A340482 -id:A340482 - cp's OEIS frontend

A128283 Numbers of the form m = p1 * p2 where for each d|m we have (d+m/d)/2 prime and p1 < p2 both prime.

21, 33, 57, 85, 93, 133, 145, 177, 205, 213, 217, 253, 393, 445, 553, 565, 633, 697, 793, 817, 865, 913, 933, 973, 1137, 1285, 1345, 1417, 1437, 1465, 1477, 1513, 1537, 1717, 1765, 1837, 1857, 1893, 2101, 2173, 2245, 2305, 2517, 2577, 2581, 2605, 2641, 2653, 2733, 2761

Offset: 1

Kok Seng Chua (chuakokseng(AT)hotmail.com), Mar 05 2007

nonn

The symmetric representation of sigma (A237593) for p1*p2, SRS(p1*p2), consists of either 4 or 3 regions. Let p1 < p2. Then 2*p1 < p2 implies that SRS(p1*p2), consists of 2 pairs of regions of widths 1 having respective sizes (p1*p2 + 1)/2 and (p1 + p2)/2; and p2 < 2*p1 implies that SRS(p1*p2) consists of 2 outer regions of width 1 and size (p1*p2 + 1)/2 and a central region of maximum width 2 of size p1 + p2 . Therefore, if SRS(p1*p2) has four regions, the area of each is a prime number (see A233562) and if it has three regions, the central area is an even semiprime (A100484). - Hartmut F. W. Hoft, Jan 09 2021

Old name was: "a(n) is the n-th smallest product of two distinct odd primes m=p1*p2 with the property that (d+m/d)/2 are all primes for each d dividing m.". - David A. Corneth, Jan 09 2021

			85=5 * 17, (5 * 17+1)/2=43, (5+17)/2=11 are both primes and 85 is in the sequence.
From _Hartmut F. W. Hoft_, Jan 09 2021: (Start)
9=3*3 is not in the sequence even though (1+9)/2 and (3+3)/2 are primes, see also A340482.
a(33) = 1537 = 29*53 is the first number for which the symmetric representation of sigma consists of three regions ( 769, 82, 769 ) with 5 units of width 2 straddling the diagonal in the central region; (1537+1)/2 = 769 and (29+53)/2 = 41 are primes. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A128281, A005383, A128284, A128285, A128286.
Cf. A100484, A233562, A237591, A237593, A249223, A262045, A340482.
Subsequence of A046388.

ppQ[s_, k_] := Last[Transpose[FactorInteger[s]]]==Table[1, k]
dQ[s_] := Module[{d=Divisors[s]}, AllTrue[Map[(d[[#]]+d[[-#]])/2&, Range[Length[d]/2]], PrimeQ]]
goodL[{m_, n_}, k_] := Module[{i=m, list={}}, While[i<=n, If[ppQ[i, k] && dQ[i], AppendTo[list, i]]; i+=2]; list]/;OddQ[m]
a128283[n_] := goodL[{1, n}, 2]
a128283[2761] (* Hartmut F. W. Hoft, Jan 09 2021 *)

Added "distinct" for clarification since 9 satisfies the divisor property. See also A340482. - Hartmut F. W. Hoft, Jan 09 2021

New name from David A. Corneth, Jan 09 2021

A340483 a(n) is the smallest product of n (not necessarily distinct) odd primes m = p_1p_2...*p_n such that (d+m/d)/2 is a prime for all d dividing m.

Original entry on oeis.org

3, 9, 105, 1365, 884037

Offset: 1

Hartmut F. W. Hoft, Jan 09 2021

This sequence differs from A128281 only in the value A128281(2)=21: a(2)=9 since, for n=2 only, the specification "distinct primes" makes a difference.

			a(2) = 9 = 3*3 since (1+9)/2=5 and (3+3)/2=3.

Cf. A128281, A128283, A340482.

cp's OEIS Frontend

A128283 Numbers of the form m = p1 * p2 where for each d|m we have (d+m/d)/2 prime and p1 < p2 both prime.

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A340483 a(n) is the smallest product of n (not necessarily distinct) odd primes m = p_1p_2...*p_n such that (d+m/d)/2 is a prime for all d dividing m.

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cp's OEIS Frontend

A128283 Numbers of the form m = p1 * p2 where for each d|m we have (d+m/d)/2 prime and p1 < p2 both prime.

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Keywords

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Programs

Mathematica

Extensions

A340483 a(n) is the smallest product of n (not necessarily distinct) odd primes m = p_1*p_2*...*p_n such that (d+m/d)/2 is a prime for all d dividing m.

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A340483 a(n) is the smallest product of n (not necessarily distinct) odd primes m = p_1p_2...*p_n such that (d+m/d)/2 is a prime for all d dividing m.