A072668 -id:A072668 - cp's OEIS frontend

A338929 a(n) is the smallest prime number p larger than A072668(n) such that p is equal to 1 (mod A072668(n)).

7, 11, 29, 17, 19, 23, 53, 29, 31, 103, 191, 41, 43, 47, 73, 101, 53, 109, 59, 311, 97, 67, 103, 71, 149, 191, 79, 83, 173, 89, 181, 283, 97, 197, 101, 103, 107, 109, 331, 113, 229, 709, 367, 311, 127, 193, 131, 269, 137, 139, 569, 293, 149, 151, 229, 463

Offset: 1

Ahmad J. Masad, Nov 15 2020

nonn

In A002808(n)-base numeral system, a(n) is the smallest prime number for which the digital root is 1.
Conjecture: As n approaches infinity, the probability that a prime number is a term in this sequence approaches 1.
Conjecture: There are infinitely many primes that are not terms in this sequence.
The sequence for all positive numbers (instead of A072668) is A034694. - Peter Munn, May 02 2023

			For n=20, A072668(20)=31, and 311 is the smallest prime number p larger than 31 such that p is equal to 1 (mod 31), so a(20)=311.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A000040, A002808, A034694, A072668, A339540.

Map[Block[{p = NextPrime[#]}, While[Mod[p, #] != 1, p = NextPrime[p]]; p] &, Select[Range[4, 78], CompositeQ] - 1] (* Michael De Vlieger, Dec 10 2020 *)

f(x) = {my(p=nextprime(x)); while ((p % x) != 1, p = nextprime(p+1)); p;}
lista(nn) = {my(list = List()); forcomposite(c=1, nn, listput(list, f(c-1));); Vec(list);} \\ Michel Marcus, Nov 17 2020

More terms from Michel Marcus, Nov 17 2020

A006093 a(n) = prime(n) - 1.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 256, 262, 268, 270

