Bob Andriesse - cp's OEIS frontend

A368958 Number of permutations of [n] where each pair of adjacent elements is coprime and does not differ by a prime.

1, 1, 2, 2, 2, 10, 4, 28, 6, 42, 40, 348, 42, 1060, 226, 998, 886, 21660, 690, 57696, 4344, 26660, 22404, 1091902, 12142, 1770008

Offset: 0

Bob Andriesse, Jan 10 2024

The number of Hamiltonian paths in a graph of which the nodes represent the numbers (1,2,3,...,n) and the edges connect each pair of nodes that are coprime and do not differ by a prime.

			a(5) = 10: 15432, 21543, 23451, 32154, 34512, 43215, 45123, 51234, 54321, 12345.
a(6) = 4: 432156, 651234, 654321, 123456.

Bob Andriesse, Combined graph of this sequence, A076220, A103839 and A367704 for a(4) to a(25).

Cf. A076220, A103839, A367704.

a[n_] := a[n] = Module[{b = 0, ps}, ps = Permutations[Range[n]]; Do[If[Module[{d}, AllTrue[Partition[pe, 2, 1], (d = Abs[#[[2]] - #[[1]]]; ! PrimeQ[d] && CoprimeQ[#[[1]], #[[2]]]) &]], b++], {pe, ps}]; b];
Table[a[n], {n, 0, 8}] (* Robert P. P. McKone, Jan 12 2024 *)

okperm(perm) = {for(k=1, #perm-1, if((isprime(abs(perm[k]-perm[k+1]))), return(0)); if(!(gcd(perm[k], perm[k+1])==1), return(0));); return(1);}
a(n) = {my(nbok = 0); for (j=1, n!, perm = numtoperm(n,j); if(okperm(perm), nbok++);); return(nbok); }

from math import gcd
from sympy import isprime
def A368958(n):
    if n<=1 : return 1
    clist = tuple({j for j in range(1,n+1) if j!=i and gcd(i,j)==1 and not isprime(abs(i-j))} for i in range(1,n+1))
    def f(p,q):
        if (l:=len(p))==n-1: yield len(clist[q]-p)
        for d in clist[q]-p if l else set(range(1,n+1))-p:
            yield from f(p|{d},d-1)
    return sum(f(set(),0)) # Chai Wah Wu, Jan 19 2024

a(14)-a(25) from Alois P. Heinz, Jan 11 2024

A367704 Number of permutations of [n] where each pair of adjacent elements differs by a prime.

Original entry on oeis.org

1, 1, 0, 0, 2, 10, 32, 96, 448, 1968, 7320, 21516, 118938, 662742, 4556360, 26950038, 155388246, 756995286, 5730299976, 38809702892, 337875402936, 2593543573702, 20560179519176, 138677553274430, 1337517942958934, 11083936316867572, 94288296012340842

Offset: 0

Bob Andriesse, Nov 27 2023

nonn

This sequence was inspired by A103839 and the PARI program is a modified version of the one in A103839.
The number of Hamiltonian paths in a graph of which the nodes represent the numbers (1,2,3,...,n) and the edges connect each pair of nodes that differ by a prime.
A076220, A103839 and this sequence are closely related, but their combined graph in the link shows an interesting difference, notably between this sequence and the two others. - Bob Andriesse, Dec 03 2023

			a(4) = 2: 2413, 3142.
a(5) = 10: 13524, 14253, 24135, 25314, 31425, 35241, 41352, 42531, 52413, 53142.

Bob Andriesse, Combined graph of this sequence, A076220 and A103839 for a(4) to a(25)

Cf. A076220, A103839.

okperm(perm) = {for (k=1, #perm -1, if (! isprime(abs(perm[k]-perm[k+1])),  return (0));  ); return (1); }
a(n) = {nbok = 0; for (j=1, n!, perm = numtoperm(n, j); if (okperm(perm), nbok++); ); return (nbok); }

a(14)-a(22) from Alois P. Heinz, Nov 27 2023

a(23)-a(26) from Martin Ehrenstein, Dec 03 2023

A365249 Composite numbers k for which A214749(k) = (k-1)/2.

