A229964 -id:A229964 - cp's OEIS frontend

A229966 Numbers n such that A229964(n) = 3.

12, 14, 22, 27, 33, 57, 85, 161, 203, 533, 689, 901, 1121, 1633, 2581, 4181, 5513, 5633, 7439, 10561, 18023, 18881, 20833, 21389, 23941, 25043, 28421, 32033, 37733, 48641, 58241, 64643, 66901, 77423, 80033, 84001, 90133, 106439, 116821, 119201, 149189, 155041

Offset: 1

Eric M. Schmidt, Oct 04 2013

nonn

Equals {12, 14, 22, 27, 57} UNION {pq | p, q prime, q = 3p+2 or (p >= 5 and q = 4p+1)}.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Rosemary Sullivan and Neil Watling, Independent Divisibility Pairs on the Set of Integers from 1 to N, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A65, 2013.

Cf. A166684, A082663, A229965, A229967, A023208, A023212.

[p * (3*p+2) for p in prime_range(10000) if (3*p+2).is_prime()] + [p * (4*p+1) for p in prime_range(5, 10000) if (4*p+1).is_prime()] + [12, 14, 22, 27, 57]

A229967 Numbers n such that A229964(n) = 4.

Original entry on oeis.org

18, 26, 28, 39, 65, 115, 119, 133, 319, 341, 377, 403, 481, 517, 629, 697, 731, 779, 799, 817, 893, 1007, 1207, 1219, 1357, 1403, 1541, 1769, 1943, 2059, 2077, 2117, 2201, 2263, 2291, 2407, 2449, 2573, 2759, 2923, 3071, 3293, 3589, 3649, 3737, 3811, 3827, 3959

Offset: 1

Eric M. Schmidt, Oct 04 2013

nonn

Equals {18, 26, 28, 39} UNION {pq | p, q prime, p >= 5 and (2p+3 <= q <= 3p-2 or (p == 2 (mod 3) and q = 4p+3))}.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Rosemary Sullivan and Neil Watling, Independent Divisibility Pairs on the Set of Integers from 1 to N, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A65, 2013.

Cf. A166684, A082663, A229965, A229966.

sum(([p * q for q in prime_range(2*p+3, 3*p-1)] for p in prime_range(5, 10000)), []) + [p * (4*p + 3) for p in prime_range(5, 10000) if (4*p+3).is_prime() and p%3==2] + [18, 26, 28, 39]

A229965 Numbers n such that A229964(n) = 1.

Original entry on oeis.org

6, 8, 10, 16, 21, 55, 253, 1081, 1711, 3403, 5671, 13861, 15931, 25651, 34453, 60031, 64261, 73153, 108811, 114481, 126253, 158203, 171991, 258121, 351541, 371953, 392941, 482653, 518671, 703891, 822403, 853471, 869221, 933661, 1034641, 1104841, 1159003

Offset: 1

Eric M. Schmidt, Oct 04 2013

nonn

Equals {6, 8, 16} UNION A156592.

Rosemary Sullivan and Neil Watling, Independent Divisibility Pairs on the Set of Integers from 1 to N, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A65, 2013.

Cf. A166684, A082663, A229966, A229967.

A166684 Numbers n such that d(n)<4.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251

Offset: 1

Juri-Stepan Gerasimov, Oct 18 2009

1 together with primes and squares of primes.
Numbers n such that A229964(n) = 0. - Eric M. Schmidt, Oct 05 2013
Numbers that cannot be written as a product of 2 distinct nonunits. - Peter Munn, May 26 2023

G. C. Greubel, Table of n, a(n) for n = 1..10000

A000430 is the main entry for this sequence.

Cf. A000005, A229964.

Select[Range[300],DivisorSigma[0,#]<4&] (* or *) Select[With[ {prs = Prime[Range[200]]},Union[Join[{1},prs,prs^2]]],#<301&] (* Harvey P. Dale, Jan 04 2012 *)

is(n)=isprime(n) || (issquare(n,&n) && isprime(n)) || n==1 \\ Charles R Greathouse IV, Dec 23 2022

from math import isqrt
from sympy import primepi
def A166684(n):
    def f(x): return n-1+x-primepi(x)-primepi(isqrt(x))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return int(m) # Chai Wah Wu, Aug 09 2024

a(n) = A000430(n-1), n>1. - R. J. Mathar, May 21 2010

Corrected (193 inserted) by R. J. Mathar, May 21 2010

A082663 Odd semiprimes pq with p < q < 2p.

Original entry on oeis.org

15, 35, 77, 91, 143, 187, 209, 221, 247, 299, 323, 391, 437, 493, 527, 551, 589, 667, 703, 713, 851, 899, 943, 989, 1073, 1147, 1189, 1247, 1271, 1333, 1363, 1457, 1517, 1537, 1591, 1643, 1739, 1763, 1829, 1891, 1927, 1961, 2021, 2173, 2183, 2257, 2279

Offset: 1

Naohiro Nomoto, May 18 2003

Numbers k such that A082647(k) = A000005(k) - 1 = 3.
A082647(p^2) = A000005(p^2) - 1 = 2, where p is odd prime.
Numbers n such that A229964(n) = 2. - Eric M. Schmidt, Oct 05 2013

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1...1000 from Vincenzo Librandi)

Cf. A000005, A056913, A046388, A082647, A229964.

f[n_]:=Last/@FactorInteger[n]=={1,1}&&FactorInteger[n][[1,1]]>2&&Floor[FactorInteger[n][[2,1]]/FactorInteger[n][[1,1]]]==1;lst={};Do[If[f[n],AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, May 19 2010 *)
pq2pQ[n_]:=Module[{fi=FactorInteger[n][[All,1]]},PrimeOmega[n]==2 && fi[[1]]< fi[[2]]< 2fi[[1]]]; Select[Range[1,2301,2],pq2pQ]//Quiet (* Harvey P. Dale, Jul 31 2021 *)

list(lim)=my(v=List()); forprime(p=3, sqrtint(lim\=1), forprime(q=p+1,min(lim\p,2*p), listput(v,p*q))); Set(v) \\ Charles R Greathouse IV, Mar 03 2021

New name based on a Jan 23 2004 comment from Vladeta Jovovic - Charles R Greathouse IV, Mar 03 2021

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A229966 Numbers n such that A229964(n) = 3.

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A229967 Numbers n such that A229964(n) = 4.

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A229965 Numbers n such that A229964(n) = 1.

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A166684 Numbers n such that d(n)<4.

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A082663 Odd semiprimes pq with p < q < 2p.

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