M. Farrokhi D. G. - cp's OEIS frontend

A367195 Primes p such that (2^(p-1)*(p+1) - 1) / p is prime.

Original entry on oeis.org

3, 5, 7, 11, 19, 29, 41, 61, 137, 181, 293, 11171

Offset: 1

M. Farrokhi D. G., Nov 09 2023

2^(p-1)*(p+1) - 1 is the p-th term of the Cunningham chain p = n_1, n_2, ..., where n_{i+1} = 2*n_i+1 for i >= 1.

a(12) from Michael S. Branicky, Nov 09 2023

A366219 Smallest positive integer whose smallest coprime divisor shift is n.

Original entry on oeis.org

1, 3, 91, 325, 2093, 1001, 12025, 1045, 4945, 6391, 189, 455, 245, 11825, 128843, 368809, 273, 1295, 14495, 37961, 252263, 91375, 595, 13013, 46189, 104951, 63875, 136345, 42237, 22253, 192647, 18655, 8225, 194545, 200629, 192907, 27625, 1911, 464783, 27797

Offset: 0

M. Farrokhi D. G., Oct 05 2023

nonn

A nonnegative number s is a coprime divisor shift of n if GCD(d + s, n) = 1 for all divisors d of n. The coprime divisor shift of n is the infimum of the set of all nonnegative coprime divisor shifts of n.

Conjecture. Every positive integer s is the coprime divisor shift of a positive integer.

			a(0) = 1 for GCD(1 + 0, 1) = 1.
a(1) = 3 for GCD(1 + 1, 3) = GCD(3 + 1, 3) = 1 but GCD(1 + 1, 2) > 1.
a(2) = 91 for GCD(d + 2, 91) = 1 for all divisors d = 1, 7, 13, 91 of 91, GCD(13 + 1, 91) > 1, and 91 is the smallest number with this property.

M. Farrokhi D. G., Table of n, a(n) for n = 0..1000

Cf. A366251, A366330.

isds(k,s)={fordiv(k,d,if(gcd(d+s, k)<>1, return(0))); 1}
findds(k)={for(s=0, k-1, if(isds(k,s), return(s))); -1}
a(n)={for(k=1, oo, if(isds(k,n) && findds(k)==n, return(k)))} \\ Andrew Howroyd, Oct 05 2023

A366330 Minimal numbers (with respect to division) with no coprime divisor shift.

Original entry on oeis.org

2, 15, 33, 51, 69, 87, 123, 141, 159, 177, 213, 249, 267, 303, 321, 339, 393, 411, 447, 501, 519, 537, 573, 591, 665, 681, 699, 717, 753, 771, 789, 807, 819, 843, 879, 933, 951, 1015, 1041, 1059, 1077, 1149, 1167, 1203, 1235, 1257, 1293, 1329, 1347, 1383

Offset: 1

M. Farrokhi D. G., Oct 07 2023

nonn

A number k has a coprime divisor shift s if GCD(d + s, n) = 1 for all divisors d of k.
A number k has a coprime divisor shift iff it is not divisible by any number in the sequence.
If k has no coprime divisor shift, then so is any multiple of k.

a(1) = 2 for GCD(2 + 0, 2) > 1 and GCD(1 + 1, 2) > 1.
a(2) = 15 for GCD(3 + 0, 15) > 1, GCD(5 + 1, 15) > 1, GCD(1 + 2, 15) > 1, and any odd number between 2 and 15 has a coprime divisor shift.

M. Farrokhi D. G., Table of n, a(n) for n = 1..10000

Cf. A044135, A044516, A366219, A366251.

A366251 Numbers with a coprime divisor shift.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 35, 37, 39, 41, 43, 47, 49, 53, 55, 57, 59, 61, 63, 65, 67, 71, 73, 77, 79, 81, 83, 85, 89, 91, 93, 95, 97, 101, 103, 107, 109, 111, 113, 115, 117, 119, 121, 125, 127, 129, 131, 133, 137, 139, 143, 145

Offset: 1

M. Farrokhi D. G., Oct 05 2023

A number k has a coprime divisor shift s if GCD(d + s, k) = 1 for all divisors d of k.
If k is in the sequence, then all divisors of k are in the sequence too.
From David A. Corneth, Oct 06 2023: (Start)
As a consequence of the above, if some m is not in the sequence then any multiple of m is not in the sequence either. So all terms are odd as 2 is not in the sequence.
To see if k is a term we can do the following:
- Check whether k is even; if so then k is not in the sequence.
- Make a list of the divisors of k. Let tau(k) be the number of divisors of k. Then for each prime p <= tau(k) if the number of residues mod p of the divisors of k is equal to p then k is not in the sequence. Otherwise by the Chinese Remainder Theorem we can find a number s such that gcd(d + s, n) = 1 for all d.
So every odd prime is a term. (End)

			1 is a term since GCD(1 + 0, 1) = 1.
2 is not a term since GCD(1 + s, 2) > 1 or GCD(2 + s, 2) > 1 for all nonnegative integers s.
3 is a term since GCD(1 + 1, 3) = GCD(3 + 1, 3) = 1.

M. Farrokhi D. G., Table of n, a(n) for n = 1..10000

Cf. A366219, A366330.