Offset: 1

N. J. A. Sloane

These are also the numbers that cannot be written as i*j + i + j (i,j >= 1). - Rainer Rosenthal, Jun 24 2001; Henry Bottomley, Jul 06 2002
The values of k for which Sum_{j=0..n} (-1)^j*binomial(k, j)*binomial(k-1-j, n-j)/(j+1) produces an integer for all n such that n < k. Setting k=10 yields [0, 1, 4, 11, 19, 23, 19, 11, 4, 1, 0] for n = [-1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], so 10 is in the sequence. Setting k=3 yields [0, 1, 1/2, 1/2] for n = [-1, 0, 1, 2], so 3 is not in the sequence. - Dug Eichelberger (dug(AT)mit.edu), May 14 2001
n such that x^n + x^(n-1) + x^(n-2) + ... + x + 1 is irreducible. - Robert G. Wilson v, Jun 22 2002
Records for Euler totient function phi.
Together with 0, n such that (n+1) divides (n!+1). - Benoit Cloitre, Aug 20 2002; corrected by Charles R Greathouse IV, Apr 20 2010
n such that phi(n^2) = phi(n^2 + n). - Jon Perry, Feb 19 2004
Numbers having only the trivial perfect partition consisting of a(n) 1's. - Lekraj Beedassy, Jul 23 2006
Numbers n such that the sequence {binomial coefficient C(k,n), k >= n } contains exactly one prime. - Artur Jasinski, Dec 02 2007
Record values of A143201: a(n) = A143201(A001747(n+1)) for n > 1. - Reinhard Zumkeller, Aug 12 2008
From Reinhard Zumkeller, Jul 10 2009: (Start)
The first N terms can be generated by the following sieving process:
  start with {1, 2, 3, 4, ..., N - 1, N};
  for i := 1 until SQRT(N) do
  (if (i is not striked out) then
  (for j := 2 * i + 1 step i + 1 until N do
  (strike j from the list)));
  remaining numbers = {a(n): a(n) <= N}. (End)
a(n) = partial sums of A075526(n-1) = Sum_{1..n} A075526(n-1) = Sum_{1..n} (A008578(n+1) - A008578(n)) = Sum_{1..n} (A158611(n+2) - A158611(n+1)) for n >= 1. - Jaroslav Krizek, Aug 04 2009
A006093 U A072668 = A000027. - Juri-Stepan Gerasimov, Oct 22 2009
A171400(a(n)) = 1 for n <> 2: subsequence of A171401, except for a(2) = 2. - Reinhard Zumkeller, Dec 08 2009
Numerator of (1 - 1/prime(n)). - Juri-Stepan Gerasimov, Jun 05 2010
Numbers n such that A002322(n+1) = n. This statement is stronger than repeating the property of the entries in A002322, because it also says in reciprocity that this sequence here contains no numbers beyond the Carmichael numbers with that property. - Michel Lagneau, Dec 12 2010
a(n) = A192134(A095874(A000040(n))); subsequence of A192133. - Reinhard Zumkeller, Jun 26 2011
prime(a(n)) + prime(k) < prime(a(k) + k) for at least one k <= a(n): A212210(a(n),k) < 0. - Reinhard Zumkeller, May 05 2012
Except for the first term, numbers n such that the sum of first n natural numbers does not divide the product of first n natural numbers; that is, n*(n + 1)/2 does not divide n!. - Jayanta Basu, Apr 24 2013
BigOmega(a(n)) equals BigOmega(a(n)*(a(n) + 1)/2), where BigOmega = A001222. Rationale: BigOmega of the product on the right hand side factorizes as BigOmega(a/2) + Bigomega(a+1) = BigOmega(a/2) + 1 because a/2 and a + 1 are coprime, because BigOmega is additive, and because a + 1 is prime. Furthermore Bigomega(a/2) = Bigomega(a) - 1 because essentially all 'a' are even. - Irina Gerasimova, Jun 06 2013
Record values of A060681. - Omar E. Pol, Oct 26 2013
Deficiency of n-th prime. - Omar E. Pol, Jan 30 2014
Conjecture: All the sums Sum_{k=s..t} 1/a(k) with 1 <= s <= t are pairwise distinct. In general, for any integers d >= -1 and m > 0, if Sum_{k=i..j} 1/(prime(k)+d)^m = Sum_{k=s..t} 1/(prime(k)+d)^m with 0 < i <= j and 0 < s <= t then we must have (i,j) = (s,t), unless d = m = 1 and {(i,j),(s,t)} = {(4,4),(8,10)} or {(4,7),(5,10)}. (Note that 1/(prime(8)+1)+1/(prime(9)+1)+1/(prime(10)+1) = 1/(prime(4)+1) and Sum_{k=5..10} 1/(prime(k)+1) = 1/(prime(4)+1) + Sum_{k=5..7} 1/(prime(k)+1).) - Zhi-Wei Sun, Sep 09 2015
Numbers n such that (prime(i)^n + n) is divisible by (n+1), for all i >= 1, except when prime(i) = n+1. - Richard R. Forberg, Aug 11 2016
a(n) is the period of Fubini numbers (A000670) over the n-th prime. - Federico Provvedi, Nov 28 2020

Archimedeans Problems Drive, Eureka, 40 (1979), 28.
Harvey Dubner, Generalized Fermat primes, J. Recreational Math., 18 (1985): 279-280.
M. Gardner, The Colossal Book of Mathematics, pp. 31, W. W. Norton & Co., NY, 2001.
M. Gardner, Mathematical Circus, pp. 251-2, Alfred A. Knopf, NY, 1979.
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..10000
Thomas F. Bloom, Unit fractions with shifted prime denominators, arXiv:2305.02689 [math.NT], 2023.
R. P. Boas & N. J. A. Sloane, Correspondence, 1974
Harvey Dubner, Generalized Fermat primes, J. Recreational Math. 18.4 (1985-1986), 279. (Annotated scanned copy)
Armel Mercier, Problem E 3065, American Mathematical Monthly, 1984, p. 649.
Armel Mercier, S. K. Rangarajan, J. C. Binz and Dan Marcus, Problem E 3065, American Mathematical Monthly, No. 4, 1987, pp. 378.
Poo-Sung Park, Additive uniqueness of PRIMES-1 for multiplicative functions, arXiv:1708.03037 [math.NT], 2017.
J. R. Rickard and J. J. Hitchcock, Problem Drive 4, Archimedeans Problems Drive, Eureka, 40 (1979), 28-29, 40. (Annotated scanned copy)
Index entries for sequences generated by sieves

