Abhimanyu Kumar - cp's OEIS frontend

User: Abhimanyu Kumar

Abhimanyu Kumar has authored 4 sequences.

A339270 a(n) is the largest m such that there is no prime except prime(n) from prime(n)-m+1 to prime(n)+m.

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

Offset: 1

Abhimanyu Kumar, Nov 29 2020

nonn

For a prime p, the degree of insulation is formally defined as D(p) = Max_{m=1..oo} U where the set U = {m: A000720(p+m) - A000720(p-m) = 1}.

This sequence is employed in defining insulated primes and highly insulated primes.

Robert Israel, Table of n, a(n) for n = 1..10000
Abhimanyu Kumar and Anuraag Saxena, Insulated primes, arXiv:2011.14210 [math.NT], 2020. See also Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 602-612.

Cf. A000040, A000720, A339148 (insulated primes), A339188 (highly insulated primes).

Related sequences: A046929.

f:= p -> min(nextprime(p)-p-1, p-prevprime(p)): f(2):= 0:
map(f@ithprime, [$1..100]); # Robert Israel, Dec 24 2020

{0}~Join~Array[Min[NextPrime[# + 1] - # - 1, # - NextPrime[# - 1, -1]] &@ Prime@ # &, 91, 2] (* Michael De Vlieger, Dec 11 2020 *)

D(p)={min(nextprime(p+1)-p-1, p-precprime(p-1))}
forprime(p=2, 1000, print1(D(p), ", "))

A339188 Highly insulated primes (see Comments for definition).

Original entry on oeis.org

23, 53, 89, 211, 293, 409, 479, 631, 797, 839, 919, 1039, 1259, 1409, 1471, 1511, 1637, 1709, 1847, 1889, 2039, 2099, 2179, 2503, 2579, 2633, 2777, 2819, 2939, 3011, 3049, 3137, 3229, 3271, 3433, 3499, 3593, 3659, 3709, 3779, 3967, 4111, 4177, 4253, 4327, 4409, 4493, 4621, 4703, 4831

Offset: 1

Abhimanyu Kumar, Nov 27 2020

nonn

Let degree of insulation D(p) for a prime p be defined as the largest m such that the prime between p-m and p+m is p only. Then the n-th insulated prime is said to be highly insulated if and only if D(A339148(n)) > D(A339148(n+1)) and D(A339148(n)) > D(A339148(n-1)).

			For the triplet (13,23,37) of insulated primes, the values of degree of insulation are D(13)=2, D(23)=4, and D(37)=3. Hence, 23 is the highly insulated prime.

François Marques, Table of n, a(n) for n = 1..10000
Abhimanyu Kumar and Anuraag Saxena, Insulated primes, arXiv:2011.14210 [math.NT], 2020. See also Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 602-612. See p. 610.

Cf. A000040, A339148 (insulated primes).

Block[{s = {0}~Join~Array[Min[NextPrime[# + 1] - # - 1, # - NextPrime[# - 1, -1]] &@ Prime@ # &, 660, 2], t}, t = Array[If[#1 < #2 > #3, #4, Nothing] & @@ Append[s[[# - 1 ;; # + 1]], #] &, Length@ s - 2, 2]; Array[If[s[[#1]] < s[[#2]] > s[[#3]], #4, Nothing] & @@ Append[t[[# - 1 ;; # + 1]], Prime@ t[[#]]] &, Length@ t - 2, 2] ] (* Michael De Vlieger, Dec 11 2020 *)

A339188(n) = { \\ Return the list of the first n highly insulated primes
  my( HighInsulated=List([]), D(p)=min(nextprime(p+1)-p-1, p-precprime(p-1)); );
  my( Dpred_ins=D(7), Pcur_ins=13, Dcur_ins=D(Pcur_ins) );
  local( Dpred=D(Pcur_ins), p=nextprime(Pcur_ins+1), Dp=D(p), Pnext=nextprime(p+1), Dnext=D(Pnext) );
  my(SearchNextInsulated() =
       until(Dp > max(Dpred,Dnext),
         Dpred = Dp; p = Pnext;  Dp = Dnext;
         Pnext = nextprime(p+1); Dnext = D(Pnext);
       );
     \\ At this point p is the first insulated prime > Dcur_ins
    );
  while(#HighInsulated max(Dpred_ins,Dp),
      Dpred_ins = Dcur_ins; Pcur_ins  = p; Dcur_ins  = Dp;
      SearchNextInsulated();
    );
    listput(HighInsulated,Pcur_ins);
  );
  return(HighInsulated);
} \\ François Marques, Dec 01 2020

A339148 Insulated primes (see Comments for definition).

