Harry E. Neel - cp's OEIS frontend

User: Harry E. Neel

Harry E. Neel has authored 5 sequences.

A348394 Primes preceding record runs of composites coprime to 30 (A007775).

7, 47, 113, 317, 523, 1327, 9551, 15683, 19609, 25471, 31397, 155921, 360653, 370261, 1349533, 1357201, 2010733, 4652353, 17051707, 20831323, 47326693, 122164747, 189695659, 191912783, 387096133, 1294268491, 1453168141, 2300942549, 3842610773, 4302407359, 10726904659, 20678048297, 22367084959, 25056082087, 42652618343, 127976334671, 182226896239

Offset: 1

Harry E. Neel, Oct 16 2021

nonn

"There are 8 potential primes modulo 30...." Using only the potential prime locations within this domain there are no consecutive integers within the domain until an integer is determined to have a prime factor, here the first such integer is 49. When an integer is determined to be composite then there is a "gap" within the succession of primes.
While the location of the first few record consecutive integers differ from established maximal gaps, they quickly become the same. It is not known if they continue to remain the same or if some variation may occur. Here the record number of composites will always be lower because the count of composites are only those that are within this domain.
Hugo van der Sanden greatly expanded the data contained in this sequence.

			The next number coprime to 30 after 7 is 11, giving a run of 0 composites.
47 is followed by 49 = 7^2 and 53 (prime), a run of 1 composite.
113 is followed by 119 = 7*17, 121 = 11^2, and 127 (prime), a run of 2 composites.
The first few entries correspond to the following table. The table contains the order in which record composites occur (n), the number of composites between successive primes (gap size), the prime preceding the record composites (1st prime), the prime following the record composites (2nd prime) and the merit of the gap (merit) rounded to 4 decimals. The merit is the gap size divided by the natural log of the 1st prime (gap size / log(1st prime)).
   n  gap size 1st prime  2nd prime   gap merit
   1,    0,        7,          11,      0.0000
   2,    1,       47,          53,      0.2597
   3,    2,      113,         127,      0.4231
   4,    3,      317,         331,      0.5209
   5,    4,      523,         541,      0.6390
   6,    8,     1327,        1361,      1.1126
   7,    9,     9551,        9587,      0.9821
   8,   10,    15683,       15727,      1.0352
   9,   12,    19609,       19661,      1.2141
  10,   13,    25471,       25523,      1.2814
  11,   18,    31397,       31469,      1.7384
  12,   22,   155921,      156007,      1.8399
  13,   24,   360653,      360749,      1.8756
  ...
  38,  125, 182226896239, 182226896713,  4.8209

C. K. Caldwell, Table of Known Maximal Gaps
P. H. Fry, J. Nesheivat and B. K. Szymanski, Computing Twin Primes and Brun's Constant: A Distributed Approach, IEEE Computer Society Press, 1998, pages 42-49.

Cf. A002386, A007775, A140378, A000040, A001223, A038510.

Block[{m = Select[Range[29], CoprimeQ[#, 30] &], s, t}, s = Reap[Array[Map[If[! PrimeQ[#], Sow[#]] &, 30 # + m] &, 2^20]][[-1, -1]]; Set[{s, t}, Transpose@ #] &@ Tally@ Array[NextPrime[s[[#]], -1] &, Length@ s]; Map[s[[FirstPosition[t, #][[1]]]] &, Union@ FoldList[Max, t]] ] (* Michael De Vlieger, Oct 25 2021 *)

isok(x) = vecsearch([1, 7, 11, 13, 17, 19, 23, 29], x%30);
nbc(n, v) = {my(i=n+1, c= v[i], nb=0); while(!isprime(c), nb++; i++; if (i>#v, return(-1)); c = v[i]); nb;}
lista(nn) = {my(v = [2..nn], m=-1, nb); v = select(x->isok(x), v); v = apply(isprime, v); for (n=1, #v-1, if (isprime(v[n]), nb = nbc(n, v); if (nb==-1, break); if (nb > m, print1(v[n], ", "); m = nb);););} \\ Michel Marcus, Oct 21 2021

A332656 Number of decompositions of 2n into unordered sums of two odd primes, including at least one twin prime.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 4, 4, 2, 3, 4, 3, 3, 5, 4, 3, 5, 3, 4, 5, 3, 5, 6, 2, 4, 6, 4, 4, 7, 4, 5, 7, 5, 4, 7, 4, 4, 7, 3, 5, 7, 4, 4, 8, 6, 6, 9, 5, 6, 9, 4, 5, 8, 3, 6, 8, 4, 2, 8, 7, 7, 10, 5, 5, 8, 4, 7, 10, 4, 7, 9, 3, 4, 11, 9, 5