a(n) = K(n, 1) and A034693(K(n, 1)) = 1 for all n. The subscript n refers to this sequence and K(n, 1) is the index in A034693. - Labos Elemer
Cf. A000040, A034694. Different from A075728.
Complement of A072668 (composite numbers minus 1), A072670(a(n))=0.
Essentially the same as A039915.
Cf. A084920, A006093, A050997, A008864, A060800, A131991, A131992, A131993.
Cf. A101301 (partial sums), A005867 (partial products).
Column 1 of the following arrays/triangles: A087738, A249741, A352707, A378979, A379010.
The last diagonal of A162619, and of A174996, the first diagonal in A131424.
Row lengths of irregular triangles A086145, A124223, A212157.

Filtered([1..280],IsPrime)-1; # Muniru A Asiru, Nov 25 2018

a006093 = (subtract 1) . a000040  -- Reinhard Zumkeller, Mar 06 2012

[NthPrime(n)-1: n in [1..100]]; // Vincenzo Librandi, Nov 17 2015

for n from 2 to 271 do if (n! mod n^2 = n*(n-1) and (n<>4) then print(n-1) fi od; # Gary Detlefs, Sep 10 2010
# alternative
A006093 := proc(n)
    ithprime(n)-1 ;
end proc:
seq(A006093(n),n=1..100) ; # R. J. Mathar, Feb 06 2019

Table[Prime[n] - 1, {n, 1, 30}] (* Vladimir Joseph Stephan Orlovsky, Apr 27 2008 *)
a[ n_] := If[ n < 1, 0, -1 + Prime @ n] (* Michael Somos, Jul 17 2011 *)
Prime[Range[60]] - 1 (* Alonso del Arte, Oct 26 2013 *)

isA006093(n) = isprime(n+1) \\ Michael B. Porter, Apr 09 2010

A006093(n) = prime(n)-1 \\ Michael B. Porter, Apr 09 2010

\\ Sieve as described in Rainer Rosenthal's comment.
m=270;s=vector(m);for(i=1,m,for(j=i,m,k=i*j+i+j;if(k<=m,s[k]=1)));for(k=1,m,if(s[k]==0,print1(k,", "))); \\ Hugo Pfoertner, May 14 2019

from sympy import prime
for n in range(1,100): print(prime(n)-1, end=', ') # Stefano Spezia, Nov 30 2018

a(n) = (p-1)! mod p where p is the n-th prime, by Wilson's theorem. - Jonathan Sondow, Jul 13 2010
a(n) = A000010(prime(n)) = A000010(A006005(n)). - Antti Karttunen, Dec 16 2012
a(n) = A005867(n+1)/A005867(n). - Eric Desbiaux, May 07 2013
a(n) = A000040(n) - 1. - Omar E. Pol, Oct 26 2013
a(n) = A033879(A000040(n)). - Omar E. Pol, Jan 30 2014

Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010
Obfuscating comments removed by Joerg Arndt, Mar 11 2010
Edited by Charles R Greathouse IV, Apr 20 2010

A072670 Number of ways to write n as i*j + i + j, 0 < i <= j.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Jun 30 2002

nonn

a(n) is the number of partitions of n+1 with summands in arithmetic progression having common difference 2. For example a(29)=3 because there are 3 partitions of 30 that are in arithmetic progressions: 2+4+6+8+10, 8+10+12 and 14+16. - N-E. Fahssi, Feb 01 2008
From Daniel Forgues, Sep 20 2011: (Start)
a(n) is the number of nontrivial factorizations of n+1, in two factors.
a(n) is the number of ways to write n+1 as i*j + i + j + 1 = (i+1)(j+1), 0 < i <= j. (End)
a(n) is the number of ways to write n+1 as i*j, 1 < i <= j. - Arkadiusz Wesolowski, Nov 18 2012
For a generalization, see comment in A260804. - Vladimir Shevelev, Aug 04 2015
Number of partitions of n into 3 parts whose largest part is equal to the product of the other two. - Wesley Ivan Hurt, Jan 04 2022

