A074173 -id:A074173 - cp's OEIS frontend

A074172 Smaller of two consecutive numbers of the form p^2*q where p and q are distinct primes.

44, 75, 98, 116, 147, 171, 244, 332, 387, 507, 548, 603, 604, 724, 844, 908, 931, 963, 1075, 1083, 1251, 1324, 1412, 1467, 1556, 1587, 1675, 1772, 2523, 2524, 2636, 2644, 2763, 3283, 3356, 3411, 3508, 3788, 3987, 4075, 4203, 4204, 4418, 4491, 4804, 4868

Offset: 1

Amarnath Murthy, Aug 30 2002

nonn

From Robert Israel, Dec 06 2018: (Start)
There are four forms of terms, for odd primes p,q,r:
4*p where 4*p+1 = q^2*r, r == 1 (mod 4)
2*p^2 where 2*p^2+1 = q^2*r, r == 3 (mod 4)
p^2*q where p^2*q+1 = 2*r^2, q == 1 (mod 4)
p^2*q where p^2*q+1 = 4*r, q == 3 (mod 4).
Are there infinitely many terms of each type?
(End)

			44 is a member as 44 = 2^2*11 and 45 = 3^2*5.

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

Cf. A054753, A074173, A074174, A178032, A308683, A141621.

filter:= proc(n) local F;
F:= map(t -> t[2],ifactors(n)[2]);
F = [2,1] or F = [1,2]
end proc:
A054753:= select(filter, {$1..10000}):
sort(convert(A054753 intersect map(`-`,A054753,1),list)); # Robert Israel, Dec 06 2018

lst={}; Do[f1=FactorInteger[n]; If[Sort[Transpose[f1][[2]]]=={1, 2}, f2=FactorInteger[n+1]; If[Sort[Transpose[f2][[2]]]=={1, 2}, AppendTo[lst, n]]], {n, 3, 10000}]; lst

isok1(n) = vecsort(factor(n)[,2]) == [1,2]~;
isok(n) = isok1(n) && isok1(n+1); \\ Michel Marcus, Sep 20 2017

More terms from T. D. Noe, Oct 04 2004

Name clarified by Sean A. Irvine, Jan 13 2025

A074174 Smallest number k such that k and k+n are of the form p^2*q where p and q are primes.

Original entry on oeis.org

44, 18, 172, 171, 45, 12, 45, 12, 236, 18, 52, 63, 50, 261, 524, 12, 28, 45, 44, 637, 404, 28, 45, 20, 20, 18, 18, 147, 63, 20, 44, 12, 12, 18, 28, 63, 116, 12, 236, 12, 75, 50, 20, 325, 18, 52, 28, 20, 50, 18, 12, 423, 45, 44, 20, 12, 18, 18, 116, 147, 63, 325, 12, 12, 52

Offset: 1

Amarnath Murthy, Aug 30 2002

nonn

			a(2) = 18 as 18 = 3^2*2 and 18 +2 =20 = 2^2*5.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A074172, A074173, A054753.

Table[k=1; found=False; While[ !found, k++; f1=FactorInteger[k]; If[Sort[Transpose[f1][[2]]]=={1, 2}, f2=FactorInteger[k+n]; If[Sort[Transpose[f2][[2]]]=={1, 2}, found=True]]]; k, {n, 100}]
snk[n_]:=Module[{k=1},While[Sort[FactorInteger[k][[All,2]]]!={1,2} || Sort[FactorInteger[k+n][[All,2]]]!={1,2},k++];k]; Array[snk,70]

Corrected and extended by T. D. Noe, Oct 04 2004

A308735 Numbers k such that k, k+2, k+4 are of the form p^2*q where p and q are distinct primes.

Original entry on oeis.org

2523, 2525, 3175, 22021, 25529, 28223, 40325, 53573, 58923, 73447, 122571, 132021, 149675, 152339, 165175, 172917, 202221, 209673, 235825, 267773, 268223, 308671, 322223, 371075, 425723, 430171, 445923, 488975, 575973, 591575

Offset: 1

Ray Chandler, Jun 24 2019

nonn

All terms are odd. See comment in A308736. - Chai Wah Wu, Jun 24 2019

			3175 =  5 *  5 * 127,
3177 =  3 *  3 * 353,
3179 = 11 * 17 *  17.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A074173, A308736.

psx = Table[{0}, {5}]; nmax = 600000; n = 1; lst = {};
While[n < nmax, n++;
  psx = RotateRight[psx];
  psx[[1]] = Sort[Last /@ FactorInteger[n]];
  If[Union[{psx[[1]], psx[[3]], psx[[5]]}] == {{1, 2}},
   AppendTo[lst, n - 4]];];
lst

A308736 Numbers n such that n, n+2, n+4, n+6 are of the form p^2*q where p and q are distinct primes.

Original entry on oeis.org

2523, 3112819, 5656019, 10132171, 12167825, 16639567, 25302173, 31995475, 35158921, 37334419, 43890719, 44816821, 47715269, 53548223, 55534523, 90526075, 90533525, 127558319, 142929025, 143167073, 144989575, 147182225

Offset: 1

Ray Chandler, Jun 24 2019

nonn

All terms are odd. Proof: if n is even then out of the 4 numbers n, n+2, n+4, n+6, 2 of them must be either both of the form 2*p^2, 2*q^2, or both of the form 4*p, 4*q. In either case, for p != q and p, q prime, the difference between these 2 numbers are more than 6, reaching a contradiction. - Chai Wah Wu, Jun 24 2019

			2523 = 3*29*29, 2525 = 5*5*101, 2527 = 7*19*19, 2529 = 3*3*281.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 123 terms from Ray Chandler)

Cf. A074173, A308735.

psx = Table[{0}, {7}]; nmax = 150000000; n = 1; lst = {};
While[n < nmax, n++;
  psx = RotateRight[psx];
  psx[[1]] = Sort[Last /@ FactorInteger[n]];
  If[Union[{psx[[1]], psx[[3]], psx[[5]], psx[[7]]}] == {{1, 2}}, AppendTo[lst, n - 6]];];
lst

from sympy import factorint
A308736_list, n, mlist = [], 3, [False]*4
while len(A308736_list) < 100:
    if mlist[0] and mlist[1] and mlist[2] and mlist[3]:
        A308736_list.append(n)
    n += 2
    f = factorint(n+6)
    mlist = mlist[1:] + [(len(f),sum(f.values())) == (2,3)] # Chai Wah Wu, Jun 24 2019, Jan 03 2022.

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A074172 Smaller of two consecutive numbers of the form p^2*q where p and q are distinct primes.

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A074174 Smallest number k such that k and k+n are of the form p^2*q where p and q are primes.

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A308735 Numbers k such that k, k+2, k+4 are of the form p^2*q where p and q are distinct primes.

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A308736 Numbers n such that n, n+2, n+4, n+6 are of the form p^2*q where p and q are distinct primes.

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