Original entry on oeis.org

25, 85, 121, 133, 145, 187, 205, 217, 221, 253, 301, 325, 361, 385, 403, 437, 445, 451, 481, 505, 529, 533, 553, 565, 625, 667, 697, 721, 745, 793, 817, 841, 865, 893, 913, 925, 973, 985, 1003, 1027, 1037, 1045, 1057, 1073, 1081, 1141, 1157, 1165, 1207, 1225

Offset: 1

Bob Andriesse, Aug 28 2023

nonn

As can be seen from A214749, for most odd composites k the smallest value of m that satisfies k-m | k^2+m is smaller than (k-1)/2. This sequence lists the exceptions. All the odd primes appear to satisfy A214749(p) = (p-1)/2.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A214749, A365248.

f(n) = my(m=1); while((n^2+m) % (n-m), m++); m; \\ A214749
lista(nn) = my(list=List()); forcomposite(c=1, nn, if (f(c) == (c-1)/2, listput(list, c))); Vec(list); \\ Michel Marcus, Sep 04 2023

from sympy import isprime
a=[]
for n in range(3,1000):
  for m in range(1,(n-1)//2+1):
   if (n**2+m)%(n-m)==0:
    if m==(n-1)/2 and not isprime(n):
     a.append(n)
    break
print(a)

from itertools import count, islice
from sympy import isprime
from sympy.abc import x, y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A365249_gen(startvalue=3): # generator of terms >= startvalue
    return filter(lambda n:not isprime(n) and min(int(x) for x,y in diop_quadratic(n*(n-y)+x*(y+1)) if x>0)==n-1>>1, count(max(startvalue+startvalue&1^1,3),2))
A365249_list = list(islice(A365249_gen(),30)) # Chai Wah Wu, Oct 06 2023

A365248 Composite numbers k that are not a prime minus one, for which A214749(k) = k/2.

Original entry on oeis.org

34, 94, 118, 142, 202, 214, 246, 274, 298, 334, 394, 402, 436, 454, 514, 526, 538, 622, 628, 634, 694, 706, 712, 754, 766, 778, 802, 814, 892, 898, 922, 934, 942, 958, 1002, 1006, 1042, 1054, 1114, 1126, 1132, 1138, 1146, 1158, 1174, 1198, 1234, 1246, 1270

Offset: 1

Bob Andriesse, Aug 28 2023

nonn

As can be seen from A214749, for most composites k that are not a prime minus one, the smallest value of m that satisfies k-m | k^2+m is smaller than k/2. This sequence lists the exceptions.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A214749, A365249.

f(n) = my(m=1); while((n^2+m) % (n-m), m++); m; \\ A214749
lista(nn) = my(list=List()); forcomposite(c=1, nn, if ((f(c) == c/2) && !isprime(c+1), listput(list, c))); Vec(list);  \\ Michel Marcus, Sep 08 2023

from sympy import isprime
a=[]
for n in range(2,1000):
  for m in range(1,n//2+1):
   if (n**2+m)%(n-m)==0:
    if m==n/2 and not isprime(n+1):
     a.append(n)
    break
print(a)

from itertools import count, islice
from sympy import isprime
from sympy.abc import x, y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A365248_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:not isprime(n+1) and min(int(x) for x,y in diop_quadratic(n*(n-y)+x*(y+1)) if x>0)==n>>1, count(max(startvalue+startvalue&1,2),2))
A365248_list = list(islice(A365248_gen(),30)) # Chai Wah Wu, Oct 06 2023

A356465 The number of unit squares enclosed by the rectangular spiral of which the n-th side has length prime(n).

