cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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.

Original entry on oeis.org

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

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Author

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

Keywords

Comments

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

Examples

			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)

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

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.

Original entry on oeis.org

3, 21, 105, 1365, 884037
Offset: 1

Views

Author

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

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Programs

  • PARI
    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

Extensions

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

Views

Author

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

Keywords

Comments

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

Examples

			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)

Links

Crossrefs

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

Programs

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

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.

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

Views

Author

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

Keywords

Comments

a(6) > 10^9.

Examples

			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.

Crossrefs

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

Extensions

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