James D. Klein - cp's OEIS frontend

A331560 Size of the palindrome-free intervals about the 196-iterates, A006960.

11, 10, 110, 110, 100, 100, 110, 1100, 1000, 11000, 11000, 11000, 11000, 10000, 10000, 10000, 11000, 110000, 110000, 100000, 100000, 1100000, 1100000, 1100000, 1000000, 1000000, 11000000, 11000000, 10000000, 10000000, 110000000, 110000000, 110000000, 100000000

Offset: 1

James D. Klein, Jan 20 2020

By empirical observation, the integers in this sequence are of the form 10*10^n and 11*10^n, n >= 0, since they are the difference of consecutive palindromes surrounding the 196-iterates. (No differences of 2 observed.)

			191 < 196 < 202, 202 - 191 = 11;
878 < 887 < 888, 888 - 878 = 10; etc.

Cf. A006960, A331556, A331557.

# Palindrome-free interval about 196 offsets. Slow brute-force
n = 196
while n < 10**15:
     m = n
     while m != int(str(m)[::-1]): m+=1
     s = m
     m = n
     while m != int(str(m)[::-1]): m-=1
     print(s-m)
     n = n + int(str(n)[::-1])

a(n) = A331556(n) + A331557(n).

a(31)-a(34) from Jinyuan Wang, Feb 29 2020

A331557 The upper (or right) offset of a 196-iterate (A006960) from the smallest palindrome greater than the iterate.

Original entry on oeis.org

6, 1, 96, 11, 48, 11, 10, 693, 732, 231, 110, 10901, 10901, 5600, 1100, 110, 1000, 12375, 108911, 96416, 99901, 470118, 110, 1089011, 999074, 110000, 2508495, 109901, 1770356, 11, 40076938, 99110000, 10901000, 56662095, 9911, 137056546, 1099890110, 545350309

Offset: 1

James D. Klein, Jan 20 2020

When normalized over (0,1) by their respective palindrome-free interval about a 196-iterate, it has been empirically observed that the frequency distribution of this sequence appears to be quite symmetric about 0.5, as well as fractal when plotting the distribution over decreasing bin sizes.

The 196-iterates referred to here come from the reverse-and-add process generating A006960.

			The first term is 6 since 202-196 = 6;
The second term is 1 since 888-887 = 1; etc.

Cf. A006960, A331556, A006960.

# Upper 196 offsets. Slow brute force
n = 196
while n < 10**15:
  m = n
  while m != int(str(m)[::-1]): m+=1
  print(m-n)
  n = n + int(str(n)[::-1])

a(n) = A331560(n) - A331556(n).

More terms from Jinyuan Wang, Feb 29 2020

A331556 The lower (or left) offset of a 196-iterate (A006960) from the largest palindrome less than the iterate.

Original entry on oeis.org

5, 9, 14, 99, 52, 89, 100, 407, 268, 10769, 10890, 99, 99, 4400, 8900, 9890, 10000, 97625, 1089, 3584, 99, 629882, 1099890, 10989, 926, 890000, 8491505, 10890099, 8229644, 9999989, 69923062, 10890000, 99099000, 43337905, 99990089, 962943454, 109890, 454649691

Offset: 1

James D. Klein, Jan 20 2020

When normalized over (0,1) by their respective palindrome-free interval about a 196-iterate, it has been empirically observed that the frequency distribution of this sequence appears to be quite symmetric about 0.5, as well as fractal when plotting the distribution over decreasing bin sizes.

The 196-iterates referred to here come from the reverse-and-add process generating A006960.

			The first term is 5 since 196-191 = 5
The second term is 9 since 887-878 = 9, etc.

Cf. A006960, A331557, A331560.

Map[Block[{k = # - 1}, While[k != IntegerReverse@ k, k--]; # - k] &, NestList[# + IntegerReverse[#] &, 196, 25]] (* brute force, or *)
Map[# - Block[{n = #, w, len, ww}, w = IntegerDigits[n]; len = Length@ w; ww = Take[w, Ceiling[len/2] ]; If[# < n, #, FromDigits@ Flatten@{#, If[OddQ@ len, Reverse@ Most@ #, Reverse@ #]} &@ If[Last@ ww == 0, MapAt[# - 1 &, Most@ ww, -1]~Join~{9}, MapAt[# - 1 &, ww, -1]]] &@ FromDigits@ Flatten@ {ww, If[OddQ@ len, Reverse@ Most@ ww, Reverse@ ww]}] &, NestList[# + IntegerReverse[#] &, 196, 37]] (* Michael De Vlieger, Jan 22 2020 *)

# Slow Brute-force
n = 196
while n < 10**15:
  m = n
  while m != int(str(m)[::-1]): m+=-1
  print(n-m, end=', ')
  n = n + int(str(n)[::-1])

a(n) = A331560(n) - A331557(n).

More terms from Michael De Vlieger, Jan 22 2020

A205601 Goldbach's problem extended to division: number of decompositions of 2n into the floor of unordered ratios of two primes, floor(q/p) = 2n, where p < 2n < q.

