0

N. J. A. Sloane

Schlaefli symbol for this polyhedron: {5,3}.
A093485 = first differences; A124388 = second differences; third differences = 27. - Reinhard Zumkeller, Oct 30 2006
One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral and icosahedral) numbers (cf. A053012). - Daniel Forgues, May 14 2010
From Peter Bala, Sep 09 2013: (Start)
a(n) = binomial(3*n,3). Two related sequences are binomial(3*n+1,3) (A228887) and binomial(3*n+2,3) (A228888). The o.g.f.'s for these three sequences are rational functions whose numerator polynomials are obtained from the fourth row [1, 4, 10, 16, 19, 16, 10, 4, 1] of the triangle of trinomial coefficients A027907 by taking every third term:
Sum_{n >= 1} binomial(3*n,3)*x^n = (x + 16*x^2 + 10*x^3)/(1-x)^4;
Sum_{n >= 1} binomial(3*n+1,3)*x^n = (4*x + 19*x^2 + 4*x^3)/(1-x)^4;
Sum_{n >= 1} binomial(3*n+2,3)*x^n = (10*x + 16*x^2 + x^3)/(1-x)^4. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.
Victor Meally, Letter to N. J. A. Sloane, no date.
Ed Pegg Jr, Dodecahedral 2024.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A027907, A035006, A060544, A093485, A124388, A228887, A228888, A295421.

Cf. A000292 (tetrahedral numbers), A000578 (cubes), A005900 (octahedral numbers), A006564 (icosahedral numbers).

a006566 n = n * (3 * n - 1) * (3 * n - 2) `div` 2
a006566_list = scanl (+) 0 a093485_list  -- Reinhard Zumkeller, Jun 16 2013

[n*(3*n-1)*(3*n-2)/2: n in [0..40]]; // Vincenzo Librandi, Dec 11 2015

A006566:=(1+16*z+10*z**2)/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation

Table[n(3n-1)(3n-2)/2,{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2011 *)
LinearRecurrence[{4,-6,4,-1},{0,1,20,84},40] (* Harvey P. Dale, Jul 24 2013 *)
CoefficientList[Series[x (1 + 16 x + 10 x^2)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 11 2015 *)

PARI
```
a(n)=n*(3*n-1)*(3*n-2)/2
```

G.f.: x(1 + 16x + 10x^2)/(1 - x)^4.
a(n) = A000292(3n-3) = A054776(n)/6 = n*A060544(n).
a(n) = C(n+2,3) + 16 C(n+1,3) + 10 C(n,3).
a(0)=0, a(1)=1, a(2)=20, a(3)=84, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Jul 24 2013
a(n) = binomial(3*n,3). a(-n) = - A228888(n). Sum_{n>=1} 1/a(n) = 1/2*( sqrt(3)*Pi - 3*log(3) ). Sum_{n>=1} (-1)^n/a(n) = 1/3*sqrt(3)*Pi - 4*log(2). - Peter Bala, Sep 09 2013
a(n) = A006564(n) + A035006(n). - Peter M. Chema, May 04 2016
E.g.f.: x*(2 + 18*x + 9*x^2)*exp(x)/2. - Ilya Gutkovskiy, May 04 2016
From Amiram Eldar, Jan 09 2024: (Start)
Sum_{n>=1} 1/a(n) = (sqrt(3)*Pi - 3*log(3))/2 (A295421).
Sum_{n>=1} (-1)^(n+1)/a(n) = (12*log(2) - sqrt(3)*Pi)/3. (End)

More terms from Henry Bottomley, Nov 23 2001

A006564 Icosahedral numbers: a(n) = n(5n^2 - 5*n + 2)/2.

Original entry on oeis.org

1, 12, 48, 124, 255, 456, 742, 1128, 1629, 2260, 3036, 3972, 5083, 6384, 7890, 9616, 11577, 13788, 16264, 19020, 22071, 25432, 29118, 33144, 37525, 42276, 47412, 52948, 58899, 65280, 72106, 79392, 87153, 95404, 104160, 113436, 123247, 133608

Offset: 1

N. J. A. Sloane

Schlaefli symbol for this polyhedron: {3,5}.

One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral and icosahedral) numbers (cf. A053012). - Daniel Forgues, May 14 2010

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., Vol. 131, No. 1 (2002), pp. 65-75.
Victor Meally, Letter to N. J. A. Sloane, no date.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000566, A053012, A175578.