			a(11)=2: 11 = 1*5 + 1 + 5 = 2*3 + 2 + 3.
From _Daniel Forgues_, Sep 20 2011 (Start)
Number of nontrivial factorizations of n+1 in two factors:
  0 for the unit 1 and prime numbers
  1 for a square: n^2 = n*n
  1 for 6 (2*3), 10 (2*5), 14 (2*7), 15 (3*5)
  1 for a cube: n^3 = n*n^2
  2 for 12 (2*6, 3*4), for 18 (2*9, 3*6) (End)

Robert Israel, Table of n, a(n) for n = 0..10000
Joseph W. Andrushkiw, Roman I. Andrushkiw and Clifton E. Corzatt, Representations of Positive Integers as Sums of Arithmetic Progressions, Mathematics Magazine, Vol. 49, No. 5 (Nov., 1976), pp. 245-248.
M. A. Nyblom and C. Evans, On the enumeration of partitions with summands in arithmetic progression, Australian Journal of Combinatorics, Vol. 28 (2003), pp. 149-159.
Vladimir Shevelev, Representation of positive integers by the form x1...xk+x1+...+xk, arXiv:1508.03970 [math.NT], 2015.

Cf. A001620, A067432, A066938, A072668, A006093, A072671, A161840, A260803, A260804.

0, seq(ceil(numtheory:-tau(n+1)/2)-1, n=1..100); # Robert Israel, Aug 04 2015

p2[n_] := 1/2 (Length[Divisors[n]] - 2 + ((-1)^(Length[Divisors[n]] + 1) + 1)/2); Table[p2[n + 1], {n, 0, 104}] (* N-E. Fahssi, Feb 01 2008 *)
Table[Ceiling[DivisorSigma[0, n + 1]/2] - 1, {n, 0, 104}] (* Arkadiusz Wesolowski, Nov 18 2012 *)

is_ok(k,i,j)=0=i&&k===i*j+i+j;
first(m)=my(v=vector(m,z,0));for(l=1,m,for(j=1,l,for(i=1,j,if(is_ok(l,i,j),v[l]++))));concat([0],v); /* Anders Hellström, Aug 04 2015 */

a(n)=(numdiv(n+1)+issquare(n+1))/2-1 \\ Charles R Greathouse IV, Jul 14 2017

a(n) = A038548(n+1) - 1.
From N-E. Fahssi, Feb 01 2008: (Start)
a(n) = p2(n+1), where p2(n) = (1/2)*(d(n) - 2 + ((-1)^(d(n)+1)+1)/2); d(n) is the number of divisors of n: A000005.
G.f.: Sum_{n>=1} a(n) x^n = 1/x Sum_{k>=2} x^(k^2)/(1-x^k). (End)
lim_{n->infinity} a(A002110(n)-1) = infinity. - Vladimir Shevelev, Aug 04 2015
a(n) = A161840(n+1)/2. - Omar E. Pol, Feb 27 2019
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 3) / 2, where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 14 2024

A060462 Integers k such that k! is divisible by k*(k+1)/2.

Original entry on oeis.org

1, 3, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94

Offset: 1

Michel ten Voorde, Apr 09 2001

nonn

k! / (k-th triangular number) is an integer.
a(n) = A072668(n) for n>0.
From Bernard Schott, Dec 11 2020: (Start)
Numbers k such that Sum_{j=1..k} j divides Product_{j=1..k} j.
k is a term iff k != p-1 with p is an odd prime (see De Koninck & Mercier reference).
The ratios obtained a(n)!/T(a(n)) = A108552(n). (End)

			5 is a term because 5*4*3*2*1 = 120 is divisible by 5 + 4 + 3 + 2 + 1 = 15.