Original entry on oeis.org

0, 2, 6, 12, 27, 59, 113, 179, 257, 359, 497, 747, 963, 1227, 1577, 1799, 2081, 2611, 3223, 3663, 4167, 4817, 5231, 5847, 6657, 7527, 8801, 9869, 10439, 11057, 11699, 12425, 14675, 16817, 18027, 19139, 20855, 22595, 23803, 25711, 27321, 29011, 31063, 32495

Offset: 0

Bob Andriesse, Aug 08 2022

nonn

The pictures in the links show how the spiral is constructed. The first segment is the small black rectangle in the center, of which the left lower corner is at the origin (0,0). It represents prime(1) = 2 (its width) and is given a height of one. The first part of the boundary of the spiral is the line between (0,0) and (2,0). Prime(2) = 3 yields the next part of the boundary, the line connecting (2,0) and (2,3). The next primes determine how many unit steps the boundary of the spiral goes left, down, right, up, etc.

Bob Andriesse, Graphical representation of the first 16 segments of the spiral with grid.
Bob Andriesse, Graphical representation of the first 44 segments of the spiral without grid.

Cf. A000040.

a[4]:=27; a[n_]:=a[n]=a[n-1]+(Prime[n]-Prime[n-2]+Prime[n-4])(Prime[n-1]-Prime[n-3]); Join[{0,2,6,12,27},Table[a[n],{n,5,45}]] (* Stefano Spezia, Aug 09 2022 *)

from sympy import prime as p
a = [0,2,6,12,27] #first 4 area values
area = 27
for n in range(5,44+1):
  darea = (p(n) - p(n-2) + p(n-4)) * (p(n-1) - p(n-3))
  area += darea
  a.append(area)
print('a(n)=',a)

a(n) = a(n-1) +(prime(n) - prime(n-2) + prime(n-4))*(prime(n-1) - prime(n-3)) for n > 4.

A339247 The primes that yield twice a prime when each bit of their binary expansion is inverted.

Original entry on oeis.org

11, 17, 37, 41, 53, 59, 89, 101, 113, 137, 149, 173, 181, 193, 197, 229, 233, 241, 251, 257, 293, 317, 353, 389, 449, 521, 541, 557, 569, 577, 601, 641, 661, 677, 709, 761, 769, 797, 809, 821, 829, 857, 877, 881, 929, 937

Offset: 1

Bob Andriesse, Nov 28 2020

Primes p such that A035327(p)/2 is prime.
Obviously the resulting prime is smaller than the starting prime.
Conjecture: Each of the primes can be created by applying this transformation T(p) to at least one other prime. T(11)=2, T(549755813881)=3, T(53)=5, T(17)=7, T(41)=11.

			a(1) = 11 = 1011_2 —inv-> 100_2 = 4 = 2 * 2.
a(2) = 17 = 10001_2 -inv-> 1110_2 = 14 = 2 * 7.
a(3) = 37 = 100101_2 -inv-> 11010_2 = 26 = 2 * 13.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A035327, A339268.

filter:= proc(n) isprime(n) and isprime((2^(1+ilog2(n))-1-n)/2) end proc:
select(filter, [seq(i,i=3..1000,2)]); # Robert Israel, Jun 27 2023

Select[Range[1000], And @@ PrimeQ[{#, (2^Ceiling @ Log2[#] - # - 1)/2}] &] (* Amiram Eldar, Dec 01 2020 *)

isok(p) = if ((p>2) && isprime(p), isprime(fromdigits(apply(x->1-x, binary(p)), 2)/2)); \\ Michel Marcus, Dec 01 2020

from sympy import isprime
from sympy import prime
for i in range(1,201):
  j=prime(i)
  xor=2**len(bin(j).strip('0b'))-1
  p=(j^xor)//2
  if isprime(p):
   print(j,end=', ')

A339268 The smallest prime that becomes 2 * prime(n), when all the bits in its binary expansion are inverted, or -1 if no such prime exists.