Cf. A000292 (tetrahedral numbers), A000578 (cubes), A005900 (octahedral numbers), A006566 (dodecahedral numbers).

a006564 n = n * (5 * n * (n - 1) + 2) `div` 2
-- Reinhard Zumkeller, Jun 16 2013

[(5*n^3-5*n^2+2*n)/2: n in [1..100]] // Vincenzo Librandi, Nov 21 2010

A006564:=(1+8*z+6*z**2)/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation

Table[n (5n^2-5n+2)/2,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1}, {1,12,48,124},40] (* Harvey P. Dale, May 26 2011 *)

a(n)=5*n^2*(n-1)/2+n \\ Charles R Greathouse IV, Oct 07 2015

a(n) = C(n+2,3) + 8*C(n+1,3) + 6*C(n,3).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) with a(0)=1, a(1)=12, a(2)=48, a(3)=124. - Harvey P. Dale, May 26 2011
G.f.: x*(6*x^2 + 8*x + 1)/(x-1)^4. - Harvey P. Dale, May 26 2011
a(n) = A006566(n) - A035006(n). - Peter M. Chema, May 04 2016
E.g.f.: x*(2 + 10*x + 5*x^2)*exp(x)/2. - Ilya Gutkovskiy, May 04 2016
Sum_{n>=1} 1/a(n) = A175578. - Amiram Eldar, Jan 03 2022

A006600 Total number of triangles visible in regular n-gon with all diagonals drawn.

Original entry on oeis.org

1, 8, 35, 110, 287, 632, 1302, 2400, 4257, 6956, 11297, 17234, 25935, 37424, 53516, 73404, 101745, 136200, 181279, 236258, 306383, 389264, 495650, 620048, 772785, 951384, 1167453, 1410350, 1716191, 2058848, 2463384, 2924000, 3462305, 4067028, 4776219, 5568786, 6479551

Offset: 3

N. J. A. Sloane

Place n equally-spaced points on a circle, join them in all possible ways; how many triangles can be seen?

			a(4) = 8 because in a quadrilateral the diagonals cross to make four triangles, which pair up to make four more.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=3..1000
Sascha Kurz, m-gons in regular n-gons
Victor Meally, Letter to N. J. A. Sloane, no date.
B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv version, which has fewer typos than the SIAM version.
B. Poonen and M. Rubinstein, Mathematica programs for these sequences
M. Rubinstein, Drawings for n=4,5,6,...
T. Sillke, Number of triangles for convex n-gon
S. E. Sommars and T. Sommars, Number of Triangles Formed by Intersecting Diagonals of a Regular Polygon, J. Integer Sequences, 1 (1998), #98.1.5.
Sequences formed by drawing all diagonals in regular polygon

Often confused with A005732.
Cf. A203016, A260417.
Row sums of A363174.
Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.

del[m_,n_]:=If[Mod[n,m]==0,1,0]; Tri[n_]:=n(n-1)(n-2)(n^3+18n^2-43n+60)/720 - del[2,n](n-2)(n-7)n/8 - del[4,n](3n/4) - del[6,n](18n-106)n/3 + del[12,n]*33n + del[18,n]*36n + del[24,n]*24n - del[30,n]*96n - del[42,n]*72n - del[60,n]*264n - del[84,n]*96n - del[90,n]*48n - del[120,n]*96n - del[210,n]*48n; Table[Tri[n], {n,3,1000}] (* T. D. Noe, Dec 21 2006 *)

a(2n-1) = A005732(2n-1) for n > 1; a(2n) = A005732(2n) - A260417(n) for n > 1. - Jonathan Sondow, Jul 25 2015

a(3)-a(8) computed by Victor Meally (personal communication to N. J. A. Sloane, circa 1975); later terms and recurrence from S. Sommars and T. Sommars.

A006601 Numbers k such that k, k+1, k+2 and k+3 have the same number of divisors.

Original entry on oeis.org

242, 3655, 4503, 5943, 6853, 7256, 8392, 9367, 10983, 11605, 11606, 12565, 12855, 12856, 12872, 13255, 13782, 13783, 14312, 16133, 17095, 18469, 19045, 19142, 19143, 19940, 20165, 20965, 21368, 21494, 21495, 21512, 22855, 23989, 26885

Offset: 1

N. J. A. Sloane

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 242, p. 67, Ellipses, Paris 2008.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B18, pp. 111-113.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Victor Meally, Letter to N. J. A. Sloane, no date.

Cf. A000005, A006558, A019273, A119479.

