A070552 -id:A070552 - cp's OEIS frontend

A361262 Numbers k such that k+i^2, i=0..6 are all semiprimes.

3238, 4162, 4537, 13918, 16837, 17857, 18673, 24553, 55477, 62353, 78457, 84358, 92878, 102838, 106813, 129838, 135853, 140002, 142822, 146722, 148318, 151957, 166177, 180013, 184213, 187933, 194338, 210637, 214393, 231757, 242698, 271198, 274393, 305677

Offset: 1

Alexandru Petrescu, Mar 06 2023

nonn

			3238 is a term because 3238=2*1619; 3239=41*79; 3242=2*1621; 3247=17*191; 3254=2*1627; 3263=13*251; 3274=2*1637.

Subsequence of A070552.

Cf. A001358 (semiprimes).

q:= n-> andmap(x-> numtheory[bigomega](x)=2, [n+i^2$i=0..6]):
select(q, [$1..400000])[];  # Alois P. Heinz, Mar 06 2023

okQ[k_] := AllTrue[Table[k+i^2, {i, 0, 6}], PrimeOmega[#] == 2&];
Select[Range[400000], okQ] (* Jean-François Alcover, Feb 02 2025 *)

isok(k) = sum(i=0, 6, bigomega(k+i^2)==2) == 7;

A367075 a(n) is the least semiprime that is the first of n consecutive semiprimes s(1) ... s(n) such that s(i) - prime(i) are all equal.

Original entry on oeis.org

4, 9, 118, 514, 1202, 9662, 46418, 198878, 273386, 717818, 717818, 270893786, 1009201118, 1009201118, 68668578806, 421210555538, 421210555538, 81550619289662, 645040014922382, 645040014922382, 645040014922382

Offset: 1

Robert Israel, Nov 05 2023

			a(3) = 118 because 118, 119, 121 are consecutive semiprimes with 118 - 2 = 119 - 3 = 121 - 5 = 116, and this is the first semiprime that works.

All terms are in A001358 and (except for the initial term 4) A070552.

P:= select(isprime, [2,seq(i,i=3..10^6,2)]):
SP:= select(t -> numtheory:-bigomega(t)=2, [$4..10^7]):
nSP:= nops(SP);
t:= 1: k0:= 1: R:= 4: tmax:= 1: d:= 2:
for k from 2 to nSP do
  if SP[k]-P[k-k0+1] = d then
    t:= t+1;
    if t > tmax then R:= R, SP[k0]; tmax:= t; fi;
  else
    t:= 1; k0:= k; d:= SP[k] - 2;
  fi
od:
R;

a(12) from David A. Corneth, Nov 05 2023
a(13)-a(15) from Daniel Suteu, Nov 18 2023
a(16)-a(18) from Martin Ehrenstein, Dec 01 2023
a(19)-a(21) from Martin Ehrenstein, Dec 03 2023

cp's OEIS Frontend

A361262 Numbers k such that k+i^2, i=0..6 are all semiprimes.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

PARI