A234103 -id:A234103 - cp's OEIS frontend

A234102 Integers of the form (pqr + 1)/2, where p, q, r are distinct primes.

53, 83, 98, 116, 128, 137, 143, 173, 179, 193, 200, 215, 218, 228, 233, 242, 278, 281, 298, 305, 308, 314, 323, 326, 332, 333, 353, 358, 371, 380, 389, 398, 403, 431, 443, 449, 452, 458, 468, 479, 485, 494, 501, 503, 508, 512, 523, 533, 543, 548, 553, 557

Offset: 1

Clark Kimberling, Dec 27 2013

			53 = (3*5*7 + 1)/2.

Cf. A046389, A234099, A234103, A234104.

t = Select[Range[1, 10000, 2], Map[Last, FactorInteger[#]] == Table[1, {3}] &]; Take[(t + 1)/2, 120] (* A234102 *)
v = Flatten[Position[PrimeQ[(t + 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A234103 *)
(w + 1)/2  (* A234104 *)    (* Peter J. C. Moses, Dec 23 2013 *)

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A234102(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(3,integer_nthroot(x,3)[0]+1),2) for b,m in enumerate(primerange(k+1,isqrt(x//k)+1),a+1)))
    return bisection(f,n,n)+1>>1 # Chai Wah Wu, Oct 18 2024

1 + A234099.

a(n) = (A046389(n)+1)/2. - Chai Wah Wu, Oct 18 2024

A234500 Integers of the form (pqr*s + 1)/2, where p, q, r, s are distinct primes.

Original entry on oeis.org

578, 683, 893, 998, 1073, 1208, 1403, 1502, 1523, 1568, 1628, 1658, 1853, 1898, 1943, 1964, 2153, 2195, 2243, 2258, 2321, 2393, 2423, 2468, 2503, 2558, 2594, 2657, 2783, 2828, 2933, 3023, 3053, 3098, 3140, 3203, 3273, 3278, 3350, 3383, 3392, 3518, 3548, 3581

Offset: 1

Clark Kimberling, Jan 01 2014

			578 = (3*5*7*11 + 1)/2.

Cf. A234099, A234103, A234104.

t = Select[Range[1, 20000, 2], Map[Last, FactorInteger[#]] == Table[1, {4}] &]; Take[(t + 1)/2, 120] (* A234500*)
v = Flatten[Position[PrimeQ[(t + 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A234501 *)
(w + 1)/2 (* A234502 *)   (* Peter J. C. Moses, Dec 23 2013 *)
With[{nn=20},Select[Union[(Times@@#+1)/2&/@Subsets[Prime[Range[2,nn]],{4}]],#<=(105Prime[nn]+1)/2&]] (* Harvey P. Dale, Oct 18 2021 *)

1 + A234105.

A234104 Primes of the form (pqr + 1)/2, where p, q, r are distinct primes.

Original entry on oeis.org

53, 83, 137, 173, 179, 193, 233, 281, 353, 389, 431, 443, 449, 479, 503, 523, 557, 587, 593, 641, 677, 773, 823, 827, 839, 853, 953, 983, 1019, 1033, 1061, 1093, 1097, 1117, 1151, 1187, 1223, 1277, 1307, 1433, 1439, 1453, 1493, 1511, 1579, 1583, 1601, 1619

Offset: 1

Clark Kimberling, Dec 27 2013

			(3*5*7 + 1)/2 = 53.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A234102, A234103, A234101.

filter:= proc(n) local s;
  if not isprime(n) then return false fi;
  s:= ifactors(2*n-1)[2];
  nops(s)=3 and map(t -> t[2],s)=[1,1,1]
end proc:
select(filter, [seq(i,i=3..1619,2)]); # Robert Israel, May 11 2020

t = Select[Range[1, 10000, 2], Map[Last, FactorInteger[#]] == Table[1, {3}] &]; Take[(t + 1)/2, 120] (* A234102 *)
v = Flatten[Position[PrimeQ[(t + 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A234103 *)
(w + 1)/2  (* A234104 *)    (* Peter J. C. Moses, Dec 23 2013 *)
Module[{nn=100},Select[(Times@@#+1)/2&/@Subsets[Prime[Range[nn]],{3}],PrimeQ[ #] && #<=5*Prime[nn]&]]//Union (* Harvey P. Dale, Jan 29 2023 *)

A234501 Products pqrs of distinct primes for which (pqrs + 1)/2 is prime.