Offset: 1

Harry E. Neel, Feb 18 2020

nonn

a(n) is the number of ways that a twin prime may be summed with another prime to provide the even number 2n. It is not known if summing primes with twin primes will provide every even number greater than 4 (or greater than or equal to 6).

			a(6) = 1 because the only way to express 2*6 = 12 as the sum of two primes, one of which is a twin prime, is 5+7. (Since the sequence counts unordered sums, 7+5 is not counted as distinct from 5+7.)
Also, 37+61 = 98 is a valid sum, 61 being a part of a twin prime pair; while 37+47 = 84 is not a valid sum because neither 37 nor 47 is a part of a twin prime pair.

Cf. A000040, A001097, A002375, A129363.

istwin(p) = isprime(p-2) || isprime(p+2);
a(n) = {n *= 2; my(nb = 0, q, v=[]); forprime(p=2, n, q = n-p; if ((q>=p) && isprime(q) && (istwin(p) || istwin(q)), nb++; v= concat(v, p));); nb;} \\ Michel Marcus, Feb 28 2020

A330761 Array read by antidiagonals: T(n,k) is the number of faces on a ring formed by connecting the ends of a prismatic rod whose cross-section is an n-sided regular polygon after applying a twist of k/n turns.

Original entry on oeis.org

2, 3, 1, 4, 1, 2, 5, 1, 1, 1, 6, 1, 2, 3, 2, 7, 1, 1, 1, 1, 1, 8, 1, 2, 1, 4, 1, 2, 9, 1, 1, 3, 1, 1, 3, 1, 10, 1, 2, 1, 2, 5, 2, 1, 2, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 14, 1, 2, 1, 2, 1, 2, 7, 2, 1, 2, 1, 2

Offset: 1

Harry E. Neel, Dec 29 2019

This sequence begins with two (2). A prismatic rod having two (biconvex) sides is the simplest three-dimensional construct for consideration here. Such a rod when twisted can only have one side and edge or two sides and edges. (Because the numbers of sides and edges are always equal, generally only sides or faces will be referenced.) The number of faces (surfaces) on rings generated by twisting (or not) prismatic rods is the greatest common divisor (GCD) of the number of sides of the rod, n, and the amount of twist, k, applied to the rod before forming a ring. That is that number of faces is equal to gcd(n,k). Because of the relationship of the number of sides of a prismatic rod and the variable of twist that may be applied, all prismatic rods that have a polygonal cross-section that is prime in number (2, 3, 5, 7, 11, etc.) and formed into rings will always have only 1 face (surface) if twisted by any amount that is not a multiple of the prime. E.g., a prismatic rod with a pentagonal cross-section will have 5 faces if left untwisted or if twisted 5/5, 10/5, etc. Any other amount of rotation will always produce 1 face.

The direction of rotation does not matter. This is an infinite sequence in the sense that primes are infinite (not to mention composites). However, it is noted that actual constructs have limitations.

			A prismatic rod having a cross-section that is an octagon will have:
8 faces if no twist is applied or if the amount of twisting is a multiple of 8: 0/8, 8/8, 16/8, etc.;
4 faces if the amount of twisting is 4/8, 12/8, etc.
2 faces if the amount of twisting is 2/8, 6/8, 10/8, etc.
1 face if the amount of twisting is 1/8, 3/8, 5/8, 7/8, 9/8, etc.
Note that the number of faces is equal to gcd(n,k) where n=number of sides of the prismatic rod and k=amount of twist applied to the rod.
T(n,k) as a table begins:
(n=number of sides of polygon; k=amount of fractional twist applied)
   n |k=0  1  2  3  4  5  6  7  8  9 10 11 12 13 ...
  ---+-------------------------------------------
   2 |  2  1  2  1  2  1  2  1  2  1  2  1  2  1...
   3 |  3  1  1  3  1  1  3  1  1  3  1  1  3  1...
   4 |  4  1  2  1  4  1  2  1  4  1  2  1  4  1...
   5 |  5  1  1  1  1  5  1  1  1  1  5  1  1  1...
   6 |  6  1  2  3  2  1  6  1  2  3  2  1  6  1...
   7 |  7  1  1  1  1  1  1  7  1  1  1  1  1  1...
   8 |  8  1  2  1  4  1  2  1  8  1  2  1  4  1...
   9 |  9  1  1  3  1  1  3  1  1  9  1  1  3  1...
  10 | 10  1  2  1  2  5  2  1  2  1 10  1  2  1...
  11 | 11  1  1  1  1  1  1  1  1  1  1 11  1  1...
  12 | 12  1  2  3  4  1  6  1  4  3  2  1 12  1...
  13 | 13  1  1  1  1  1  1  1  1  1  1  1  1 13...
  ...