Jean-Marie De Koninck & Armel Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 181 pp. 31 and 163, Ellipses, Paris, 2004.
Joseph D. E. Konhauser et al., Which Way Did The Bicycle Go?, Problem 98, pp. 29; 145-146, MAA Washington DC, 1996.

Harry J. Smith, Table of n, a(n) for n = 1..2001 [offset adapted by _Georg Fischer_, Jan 04 2021]

Cf. A000142, A000217, A072668, A108552.

for n from 1 to 300 do if n! mod (n*(n+1)/2) = 0 then printf(`%d,`,n) fi:od:

Select[Range[94], Mod[#!, #*(# + 1)/2] == 0 &] (* Jayanta Basu, Apr 24 2013 *)

{ f=1; t=0; n=-1; for (m=1, 4000, f*=m; t+=m; if (f%t==0, write("b060462.txt", n++, " ", m)); if (n==2000, break); ) } \\ Harry J. Smith, Jul 05 2009

from sympy import composite
def A060462(n): return composite(n-1)-1 if n>1 else 1 # Chai Wah Wu, Aug 02 2024

Corrected and extended by Henry Bottomley and James Sellers, Apr 11 2001

Offset corrected by Alois P. Heinz, Dec 11 2020

A055926 Numbers k such that {largest m such that 1, 2, ..., m divide k} is different from {largest m such that m! divides k}; numbers k which are either odd multiples of 12 or the largest m such that (m-1)! divides k is a composite number > 5.

Original entry on oeis.org

12, 36, 60, 84, 108, 120, 132, 156, 180, 204, 228, 240, 252, 276, 300, 324, 348, 360, 372, 396, 420, 444, 468, 480, 492, 516, 540, 564, 588, 600, 612, 636, 660, 684, 708, 732, 756, 780, 804, 828, 840, 852, 876, 900, 924, 948, 960, 972, 996, 1020, 1044, 1068

Offset: 1

Leroy Quet, Jul 16 2000

From Antti Karttunen, Nov 20 - Dec 06 2013: (Start)
This sequence has several interpretations:
Numbers k such that A055874(k) differs from A055881(k). [Leroy Quet's original definition of the sequence. Note that A055874(k) >= A055881(k) for all k.]
Numbers k such that {largest m such that m! divides k^2} is different from {largest m such that m! divides k}, i.e., numbers k for which A232098(k) > A055881(k).
Numbers k which are either 12 times an odd number (A073762) or the largest m such that (m-1)! divides k is a composite number > 5 (A232743).
Please see my attached notes for the proof of the equivalence of these interpretations.
Additional implications based on that proof:
A232099 is a subset of this sequence.
A055881(a(n))+1 is always composite. In the range n = 1..17712, only values 4, 6, 8, 9 and 10 occur.
The new definition can be also rephrased by saying that the sequence contains all the positive integers k whose factorial base representation of (A007623(k)) either ends as '...200' (in which case k is an odd multiple of 12, 12 = '200', 36 = '1200', 60 = '2200', ...) or the number of trailing zeros + 2 in that representation is a composite number greater than or equal to 6, e.g. 120 = '10000' (in other words, A055881(k) is one of the terms of A072668 after the initial 3). Together these conditions also imply that all the terms are divisible by 12.
(End)

			12 is included because 3! is the largest factorial to divide 12, but 1, 2, 3 and 4 all divide 12. Equally, 12 is included because it is one of the terms of A073762, or equally, because its factorial base representation ends with digits '...200': A007623(12) = 200.
840 (= 3*5*7*8) is included because the largest factorial which divides 840 is 5! (840 = 7*120), but all positive integers up to 8 divide 840. Equally, 840 is included because it is one of the terms of A232743 as 5+1 = 6 is a composite number larger than 5. Note that A007623(840) = 110000.

Antti Karttunen, Table of n, a(n) for n = 1..17712
Antti Karttunen, A proof for the equivalence of three alternative definitions of A055926
Wikipedia, Wilson's theorem (See the section "Composite modulus")

Union of A073762 and A232743. Equivalently, setwise difference of A232742 and A017593. Subset: A232099.

