A128283 -id:A128283 - cp's OEIS frontend

A128281 a(n) is the least product of n distinct odd primes m=p_1p_2...*p_n, such that (d+m/d)/2 are all primes for each d dividing m.

3, 21, 105, 1365, 884037

Offset: 1

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

From Iain Fox, Aug 26 2020: (Start)
a(6) > 10^9 if it exists.
All terms are members of A076274 since the definition requires that (1+m)/2 be prime.
The number of prime factors of m congruent to 3 (mod 4) must be even except for n=1.
(End)
a(6) > 2*10^11 if it exists. - David A. Corneth, Aug 27 2020
a(n) >= A070826(n+1) by definition of the sequence. - Iain Fox, Aug 28 2020

			105=3*5*7, (3*5*7+1)/2=53, (3+5*7)/2=19, (5+3*7)/2=13, (7+3*5)/2=11 are all primes and 105 is the least such number which is the product of 3 primes, so a(3)=3.

Subsequence of A076274.
Lower bound: A070826.
Cf. A002145, A128276, A128282, A128283, A128284, A128285, A128286, A168352.

a(n)=if(n==1, return(3)); my(p=prod(k=1, n, prime(k+1))); forstep(m=p+if(p%4-1, 2), +oo, 4, if(bigomega(m)==n && omega(m)==n, fordiv(m, d, if(!isprime((d+m/d)/2), next(2))); return(m))) \\ Iain Fox, Aug 27 2020

Definition corrected by Iain Fox, Aug 25 2020

A128284 Numbers of the form m = p1 * p2 * p3 where for each d|m we have (d+m/d)/2 prime and p1 < p2 < p3 each prime.

Original entry on oeis.org

105, 165, 273, 345, 357, 385, 777, 897, 1045, 1173, 1353, 1653, 1677, 1705, 2193, 2233, 2373, 2905, 3157, 3237, 3333, 3417, 3445, 3553, 3565, 3945, 4053, 4585, 4953, 5665, 5817, 6097, 6513, 6693, 7077, 7833, 8437, 8565, 8845, 10153, 11005, 11433

Offset: 1

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

nonn

The symmetric representation of sigma (cf. A237593), SRS(a(n)), of any number in this sequence has between 4 and 8 regions with 3 regions impossible because p1 < p2 < p3 implies 2*p3 < p1*p2. When there are 8 regions they all have width 1 and their areas are the prime numbers (d+a(n)/d)/2 for the 4 respective pairs of divisors of a(n). In general, the areas of the regions in SRS(a(n)) need not be prime, except for the two symmetric outer regions (n+1)/2. - Hartmut F. W. Hoft, Jan 09 2021

			165=3*5*11 and (3*5*11+1)/2=83, (3+5*7)/2=19, (5+3*7)/2=13, (7+3*5)/2=11 are all primes, so 165 is a term.
From _Hartmut F. W. Hoft_, Jan 09 2021: (Start)
a(1) = 105 = 3*5*7 and SRS(a(1)) consists of four regions with areas ( 53, 43, 43, 53 ); the center areas have maximum width 2 and represent the sum of primes (3+35)/2 + (5+21)/2 + (7+15)/2 = 43.
a(17) = 2373 = 3*7*17 is the first number in the sequence whose symmetric representation of sigma consists of 8 regions, all of width 1 and the respective symmetric regions have areas:  (2373 + 1)/2 = 1187, (791 + 3)/2 = 397, (339 + 7)/2 = 173, (21 + 113)/2 = 67. (End)

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

Cf. A128281, A005383, A128283, A128285, A128286.
Cf. A237048, A237591, A237593, A249223, A262045.
Subsequence of A046389.

(* function goodL[] is defined in A128283 *)
a128284[n_] := goodL[{1, n}, 3]
a128284[11433] (* Hartmut F. W. Hoft, Jan 09 2021 *)

A128285 Numbers of the form m = p1 * p2 * p3 * p4 where for each d|m we have (d+m/d)/2 prime and p1 < p2 < p3 < p4 each prime.

Original entry on oeis.org

1365, 4305, 10465, 11685, 15873, 27105, 31845, 35245, 50065, 54033, 58765, 74965, 84513, 91977, 95557, 95613, 96033, 104377, 113997, 114405, 117957, 118105, 126357, 127605, 136437, 170905, 197985, 209605, 215373, 226185, 248385, 277797

Offset: 1

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

nonn

			1365=3 * 5 * 7 * 13 and (3 * 5 * 7 * 13+1)/2, (3+5 * 7 * 13)/2, (5+3 * 7 * 13)/2, (7+3 * 5 * 13)/2, (13+3 * 5 * 7)/2, (3 * 5+7.13)/2, (3 * 7+5 * 13)/2, (3 * 13+5 * 7)/2 are all primes and 1365 is the smallest such integer which is the product of 4 primes, so 1365 is in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..3114

Cf. A128281, A005383, A128283, A128284, A128286.

Subsequence of A046390.

New name from David A. Corneth, Jan 09 2021

A233562 Products pq of distinct primes such that (pq + 1)/2 is a prime.

Original entry on oeis.org

21, 33, 57, 85, 93, 133, 141, 145, 177, 201, 205, 213, 217, 253, 301, 381, 393, 445, 453, 481, 501, 537, 553, 565, 633, 697, 717, 745, 793, 817, 865, 913, 921, 933, 973, 1041, 1081, 1137, 1141, 1261, 1285, 1293, 1317, 1345, 1401, 1417, 1437, 1465, 1477, 1501

Offset: 1

Clark Kimberling, Dec 14 2013

This sequence is a subsequence of A128283 since the condition that (p+q)/2 be prime is not required here. The smallest number not in A128283 is 141=3*47 since (3+47)/2=25. - Hartmut F. W. Hoft, Oct 31 2020

			21 = 3*7 is the least product of distinct primes p and q for which (p*q + 1)/2 is a prime, so a(1) = 21.