Martin Gardner, Mathematical Games, A Möbius band has a finite thickness, and so it is actually a twisted prism, Scientific American, August, 1978, 18-25.

Subtable of A109004.

T(n, k) = gcd(n, k).

A330670 Squares of primes congruent to 1 (mod 30).

Original entry on oeis.org

121, 361, 841, 961, 1681, 3481, 3721, 5041, 6241, 7921, 10201, 11881, 17161, 19321, 22201, 22801, 32041, 32761, 36841, 39601, 44521, 52441, 57121, 58081, 63001, 72361, 73441, 78961, 96721, 109561, 121801, 128881, 143641, 151321, 166801, 167281, 175561

Offset: 1

Harry E. Neel, Dec 24 2019

nonn

Sequence lists squares of prime numbers that end in 1 or 9.

			Prime 11*11=121 and 19*19=361; 121 and 361 are terms of this sequence.
Prime 13*13=169 and 17*17=289; 169 and 289 are not terms of this sequence.

Intersection of A001248 and A128470.

Select[Range[360], PrimeQ[#] && Mod[#^2, 30] == 1 &]^2 (* Amiram Eldar, Dec 28 2019 *)

isok(m) = issquare(m) && isprime(sqrtint(m)) && ((m % 30) == 1); \\ Michel Marcus, Dec 26 2019

A329262 Prime pairs of the form (30k - 7, 30k + 7).

Original entry on oeis.org

23, 37, 53, 67, 83, 97, 113, 127, 263, 277, 293, 307, 353, 367, 383, 397, 443, 457, 563, 577, 593, 607, 743, 757, 773, 787, 863, 877, 953, 967, 983, 997, 1103, 1117, 1223, 1237, 1283, 1297, 1433, 1447, 1553, 1567, 1583, 1597, 1613, 1627

Offset: 1

Harry E. Neel, Nov 09 2019

The terms of this sequence are created by pairing the terms of the primes when the terms 30k - 7 and 30k + 7 are both prime. As has been pointed out, it is only a guess as to whether, or not, that (like the twin primes, etc.) there is or is not an infinite number of these pairings.

			As 4 * 30 - 7 = 113 and 4 * 30 + 7 = 127 are prime, both 113 and 127 are in the sequence. - _David A. Corneth_, Nov 10 2019

Cf. A140446, A153417, A007510.

Odd- (resp. even-) indexed terms are a subsequence of A132235 (resp. A132231).

&cat[[30*k-7] cat [30*k+7]:k in [1..60]|IsPrime(30*k-7) and IsPrime(30*k+7)]; // Marius A. Burtea, Nov 17 2019

Select[Prime[Range[1000]], MemberQ[{7, 23}, Mod[#, 30]] &] (* Jinyuan Wang, Nov 16 2019 *)
Flatten[Select[Table[30n + {-7, 7}, {n, 90}], PrimeQ[#[[1]]] && PrimeQ[#[[2]]] &]] (* Alonso del Arte, Dec 07 2019 *)

first(n) = n+=(n%2); my(res=List(),todo=n); for(i=1,oo, if(isprime(30*i-7) && isprime(30*i+7), listput(res,30*i-7); listput(res,30*i+7); todo-=2; if(todo<=0, return(res)))) \\ David A. Corneth, Nov 10 2019

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User: Harry E. Neel

A348394 Primes preceding record runs of composites coprime to 30 (A007775).

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A332656 Number of decompositions of 2n into unordered sums of two odd primes, including at least one twin prime.

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A330761 Array read by antidiagonals: T(n,k) is the number of faces on a ring formed by connecting the ends of a prismatic rod whose cross-section is an n-sided regular polygon after applying a twist of k/n turns.

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A330670 Squares of primes congruent to 1 (mod 30).

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A329262 Prime pairs of the form (30k - 7, 30k + 7).

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