Cf. A055874, A055881, A072668, A232098, A232100, A232741, A232744, A232745.

More terms from Antti Karttunen, Dec 01 2013

A066938 Primes of the form p*q+p+q, where p and q are primes.

Original entry on oeis.org

11, 17, 23, 31, 41, 47, 53, 59, 71, 79, 83, 89, 107, 113, 127, 131, 151, 167, 179, 191, 227, 239, 251, 263, 269, 271, 293, 311, 359, 383, 419, 431, 439, 443, 449, 479, 491, 503, 521, 587, 593, 599, 607, 631, 647, 659, 683, 701, 719, 727, 743, 773, 809, 827

Offset: 1

Reinhard Zumkeller, Jan 24 2002

nonn

For p not equal to q, either p*q or p+q is odd, so their sum is odd.
The representation is ambiguous, e.g. 2*7+2+7 = 23 = 3*5+3+5.
Complement of A198273 with respect to A000040. - Reinhard Zumkeller, Oct 23 2011
None of these primes are in A158913 since if p*q+p+q is a prime, then sigma(p*q+p+q) = sigma(p*q). - Amiram Eldar, Nov 15 2021

			59 is in the sequence because 59 = 2 * 19 + 2 + 19.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A054973, A067432, A072668, A072673, A088709, A158913, A198273, A198277.

a066938 n = a066938_list !! (n-1)
a066938_list = map a000040 $ filter ((> 0) . a067432) [1..]
-- Reinhard Zumkeller, Oct 23 2011

nn = 1000; n2 = PrimePi[nn/3]; Select[Union[Flatten[Table[(Prime[i] + 1) (Prime[j] + 1) - 1, {i, n2}, {j, n2}]]], # <= nn && PrimeQ[#] &]

is(n)=fordiv(n+1,d,my(p=d-1,q=(n+1)/d-1); if(isprime(p) && isprime(q), return(isprime(n)))); 0 \\ Charles R Greathouse IV, Jul 23 2013

A067432(A049084(a(n))) > 0. - Reinhard Zumkeller, Oct 23 2011

A054973(a(n)+1) >= 2. - Amiram Eldar, Nov 15 2021

Edited by Robert G. Wilson v, Feb 01 2002

A079696 Numbers one more than composite numbers.

Original entry on oeis.org

5, 7, 9, 10, 11, 13, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 36, 37, 39, 40, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 63, 64, 65, 66, 67, 69, 70, 71, 73, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96

Offset: 1

Vladeta Jovovic, Jan 31 2003

From Hieronymus Fischer, Mar 27 2014: (Start)
Numbers m such that m == 1 mod j and m > j^2 for any j > 1.
Example: m == 6 mod 10 is a term for m > 6, since m = 6 + 10k = 1 + (2k+1)*5, and m > (2k+1)^2 (for k := 1, m = 16), and m > 5^2 (for k > 1, m > 16).
A187813 and this sequence have no terms in common; this means that for each term a(n) there exists a base b such that the base-b digit sum is b.
Example: m = 1 + 3k, k > 3, is a term, since m > 3(1+3) > 3^2, thus the base-b-digit sum of (m) is = b for any b > 1 (here the base b is k+1 since 1+3k = 2(k+1) + k-1).
In general: Given a term a(n) there are p and q with p >= q > 1 such that a(n) = 1 + p*q. With b := p + 1 we get a(n) = (q-1)*b + b - (q-1), where 1 <= q-1 < b, which implies that the base-b digital sum of a(n) is = q-1 + b - (q-1) = b.
This sequence is the complement of the disjunction of A187813 with A239708. This means that a number m is a term if and only if there is a base b > 2 such that the base-b digit sum of m is b.
(End)

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A072668.

Cf. A007953, A187813, A239703, A239708.

from sympy import composite
def A079696(n): return composite(n)+1 # Chai Wah Wu, Mar 19 2025

a(n) = A002808(n) + 1.