Cf. A233561, A046388.

Cf. A128283.

t = Select[Range[1, 7000, 2], Map[Last, FactorInteger[#]] == Table[1, {2}] &]; Take[(t + 1)/2, 120] (* A234096 *)
v = Flatten[Position[PrimeQ[(t + 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A233562 *)
(w + 1)/2 (* A234098 *)    (* Peter J. C. Moses, Dec 23 2013 *)
With[{nn=50},Take[Union[Select[Times@@@Subsets[Prime[Range[2nn]],{2}], PrimeQ[ (#+1)/2]&]],nn]] (* Harvey P. Dale, Mar 24 2015 *)

A128286 a(n) is the n-th smallest product of 5 odd primes m = p1p2p3p4p5 such that (d+m/d)/2 are all primes for each d dividing m.

Original entry on oeis.org

884037, 1137565, 2398377, 123156993, 681714273, 2347722213, 7283144845, 7794246057, 8953447917, 10287992785, 13749228493, 38108016453, 38901676405, 70918253385, 71809744693, 120418624965, 148282565865, 150721729873

Offset: 1

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

a(6) > 10^9.

			884037 = 3*7*11*43*89 and (1 + 884037)/2, (3 + 7*11*43*89)/2,
(7 + 3*11*43*89)/2, (11 + 3*7*43*89)/2, (43 + 3*7*11*89)/2, (89 + 3*7*11*43)/2,
(3*7 + 11*43*89)/2, (3*11 + 7*43*89)/2, (3*43 + 7*11*89)/2,(3*89 + 7*11*43)/2,
(7*11 + 3*43*89)/2, (7*43 + 3*7*89)/2, (7*89 + 3*7*43)/2, (11*43 + 3*7*89)/2,
(11*89 + 3*7*43)/2, (43*89 + 3*7*11)/2 are all primes and 884037 is the smallest such integer, so a(1) = 884037.

Cf. A128281, A005383, A128283, A128284, A128285.

a(6)-a(18) from Donovan Johnson, Oct 12 2008

A340482 Numbers that are the product of two not necessarily distinct odd primes pq with the property that (pq+1)/2 and (p+q)/2 are primes.

Original entry on oeis.org

9, 21, 25, 33, 57, 85, 93, 121, 133, 145, 177, 205, 213, 217, 253, 361, 393, 445, 553, 565, 633, 697, 793, 817, 841, 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

Offset: 1

Hartmut F. W. Hoft, Jan 09 2021

nonn

For the squares p^2 in this sequence the area of the central region of the three regions in the symmetric representation of sigma(p^2) is equal to p.

p^2 is a term iff p is in A048161, and this subsequence of p^2 is A263951. - Bernard Schott, Jan 10 2021

			a(1) = 9 = 3*3 is the first number for which SRS(a(1)) consists of three regions ( 5, 3, 5 ).
a(6) = 85 = 5*17, both (1+85)/2 = 43 and (5+17)/2 = 11 are primes, and SRS(a(6)) consists of the 4 regions ( 43, 11, 11, 43 ).

Union of A128283 and A263951.
Subsequence of A046315 (all odd semiprimes).
Cf. A048161, A070552, A128281, A237591, A237593, A249223, A262045, A340482.

dQ[s_] := Module[{d=Divisors[s]}, AllTrue[Map[(d[[#]]+d[[-#]])/2&, Range[Length[d]/2]], PrimeQ]]
a340482[n_] := Select[Range[n], PrimeOmega[#]==2&&dQ[#]&]
a340482[2700]

isok(m) = if ((m % 2) && (bigomega(m)==2), if (issquare(m), isprime((m+1)/2), my(p=factor(m)[1,1], q=factor(m)[2,1]); isprime((p*q+1)/2) && isprime((p+q)/2))); \\ Michel Marcus, Jan 10 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

A128281 a(n) is the least product of n distinct odd primes m=p_1*p_2*...*p_n, such that (d+m/d)/2 are all primes for each d dividing m.

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A128284 Numbers of the form m = p1 * p2 * p3 where for each d|m we have (d+m/d)/2 prime and p1 < p2 < p3 each prime.

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A128285 Numbers of the form m = p1 * p2 * p3 * p4 where for each d|m we have (d+m/d)/2 prime and p1 < p2 < p3 < p4 each prime.

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A233562 Products p*q of distinct primes such that (p*q + 1)/2 is a prime.

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A128286 a(n) is the n-th smallest product of 5 odd primes m = p1*p2*p3*p4*p5 such that (d+m/d)/2 are all primes for each d dividing m.

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A340482 Numbers that are the product of two not necessarily distinct odd primes p*q with the property that (p*q+1)/2 and (p+q)/2 are primes.

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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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A128281 a(n) is the least product of n distinct odd primes m=p_1p_2...*p_n, such that (d+m/d)/2 are all primes for each d dividing m.

A233562 Products pq of distinct primes such that (pq + 1)/2 is a prime.

A128286 a(n) is the n-th smallest product of 5 odd primes m = p1p2p3p4p5 such that (d+m/d)/2 are all primes for each d dividing m.

A340482 Numbers that are the product of two not necessarily distinct odd primes pq with the property that (pq+1)/2 and (p+q)/2 are primes.

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.