A084970 -id:A084970 - cp's OEIS frontend

A255416 Row 6 of Ludic array A255127.

13, 73, 133, 197, 263, 325, 385, 449, 511, 571, 641, 701, 761, 823, 887, 947, 1013, 1075, 1139, 1199, 1261, 1327, 1387, 1447, 1513, 1573, 1637, 1703, 1763, 1825, 1889, 1951, 2011, 2071, 2141, 2201, 2261, 2327, 2387, 2453, 2515, 2575, 2639, 2699, 2767, 2827, 2887, 2953, 3013, 3073, 3143, 3203, 3265, 3325, 3389, 3451, 3511, 3581, 3641, 3701

Offset: 1

Antti Karttunen, Feb 22 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001

Row 6 of A255127. See A255415 for row 5 and A255417 for row 7.

Cf. A084970, A255407.

my(L=[x+2^(x%2)|x<-[1..10^4]*3], m(n,k)=2^(n\/k*k)\(2^k-1)); for(i=3, 5, L=vecextract(L, 2^#L-m(#L, L[1])-1)); L255416=vecextract(L, m(#L, L[1]));
A255416(n)=n--\480*30030+L255416[n%480+1] \\ M. F. Hasler, Nov 17 2024

def A255416(n):
    try: n-=1; return A255416.L[n]
    except IndexError: return n//480*30030 + A255416.L[n%480]
    except AttributeError: L = [3*x+5-(x&1) for x in range(10**4)]
    for k in L[:3]: L = [x for i,x in enumerate(L) if i%k]
    A255416.L = L[::13]; return n//480*30030 + A255416.L[n%480]
# M. F. Hasler, Nov 17 2024

(define (A255416 n) (A255127bi 6 n)) ;; Code for A255127bi given in A255127.

a(n) = A255407(A084970(n)).

a(n) = a(n-480) + 30030 = 30030*floor((n-1)/480) + a((n-1)%480 + 1), where % is the modulo or remainder operator. - M. F. Hasler, Nov 10 2024 and Nov 17 2024

A008367 Composite but smallest prime factor >= 17.

Original entry on oeis.org

289, 323, 361, 391, 437, 493, 527, 529, 551, 589, 629, 667, 697, 703, 713, 731, 779, 799, 817, 841, 851, 893, 899, 901, 943, 961, 989, 1003, 1007, 1037, 1073, 1081, 1121, 1139, 1147, 1159, 1189, 1207, 1219, 1241, 1247, 1271, 1273, 1333, 1343, 1349, 1357, 1363, 1369, 1387, 1403, 1411, 1457

Offset: 1

N. J. A. Sloane

nonn

Composite numbers k such that k^720 mod 30030 = 1. - Gary Detlefs, May 02 2012

The asymptotic density of this sequence is 192/1001. - Amiram Eldar, Mar 22 2021

T. D. Noe, Table of n, a(n) for n = 1..10000

Intersection of A002808 and A008366.
Cf. A038511, A084969, A084970.
Cf. A287391.

Filtered([17..1500],n->PowerMod(n,720,30030)=1 and not IsPrime(n)); # Muniru A Asiru, Nov 24 2018

for i from 1 to 2000 do if gcd(i,30030) = 1 and not isprime(i) then print(i); fi; od;

Select[ Range[ 1500 ], (GCD[ #1, 30030 ]==1&&!PrimeQ[ #1 ])& ]
Select[Range[2000], ! PrimeQ[#] && FactorInteger[#][[1, 1]] >= 17 &] (* T. D. Noe, Mar 16 2013 *)

is(n)={gcd(n,30030)==1 && !ispseudoprime(n)} \\ M. F. Hasler, Oct 04 2018

For 1 <= n < 107, a(n) = A287391(n+2); then a(107) = 2329, a(108) = 2363 are not in A287391, but again a(n) = A287391(n) for 108 < n < 120. - M. F. Hasler, Oct 04 2018

A251758 Let n>=2 be a positive integer with divisors 1 = d_1 < d_2 < ... < d_k = n, and s = d_1d_2 + d_2d_3 + ... + d_(k-1)*d_k. The sequence lists the values a(n) = floor(n^2/s).

Original entry on oeis.org

2, 3, 1, 5, 1, 7, 1, 2, 1, 11, 1, 13, 1, 2, 1, 17, 1, 19, 1, 2, 1, 23, 1, 4, 1, 2, 1, 29, 1, 31, 1, 2, 1, 4, 1, 37, 1, 2, 1, 41, 1, 43, 1, 2, 1, 47, 1, 6, 1, 2, 1, 53, 1, 4, 1, 2, 1, 59, 1, 61, 1, 2, 1, 4, 1, 67, 1, 2, 1, 71, 1, 73, 1, 2, 1, 6, 1, 79, 1, 2, 1