Original entry on oeis.org

0, 1, 3, 5, 4, 5, 10, 5, 10, 16, 12, 17, 18, 16, 19, 27, 23, 22, 34, 27, 34, 39, 39, 45, 51, 41, 50, 51, 44, 57, 68, 71, 63, 74, 63, 76, 87, 84, 89, 104, 94, 108, 111, 99, 117, 116, 120, 104, 126, 114, 133, 146, 149, 146, 166, 148, 190, 178, 182, 170, 179, 173

Offset: 1

James D. Klein, Jan 29 2012

nonn

			For n = 3, a(n) = 3 because 6 is the floor of 13/2, 19/3, and 31/5. - _T. D. Noe_, Jan 31 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A002375, A202472.

Table[Length[Flatten[Table[Select[2*n*p + Range[p - 1], PrimeQ], {p, Prime[Range[PrimePi[2*n - 1]]]}]]], {n, 62}] (* T. D. Noe, Jan 31 2012 *)

a(n)=n*=2;my(s,t);forprime(p=2,n-1,t=n*p;while(n==(t=nextprime(t+1))\p,s++));s \\ Charles R Greathouse IV, Jan 30 2012

a(21)-a(62) from Charles R Greathouse IV, Jan 31 2012

A202472 Goldbach's Problem extended to subtraction: number of decompositions of 2n into unordered differences of two primes, p, q, where p < 2n < q.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 2, 3, 3, 3, 2, 6, 4, 3, 6, 3, 4, 6, 4, 5, 8, 4, 4, 7, 6, 4, 9, 8, 4, 11, 5, 5, 11, 6, 8, 9, 4, 7, 11, 7, 4, 13, 7, 5, 15, 7, 8, 13, 8, 9, 11, 7, 7, 13, 10, 5, 13, 7, 7, 19, 9, 8, 17, 9, 10, 16, 9, 9, 15, 12, 7, 19, 9, 7, 19, 9, 12, 17, 8, 14

Offset: 1

James D. Klein, Dec 19 2011

nonn

T. D. Noe, Table of n, a(n) for n = 1..10000

Extension of A002375.

Bisection of A092953.

Table[Length[Select[Prime[Range[PrimePi[2*n]]], PrimeQ[2*n + #] &]], {n, 100}] (* T. D. Noe, Apr 16 2013 *)

a(n)=my(s);forprime(p=2,2*n,s+=isprime(2*n+p));s \\ Charles R Greathouse IV, Dec 19 2011
(C++)
#include 
using namespace std;
int main()
{ int p[25] = {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97};
  int count, istart = 2;
  for(int n=1; n<=25; n++)
  {
      if(2*n>p[istart]) istart++;
      count = 0;
      for(int j=1; p[j]<2*n; j++)
        for(int i=istart; p[i]-p[j]<=2*n; i++)
          if(p[i]-p[j]==2*n) count++;
      cout << n << ". " << count << endl;
  }
    return 0;
} // code for the first 25 integers, James D. Klein, Dec 21 2011

a(n) = A092953(2*n). - Bill McEachen, May 24 2024

A004140 Number of nonempty labeled simple graphs on nodes chosen from an n-set.

Original entry on oeis.org

0, 1, 4, 17, 112, 1449, 40068, 2350601, 286192512, 71213783665, 35883905263780, 36419649682706465, 74221659280476136240, 303193505953871645562969, 2480118046704094643352358500, 40601989176407026666590990422105, 1329877330167226219547875498464516480

Offset: 0

N. J. A. Sloane, Colin Mallows, James D. Klein

We are given n labeled points, we choose k (1 <= k <= n) of them and construct a simple (but not necessarily connected) graph on these k nodes in 2^C(k,2) ways.

a(n) is the number of (non-null) subgraphs of the complete graph with n vertices. - Maharshee K. Shah, Sep 08 2024

			n=2: there are 4 graphs: {o}, {o}, {o o}, {o-o}
......................... 1 .. 2 .. 1 2 .. 1 2

Alois P. Heinz, Table of n, a(n) for n = 0..50

Cf. A006896.

a:= n-> add (binomial(n, k)*2^(k*(k-1)/2), k=1..n):
seq (a(n), n=0..20);  # Alois P. Heinz, Oct 09 2012

nn=20;s=Sum[2^Binomial[n,2]x^n/n!,{n,0,nn}];Range[0,nn]!CoefficientList[ Series[(s-1) Exp[x],{x,0,nn}],x]  (* Geoffrey Critzer, Oct 09 2012 *)

a(n)=sum(k=1,n,binomial(n,k)*2^((k^2-k)/2))

a(n) = Sum_{k=1..n} binomial(n, k)*2^(k(k-1)/2).
E.g.f.: exp(x)*(A(x)-1), where A(x) is e.g.f. for A006125. - Geoffrey Critzer, Oct 09 2012
a(n) ~ 2^(n*(n-1)/2). - Vaclav Kotesovec, Nov 15 2014

cp's OEIS Frontend

User: James D. Klein

A331560 Size of the palindrome-free intervals about the 196-iterates, A006960.

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A331557 The upper (or right) offset of a 196-iterate (A006960) from the smallest palindrome greater than the iterate.

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A331556 The lower (or left) offset of a 196-iterate (A006960) from the largest palindrome less than the iterate.

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A205601 Goldbach's problem extended to division: number of decompositions of 2n into the floor of unordered ratios of two primes, floor(q/p) = 2n, where p < 2n < q.

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A202472 Goldbach's Problem extended to subtraction: number of decompositions of 2n into unordered differences of two primes, p, q, where p < 2n < q.

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A004140 Number of nonempty labeled simple graphs on nodes chosen from an n-set.

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