Original entry on oeis.org

7, 13, 23, 37, 53, 67, 89, 103, 113, 131, 139, 157, 173, 181, 193, 211, 233, 277, 293, 337, 359, 389, 409, 421, 449, 479, 491, 509, 547, 577, 607, 631, 653, 691, 709, 751, 761, 797, 811, 823, 839, 863, 887, 919, 953, 983

Offset: 1

Abhimanyu Kumar, Nov 25 2020

nonn

Let the degree of insulation D(p) for a prime p be defined to be the largest m such that the only prime between p-m and p+m inclusive is p. Then the n-th prime is said to be insulated if and only if D(prime(n)) > D(prime(n+1)) and D(prime(n)) > D(prime(n-1)).

			For the prime triplet (19,23,29), we have D(19)=2, D(23)=4, and D(29)=1. Hence, 23 is an insulated prime.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Abhimanyu Kumar and Anuraag Saxena, Insulated primes, arXiv:2011.14210 [math.NT], 2020. See also Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 602-612.

Cf. A000040, A000720.

Select[Prime@ Range[2, 166], And[#2 > #1, #2 > #3] & @@ Map[Min[NextPrime[# + 1] - # - 1, # - NextPrime[# - 1, -1]] &, {NextPrime[# + 1], #, NextPrime[# - 1, -1]}] &] (* Michael De Vlieger, Mar 17 2021 *)

D(p)={min(nextprime(p+1)-p-1, p-precprime(p-1))}
ok(p)={my(d=D(p)); d>D(nextprime(p+1)) && d>D(precprime(p-1))}
forprime(p=3, 1000, if(ok(p), print1(p, ", "))) \\ Andrew Howroyd, Nov 25 2020

a(n) ~ 19.18*n^1.093 (heuristically accurate fit for n up to one million).

a(n) ~ c*n^(1+epsilon) (conjectured for some constant c,epsilon as n->oo).

A339095 Triangle read by rows: T(n,k) is the number of partitions of n with product of parts equal to k, 1 <= k <= A000792(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 0, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 2, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 0, 3, 0, 0, 1, 3, 0, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 0, 3, 0, 1, 1, 3, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 1

Offset: 1

Abhimanyu Kumar, Nov 23 2020

The product of the parts of a partition is called its norm.

			For n=6 the partitions and their counts for each norm are given in the table below.
  Relevant partition(s)  | Norm | Count
  1+1+1+1+1+1+1          | 1    | 1
  2+1+1+1+1              | 2    | 1
  3+1+1+1                | 3    | 1
  4+1+1, 2+2+1+1         | 4    | 2
  5+1                    | 5    | 1
  6, 3+2+1               | 6    | 2
  4+2, 2+2+2             | 8    | 2
  3+3                    | 9    | 1
The number of partitions of 6 with norm value 4 are 2, expressed as T(6,4)=2. Similarly, T(6,7)=0 because there is no partition of 6 with norm 7.
So the 6th row is 1, 1, 1, 2, 1, 2, 0, 2, 1.
First few rows of the array are:
  1;
  1, 1;
  1, 1, 1;
  1, 1, 1, 2;
  1, 1, 1, 2, 1, 1;
  1, 1, 1, 2, 1, 2, 0, 2, 1;
  1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 2;
  1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 0, 3, 0, 0, 1, 3, 0, 1;
  ...

Abhimanyu Kumar and Meenakshi Rana, On the treatment of partitions as factorization and further analysis, Journal of the Ramanujan Mathematical Society 35(3), 263-276 (2020).

Andrew V. Sills and Robert Schneider, The product of parts or "norm" of a partition, arXiv:1904.08004 [math.NT], 2019.

Cf. A000041 (row sums), A000792 (row lengths), A001055, A118851, A212721.

row(n) = {my(list = List()); forpart(p=n, listput(list, vecprod(Vec(p)));); my(vlist = Vec(list)); my(v = vector(vecmax(vlist))); for (i=1, #vlist, v[vlist[i]]++); v;} \\ Michel Marcus, Nov 26 2020

Let the number of partitions of n having the norm value k refer to the norm counting function T(n,k). The following properties hold true:
Max_{n=1..oo} T(n,k) = A001055(k).
Sum_{k=1..A000792(n)} T(n,k) = A000041(n).
Sum_{n>=1} T(n+1,k) - T(n,k) = A001055(k) - 1.
G.f.: Sum_{n>=0} Sum_{k>=1} ((q^n)/(k^s))*T(n,k) = Product_{m>=1}(1-((q^m)/(m^s)))^(-1).
n*Sum_{k=1..A000792(n)} T(n,k) = Sum_{m=1..n} (Sum_{k=1..A000792(n-m)} T(n-m,i)*k^(-s))*(Sum_{d|m} (d/m)^(d*s-1))

More terms from Michel Marcus, Nov 26 2020

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User: Abhimanyu Kumar

A339270 a(n) is the largest m such that there is no prime except prime(n) from prime(n)-m+1 to prime(n)+m.

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A339188 Highly insulated primes (see Comments for definition).

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A339148 Insulated primes (see Comments for definition).

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A339095 Triangle read by rows: T(n,k) is the number of partitions of n with product of parts equal to k, 1 <= k <= A000792(n).

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