Offset: 2

Michel Lagneau, Dec 08 2014

nonn

s is always less than n^2 and if n is a prime number then s divides n^2.
For n >= 2, the sequence has the following properties:
a(n) = n if n is prime.
a(n) = 1 if n is in A005843 and > 2;
a(n) <= 2 if n is in A016945 and > 3;
a(n) <= 4 if n is in A084967 and > 5;
a(n) <= 6 if n is in A084968 and > 7;
a(n) = 8: <= 35336848261, ...;
a(n) <= 10 if n is in A084969 and > 11;
a(n) <= 12 if n is in A084970 and > 13;
a(n) = 14: 6678671, ...;
This is different from A250480 (a(n) = n for all prime n, and a(n) = A020639(n) - 1 for all composite n), which thus satisfies the above conditions exactly, while with this sequence A020639(n)-1 gives only the guaranteed upper limit for a(n) at composite n. Note that the first different term does not occur until at n = 2431 = 11*13*17, for which a(n) = 9. (See the example below.)
Conjecture: Terms x, where a(x)=n, x=p#k/p#j, p#i is the i-th primorial, k>j is suitable large k and j is the number of primes less than n. As an example, n=9, x = p#7/p#4 = 2431. For n=10, x = p#6/p#4 = 143 although 121 = 11^2 is the least x where a(x)=10 (see formula section). For n=8, x = p#12/p#4, p#13/p#4, p#14/p#4, p#15/p#4, p#16/p#4, etc. But is p#12/p#4 the least such x? - Robert G. Wilson v, Dec 18 2014
n^2/s is only an integer iff n is prime. - Robert G. Wilson v, Dec 18 2014
First occurrence of n >= 1: 4, 2, 3, 25, 5, 49, 7, ??? <= 35336848261, 2431, 121, 11, 169, 13, 6678671, 7429, 289, 17, 361, 19, 31367009, 20677, 529, 23, ..., . - Robert G. Wilson v, Dec 18 2014

			For n = 2431 = 11*13*17, we have (as the eight divisors of 2431 are [1, 11, 13, 17, 143, 187, 221, 2431]) a(n) = floor((2431*2431) / ((1*11)+(11*13)+(13*17)+(17*143)+(143*187)+(187*221)+(221*2431))) = floor(5909761/608125) = floor(9.718) = 9.

Michel Lagneau, Table of n, a(n) for n = 2..10000
International Mathematical Olympiad, Problems, IMO-2002, Problem 4.

Cf. A000040 (prime numbers), A005843 (even numbers), A016945 (6n+3), A084967 (GCD( 5k, 6) =1), A084968 (GCD( 7k, 30) =1), A084969 (GCD( 11k, 30) =1), A084970 (Numbers whose smallest prime factor is 13).
Cf. also A020639 (the smallest prime divisor), A055396 (its index) and arrays A083140 and A083221 (Sieve of Eratosthenes).
Differs from A250480 for the first time at n = 2431, where a(2431) = 9, while A250480(2431) = 10.
Cf. A078730 (sum of products of two successive divisors of n).

with(numtheory):nn:=100:
for n from 2 to nn do:
   x:=divisors(n):n0:=nops(x):s:=sum('x[i]*x[i+1]','i'=1..n0-1):
   z:=floor(n^2/s):printf(`%d, `,z):
od:

f[n_] := Floor[ n^2/Plus @@ Times @@@ Partition[ Divisors@ n, 2, 1]]; Array[f, 81, 2] (* Robert G. Wilson v, Dec 18 2014 *)

a(n) <= A250480(n), and especially, for all composite n, a(n) < A020639(n). [Cf. the Comments section above.] - Antti Karttunen, Dec 09 2014
From Robert G. Wilson v, Dec 18 2014: (Start)
a(n) = floor(n^2/A078730(n));
a(n) = n iff n is prime. (End)

Comments section edited by Antti Karttunen, Dec 09 2014

Instances of n for which a(n) = 8 and 14 found by Robert G. Wilson v, Dec 18 2014

cp's OEIS Frontend

A255416 Row 6 of Ludic array A255127.

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A008367 Composite but smallest prime factor >= 17.

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A251758 Let n>=2 be a positive integer with divisors 1 = d_1 < d_2 < ... < d_k = n, and s = d_1d_2 + d_2d_3 + ... + d_(k-1)*d_k. The sequence lists the values a(n) = floor(n^2/s).

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cp's OEIS Frontend

A255416 Row 6 of Ludic array A255127.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Scheme

Formula

A008367 Composite but smallest prime factor >= 17.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Formula

A251758 Let n>=2 be a positive integer with divisors 1 = d_1 < d_2 < ... < d_k = n, and s = d_1*d_2 + d_2*d_3 + ... + d_(k-1)*d_k. The sequence lists the values a(n) = floor(n^2/s).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A251758 Let n>=2 be a positive integer with divisors 1 = d_1 < d_2 < ... < d_k = n, and s = d_1d_2 + d_2d_3 + ... + d_(k-1)*d_k. The sequence lists the values a(n) = floor(n^2/s).