Other runs of equidivisor numbers: A005237 (runs of 2), A005238 (runs of 3), A049051 (runs of 5), A049052 (runs of 6), A049053 (runs of 7).

import Data.List (elemIndices)
a006601 n = a006601_list !! (n-1)
a006601_list = map (+ 1) $ elemIndices 0 $
   zipWith3 (((+) .) . (+)) ds (tail ds) (drop 2 ds) where
   ds = map abs $ zipWith (-) (tail a000005_list) a000005_list
-- Reinhard Zumkeller, Jan 18 2014

f[n_]:=Length[Divisors[n]]; lst={};Do[If[f[n]==f[n+1]==f[n+2]==f[n+3],AppendTo[lst,n]],{n,8!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 14 2009 *)
dsQ[n_]:=Length[Union[DivisorSigma[0,Range[n,n+3]]]]==1; Select[Range[ 30000],dsQ] (* Harvey P. Dale, Nov 23 2011 *)
Flatten[Position[Partition[DivisorSigma[0,Range[27000]],4,1],?(Union[ Differences[ #]]=={0}&),{1},Heads->False]] (* Faster, because the number of divisors for each number is only calculated once *) (* _Harvey P. Dale, Nov 06 2013 *)
SequencePosition[DivisorSigma[0,Range[27000]],{x_,x_,x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 03 2017 *)

is(n)=my(t=numdiv(n)); numdiv(n+1)==t && numdiv(n+2)==t && numdiv(n+3)==t \\ Charles R Greathouse IV, Jun 25 2017

More terms from Olivier Gérard

A054471 Smallest prime p having n different cycles in decimal expansions of k/p, k=1..p-1.

Original entry on oeis.org

7, 3, 103, 53, 11, 79, 211, 41, 73, 281, 353, 37, 2393, 449, 3061, 1889, 137, 2467, 16189, 641, 3109, 4973, 11087, 1321, 101, 7151, 7669, 757, 38629, 1231, 49663, 12289, 859, 239, 27581, 9613, 18131, 13757, 33931, 9161, 118901, 6763, 18233

Offset: 1

Robert G. Wilson v, 1994; Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), May 22 2000

First cyclic number of n-th degree (or n-th order): the reciprocals of these numbers belong to one of n different cycles. Each cycle has (a(n) - 1)/n digits.
From Robert G. Wilson v, Aug 21 2014: (Start)
recursive by indices:
 1,    7,        211,            79337,    634776923741, ...
 2,    3,        103,          2368589, 785245568161181, ...
 4,   53,     135257,    2332901103899, ...
 5,   11,        353,          3795457, 693814982285339, ...
 6,   79,      26861,      23947548497, ...
 8,   41,     118901,    1015118238709, ...
 9,   73,     142789,     267291583927, ...
10,  281,    3097183,   66880786504811, ...
12,   37,      18131,     105385168331, ...
13, 2393,   11160953, 7140939250711817, ...
14, 4999, 2148340247,          > 10^19,
... .
(End)

John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 162.
M. Gardner, Mathematical Circus, Cambridge University Press (1996).

Robert G. Wilson v, Table of n, a(n) for n = 1..65000 (first 1000 terms from _T. D. Noe_)
H. Richter, The period length of the decimal expansion of a fraction
Index entries for sequences related to decimal expansion of 1/n

First time n appears in A006556.
Cf. A006883, A097443, A055628, A056157, A056210, A056211, A056212, A056213, A056214, A056215, A056216, A056217, A098680, which are sequences of primes p where the period of the reciprocal is (p-1)/n for n=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13.
Cf. A101208, A101209 (similar sequences for base 2 and base 3).

a[n_Integer] := Block[{m = If[ OddQ@ n, 2n, n]}, p = m +1; While[ !PrimeQ@ p || p != 1 + n*MultiplicativeOrder[10, p], p = p += m]; p]; a[1] = 7; a[4] = 53; Array[f, 50] (* Robert G. Wilson v, Apr 19 2005; revised Aug 20 2014 and Feb 14 2025 *)

More terms from David W. Wilson, May 22 2000

cp's OEIS Frontend

A076517 Partial sums of A060370 (cf. A006556).

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A334904 a(n) is the least integer b such that the fractions (b^0)/p, (b^1)/p, ..., (b^(r-1))/p where p is the n-th prime, produce the A006556(n) distinct cycles.

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A005384 Sophie Germain primes p: 2p+1 is also prime.

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A005231 Odd abundant numbers (odd numbers m whose sum of divisors exceeds 2m).

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A005114 Untouchable numbers, also called nonaliquot numbers: impossible values for the sum of aliquot parts function (A001065).

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A006566 Dodecahedral numbers: a(n) = n*(3*n - 1)*(3*n - 2)/2.

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A006566 Dodecahedral numbers: a(n) = n(3n - 1)(3n - 2)/2.

A006564 Icosahedral numbers: a(n) = n(5n^2 - 5*n + 2)/2.