Original entry on oeis.org

11, 549755813881, 53, 17, 41, 37, 16349, 89, 977, 197, 193, 181, 173, 937, 929, 149, 137, 389, 1913, 881, 877, 353, 857, 3917, 317, 821, 3889, 809, 293, 797, 257, 761, 3821, 65257, 3797, 3793, 709, 1721, 3761, 677, 1048217, 661, 641, 3709, 3701, 3697, 601, 577

Offset: 1

Bob Andriesse, Nov 29 2020

Conjecture: a(n) > 0 for all n.

a(6705), for prime(6705) = 67289, is either -1 or greater than (2^62500 - 1) * 262144 + 127565, which has 18820 digits. - Michael S. Branicky, Dec 05 2020

			a(1) = 11, because 11 = 1011_2 -inv-> 0100_2 = 4 = 2 * 2.
a(2) = 549755813881, because 549755813881 = 111111111111111111111111111111111111001_2 -inv-> 110_2 = 6 = 2 * 3. No smaller prime generates 3.
a(3) = 53, because 53 = 110101_2 -inv-> 001010_2 = 10 = 2 * 5.

Michael S. Branicky, Table of n, a(n) for n = 1..6704

Cf. A035327, A339247.

a[n_] := Module[{q = 2*Prime[n], m, r}, m = 2^Ceiling@Log2[q]; r = m - q - 1; While[r < q || ! PrimeQ[r], r += m; m *= 2]; r]; Array[a, 48] (* Amiram Eldar, Dec 04 2020 *)

a(n) = {my(b = apply(x->1-x, binary(2*prime(n))), e=#b, q=fromdigits(b, 2)+2^e); while (!isprime(q), e++; q+=2^e; q); q;} \\ Michel Marcus, Dec 04 2020

from sympy import isprime
from sympy import prime
for i in range(1,50):
  d=2*prime(i)
  l=len(bin(d).lstrip('0b'))
  xor=2**l-1
  p=d^xor+2**l
  while not isprime(p):
   l+=1
   p+=2**l
  print(p,end=', ')

A309655 The smallest possible nonnegative difference between the sum of the first n primes (A007504) and the sum of any number of the directly following and consecutive primes.

Original entry on oeis.org

0, 2, 0, 3, 6, 15, 5, 16, 25, 3, 20, 39, 13, 36, 61, 17, 50, 6, 39, 76, 14, 53, 102, 28, 75, 132, 46, 101, 158, 46, 99, 174, 64, 145, 27, 114, 193, 51, 144, 239, 93, 194, 24, 135, 244, 74, 179, 294, 116, 253, 43, 162, 291, 61, 196, 337, 101, 250, 395, 139, 282, 427, 149, 324

Offset: 0

Bob Andriesse, Aug 11 2019

Conjecture: a(0)=0, a(2)=0 and a(532)=0 are the only zeros in the sequence. a(n) has been computed for primes < 10^10. - Bob Andriesse, Oct 07 2020

			a(2) = 2 + 3 - 5 = 0;
a(3) = 2 + 3 + 5 - 7 = 3;
a(6) = 2 + 3 + 5 + 7 + 11 + 13 - (17 + 19) = 5.
a(532)=0 because A007504(733) = 2*A007504(532).

David Radcliffe, Table of n, a(n) for n = 0..10000

Cf. A007504, A309714.

#Lists a(1)...a(100)
from sympy import prime
sumP=0
for i in range(1,101):
  sumP+=prime(i)
  j=i+1
  diff=sumP
  while diff-prime(j) >=0:
   diff-=prime(j)
   j+=1
  print(diff, end=', ')

from itertools import islice
from gmpy2 import next_prime
def a309655_gen():
    lower_prime = upper_prime = difference = 0
    while True:
        while difference >= 0:
            upper_prime = next_prime(upper_prime)
            if difference < upper_prime:
                yield int(difference)
            difference -= upper_prime
        lower_prime = next_prime(lower_prime)
        difference += 2 * lower_prime
print(list(islice(a309655_gen(), 64))) # David Radcliffe, Jun 10 2025

A309714 The smallest possible nonnegative difference between the sum of the first n positive integers (A000217) and the sum of any number of the directly following and consecutive integers.