Original entry on oeis.org

1365, 3045, 4305, 4485, 4785, 4845, 5005, 5313, 6045, 6405, 7161, 7665, 8265, 8745, 9165, 9345, 9933, 10005, 10101, 10465, 10545, 10605, 10965, 11305, 11685, 12441, 12597, 13585, 14385, 14421, 14973, 15045, 15405, 15645, 15873, 16185, 16485, 17085, 17385

Offset: 1

Clark Kimberling, Jan 01 2014

			1365 = 3*5*7*13, and (1365+1)/2 = 683, a prime.

Cf. A234500, A234502, A234498, A234103.

t = Select[Range[1, 20000, 2], Map[Last, FactorInteger[#]] == Table[1, {4}] &]; Take[(t + 1)/2, 120] (* A234500*)
v = Flatten[Position[PrimeQ[(t + 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A234501 *)
(w + 1)/2 (* A234502 *)   (* Peter J. C. Moses, Dec 23 2013 *)

A333040 Even numbers m such that sigma(m) < sigma(m-1).

Original entry on oeis.org

46, 106, 118, 166, 226, 274, 298, 316, 346, 358, 406, 466, 514, 526, 562, 586, 622, 694, 706, 766, 778, 826, 838, 862, 886, 946, 1006, 1114, 1126, 1156, 1186, 1198, 1282, 1306, 1366, 1396, 1426, 1486, 1522, 1546, 1576, 1594, 1618, 1702, 1726, 1756

Offset: 1

Bernard Schott, Mar 22 2020

nonn

The even terms of A333039 represent about only 7% of the data, so they are proposed in this sequence. Some of these integers are semiprimes with for example these two families:
1) m = 2*p with p prime of the form k^2+k+3 is in A027753. The first few terms are: 46, 118, 226, 766, ... but not all the integers of this form are terms; the first 3 counterexamples are 6, 10, 1018 (see examples).
2) m = 2*p with p prime = (r*s*t+1)/2 and 2A234103. The first few terms are: 106, 166, 274, 346, 358, ... but not all the integers of this form are terms; the first 3 counterexamples are 386, 898 and 958 (see examples).
There is also this subsequence of even m = 2^2*p where p prime, congruent to 34 mod 45, is in A142330. The first few terms are: 316, 1396, 1756, 2416, ... but not all the integers of this form are terms; the first counterexample that comes from the 8th term of A142330 is 5716.
Even (and odd) numbers such that sigma(m) = sigma(m-1) are in A231546.

			166 = 2*83 and 165 = 3*5*11, as 252 = sigma(166) < sigma(165) = 288, hence 166 is a term.
386 = 2*193 and 385 = 5*7*11, as 582 = sigma(386) > sigma(385)= 576, hence 386 is not a term.
766 = 2*383 where 383 = 19^2+19+3 and 765 = 3^2*5*13, as 1152 = sigma(766) < sigma(765) = 1404, hence 766 is a term.
1018 = 2*509 where 509 = 22^2+22+3, and 1017 = 3^2*113, as 1530 = sigma(1018) > sigma(1017) = 1482, hence 1018 is not a term.

J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 620 pp. 82, 280, Ellipses Paris 2004.

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A005843 and A333039.
Subsequence of A333038.
Cf. A231546.

filter:= n -> numtheory:-sigma(n) < numtheory:-sigma(n-1):
select(filter, [seq(i,i=2..2000,2)]); # Robert Israel, Mar 29 2020

Select[2 * Range[1000], DivisorSigma[1, #] < DivisorSigma[1, #-1] &] (* Amiram Eldar, Mar 24 2020 *)

isok(m) = !(m%2) && (sigma(m) < sigma(m-1)); \\ Michel Marcus, Mar 22 2020

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A234102 Integers of the form (p*q*r + 1)/2, where p, q, r are distinct primes.

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A234500 Integers of the form (p*q*r*s + 1)/2, where p, q, r, s are distinct primes.

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A234104 Primes of the form (p*q*r + 1)/2, where p, q, r are distinct primes.

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A234501 Products p*q*r*s of distinct primes for which (p*q*r*s + 1)/2 is prime.

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A333040 Even numbers m such that sigma(m) < sigma(m-1).

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A234102 Integers of the form (pqr + 1)/2, where p, q, r are distinct primes.

A234500 Integers of the form (pqr*s + 1)/2, where p, q, r, s are distinct primes.

A234104 Primes of the form (pqr + 1)/2, where p, q, r are distinct primes.

A234501 Products pqrs of distinct primes for which (pqrs + 1)/2 is prime.