A239703(a(n)) > 0. - Hieronymus Fischer, Apr 10 2014

Edited by Charles R Greathouse IV, Mar 19 2010

A232743 Numbers n for which the largest m such that (m-1)! divides n is a composite number > 5.

Original entry on oeis.org

120, 240, 360, 480, 600, 840, 960, 1080, 1200, 1320, 1560, 1680, 1800, 1920, 2040, 2280, 2400, 2520, 2640, 2760, 3000, 3120, 3240, 3360, 3480, 3720, 3840, 3960, 4080, 4200, 4440, 4560, 4680, 4800, 4920, 5040, 5160, 5280, 5400, 5520, 5640, 5880, 6000, 6120, 6240

Offset: 1

Antti Karttunen, Dec 01 2013

nonn

Numbers n for which A055881(n)>4 and is one of the terms of A072668.
Numbers n for which two plus the number of the trailing zeros in their factorial base representation A007623(n) is a composite number larger than 5.
All terms are multiples of 120. Specifically, these are all those terms of A232742 which are divisible by 120 (or equally: 24).
Please see also the comments in A055926, whose subset this sequence is.

			120 is included because A055881(120)=5 and 5+1 is a composite number larger than 5. Note that A007623(120) = '10000', with four trailing zeros.
720 is the first missing multiple of 120, as A055881(720)=6 and 7 is a prime, not composite, so 720 is not included in this sequence. Note that A007623(720) = '100000', with five trailing zeros, and 5+2 is not a composite.
120960 (= 3*8!) is included because A055881(120960)=8 and 9 is a composite number larger than 5. Note that A007623(120960) = '30000000', with seven trailing zeros.

Antti Karttunen, Table of n, a(n) for n = 1..25920

Subset of both A232742 and A055926.

Cf. also A007623, A055881, A230403, A232744-A232745.

A232742 Numbers n for which the largest m such that (m-1)! divides n is a composite.

Original entry on oeis.org

6, 12, 18, 30, 36, 42, 54, 60, 66, 78, 84, 90, 102, 108, 114, 120, 126, 132, 138, 150, 156, 162, 174, 180, 186, 198, 204, 210, 222, 228, 234, 240, 246, 252, 258, 270, 276, 282, 294, 300, 306, 318, 324, 330, 342, 348, 354, 360, 366, 372, 378, 390, 396, 402, 414

Offset: 1

Antti Karttunen, Dec 01 2013

nonn

Numbers n for which A055881(n) is one of the terms of A072668.
Equally: numbers n for which {the number of the trailing zeros in their factorial base representation A007623(n)} + 2 is a composite number.
All terms are divisible by 6.
The sequence can be described in the following manner: Sequence includes all multiples of 3!, except that it excludes from those the multiples of 4! (24), except that it includes the multiples of 5! (120), except that it excludes the multiples of 6! (720), except that it includes the multiples of 7! (5040) (and also those of 8! and 9!, because here 8+1 = 9 is the first odd composite), of which it however excludes the multiples of 10!, except that it includes the multiples of 11!, but excludes the multiples of 12!, but includes the multiples of 13! (and 14! and 15!, because 14-16 are all composites), but excludes the multiples of 16!, and so on, ad infinitum.

			6 is included because A055881(6)=3 and 3+1 is a composite number.
24 is the first excluded multiple of 6, as A055881(24)=4 and 5 is a prime, not composite, so 24 is not included in this sequence.
120 is the first included multiple of 24, as A055881(120)=5 and 6 is a composite.

Antti Karttunen, Table of n, a(n) for n = 1..10656

Complement: A232741. Subset: A232743.

Cf. A055926, A232744-A232745, A007623, A055881, A230403.

cp's OEIS Frontend

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A072670 Number of ways to write n as i*j + i + j, 0 < i <= j.

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A060462 Integers k such that k! is divisible by k*(k+1)/2.

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A055926 Numbers k such that {largest m such that 1, 2, ..., m divide k} is different from {largest m such that m! divides k}; numbers k which are either odd multiples of 12 or the largest m such that (m-1)! divides k is a composite number > 5.

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A066938 Primes of the form p*q+p+q, where p and q are primes.

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