Original entry on oeis.org

1, 0, 2, 5, 2, 6, 1, 6, 12, 5, 12, 3, 11, 0, 9, 19, 6, 17, 2, 14, 27, 10, 24, 5, 20, 36, 15, 32, 9, 27, 2, 21, 41, 14, 35, 6, 28, 51, 20, 44, 11, 36, 1, 27, 54, 17, 45, 6, 35, 65, 24, 55, 12, 44, 77, 32, 66, 19, 54, 5, 41, 78, 27, 65, 12, 51, 91, 36, 77, 20, 62

Offset: 1

Bob Andriesse, Aug 13 2019

nonn

a(n) = 0 if a positive integer m exists, such that m * (m + 1) = 2 * n * (n + 1). Let k = m - n, then n = (2 * k - 1 + sqrt(8 * k^2 + 1)) / 2. All k for which 8 * k^2 + 1 is a perfect square (A001109) yield a value for n for which a(n) = 0.

a(A053141(n)) = 0 for all n.

			a(2) = 1 + 2 - 3 = 0;
a(3) = 1 + 2 + 3 - 4 = 2;
a(7) = 1 + 2 + 3 + 4 + 5 + 6 + 7 - (8 + 9 + 10) = 1.
a(A053141(2)) = a(14) = 0, because A000217(20) = 2 * A000217(14).

Cf. A000217, A001109, A001652, A053141, A309655.

a(n) = {my(t=n*(n+1)/2, k = n+1); while(t >= k, t -= k; k++); t;} \\ Michel Marcus, Aug 16 2019

A308416 Values of m for which 2*p + m cannot be a square when p is a prime.

Original entry on oeis.org

1, 4, 8, 9, 13, 16, 17, 20, 24, 25, 28, 29, 33, 36, 37, 40, 41, 44, 48, 49, 52, 53, 56, 57, 61, 64, 65, 68, 69, 72, 73, 76, 80, 81, 84, 85, 88, 89, 92, 93, 97, 100, 101, 104, 105, 108, 109, 112, 113, 116, 120, 121, 124, 125, 128, 129, 132, 133, 136, 137, 141, 144, 145, 148, 149

Offset: 1

Bob Andriesse, May 25 2019

nonn

m = i^2 + 4*j is a term for i > 0, 0 <= j < i. Proof: If p = 2, then i^2 < 2*p + m < (i+2)^2. Therefore (i+1)^2 = 4 + i^2 + 4*j, which leads to a contradiction. If p > 2 is such that 2*p + i^2 + 4*j = k^2, then k + i and k - i are both even numbers. Therefore 4 | 2*p + 4*j, which is also a contradiction.

The terms of this sequence can be obtained by starting with A042948 (numbers congruent to 0 or 1 mod 4) and deleting the terms of A028347 (n^2 - 4).

Cf. A042948, A028347.

a=[]
a.append(0) #Offset starts at 1
iMax=15 #Example value
for i in range(1,iMax+1):
  for j in range(0,i):
   m=i*i+j*4
   a.append(m)
a.sort()

Conjecture: for k > 0 and 1 <= j <= k, a(2k^2-2j+1) = 4k^2+4k-4j-3, a(2k^2-2j+2) = 4k^2+4k-4j, a(2k^2+2k-2j+1) = 4k^2+8k-4j, a(2k^2+2k-2j+2) = 4k^2+8k-4j+1. - Jinyuan Wang, Jul 23 2019

cp's OEIS Frontend

User: Bob Andriesse

A368958 Number of permutations of [n] where each pair of adjacent elements is coprime and does not differ by a prime.

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A367704 Number of permutations of [n] where each pair of adjacent elements differs by a prime.

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PARI

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A365249 Composite numbers k for which A214749(k) = (k-1)/2.

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A365248 Composite numbers k that are not a prime minus one, for which A214749(k) = k/2.

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A356465 The number of unit squares enclosed by the rectangular spiral of which the n-th side has length prime(n).

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A339247 The primes that yield twice a prime when each bit of their binary expansion is inverted.

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A339268 The smallest prime that becomes 2 * prime(n), when all the bits in its binary expansion are inverted, or -1 if no such prime exists.

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