CoprimeDivisorShift := function(n)
local shift, divisors;
shift := 0;
if not IsPrimeInt(n) and First(PrimeDivisors(n), p -> CoprimeDivisorShift(n / p) = infinity) <> fail then
	shift := infinity;
fi;
if shift < infinity then
	divisors := DivisorsInt(n);
	while shift < n and First(divisors, d -> GcdInt(d + shift, n) > 1) <> fail do
		shift := shift + 1;
	od;
	if shift = n then
		shift := infinity;
	fi;
fi;
return shift;
end;

aList := proc(len) local isds, findds;
isds := (k, s) -> andmap(d -> igcd(d + s, k) = 1, NumberTheory:-Divisors(k));
findds := k -> ormap(s -> isds(k, s), [seq(1..k)]);
select(k -> findds(k), [seq(1..len)]) end:
aList(133);  # Peter Luschny, Oct 06 2023
# More efficient, after David A. Corneth:
isa := proc(n) local d, p, t; p := 3;
if irem(n, 2) = 0 then return false fi;
d := NumberTheory:-Divisors(n);
while p < nops(d) do
   {seq(irem(t, p), t = d)};
   if nops(%) = p then return false fi;
   p := nextprime(p);
od: true end:
aList := len -> select(isa, [seq(1..len)]):
aList(145);  # Peter Luschny, Oct 07 2023

isa[n_] := Module[{d, p, t}, p = 3; If[Mod[n, 2] == 0, Return[False]]; d = Divisors[n]; While[p < Length[d], If[Length[Union@Table[Mod[t, p], {t, d}]] == p, Return[False]]; p = NextPrime[p]]; True];
aList[len_] := Select[Range[len], isa];
aList[145] (* Jean-François Alcover, Oct 27 2023, after Peter Luschny *)

isds(k,s)={fordiv(k, d, if(gcd(d+s, k)<>1, return(0))); 1}
findds(k)={for(s=1, k, if(isds(k,s), return(s))); 0}
select(k->findds(k), [1..150]) \\ Andrew Howroyd, Oct 05 2023

is(n) = {
	if(!bitand(n, 1), return(0));
	my(d = divisors(n));
	forprime(p = 3, #d,
		if(#Set(d % p) == p,
			return(0)
		)
	); 1	
} \\ David A. Corneth, Oct 06 2023

A348358 Primes which are not the concatenation of smaller primes (in base 10 and allowing leading 0's).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 47, 59, 61, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 127, 131, 139, 149, 151, 157, 163, 167, 179, 181, 191, 199, 239, 251, 263, 269, 281, 349, 401, 409, 419, 421, 431, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499

Offset: 1

M. Farrokhi D. G., Oct 14 2021

This is the sequence of numbers that are neither a product of smaller primes nor a concatenation of smaller primes (in base 10).
This sequence differs from A238647. The prime 227 is in A238647 but not in this sequence for it is the concatenation of primes 2, 2, 7 (in base 10).
Conjecture. If p > 7 is a prime, then there exists a base b such that p in base b is the concatenation of smaller primes in base b.

			The prime 127 is in the sequence because the only expressions of 127 as concatenation of smaller numbers are 1 U 2 U 7, 1 U 27, and 12 U 7 (in base 10) but 1 and 12 are not primes.
The prime 271 is not in the sequence because it is the concatenation of primes 2 and 71 (in base 10).
The prime 307 is not in the sequence because it is the concatenation of primes 3 and 07 (in base 10).

Cf. A141033, A141409, A238647, A342244, A348358.

Select[Prime@Range@100,Union[And@@@PrimeQ[FromDigits/@#&/@Union@Select[Flatten[Permutations/@Subsets[Most@Rest@Subsequences[d=IntegerDigits@#]],1],Flatten@#==d&]]]=={False}||Length@d==1&] (* Giorgos Kalogeropoulos, Oct 15 2021 *)

from sympy import isprime, primerange
def cond(n): # n is not a concatenation of smaller primes
    if n%10 in {4, 6, 8}: return True
    d = str(n)
    for i in range(1, len(d)):
        if isprime(int(d[:i])):
             if isprime(int(d[i:])) or not cond(int(d[i:])):
                 return False
    return True
def aupto(lim): return [p for p in primerange(2, lim+1) if cond(p)]
print(aupto(490)) # Michael S. Branicky, Oct 15 2021

A337706 Possible values for the product of the orders of all elements of a finite group.

Original entry on oeis.org

1, 2, 8, 9, 32, 72, 128, 512, 625, 648, 2048, 6561, 8192, 20000, 32768, 41472, 52488, 117649, 131072, 524288, 2654208, 3359232, 4782969, 8388608, 12500000, 15059072, 33554432, 134217728, 214990848, 536870912, 2147483648, 2448880128

Offset: 1

M. Farrokhi D. G., Sep 16 2020

nonn

The product of the orders of all elements of a finite group G is denoted by P(G).
P(A X B) = P(A)^|B|P(B)^|A| for finite groups A and B of coprime orders.
P(G) < P(C_n) for every noncyclic finite group G of order n.

			P(C_6) = 1*2*3*3*6*6 = 648.

M. Farrokhi D. G., Table of n, a(n) for n = 1..2733
M. Garonzi and M. Patassini, Inequalities detecting structural properties of a finite group, Comm. Algebra (2017) 45(2), 677-687.
M. Herzog, P. Longobardi, and M. Maj, Properties of finite and periodic groups determined by their element of orders (a survey), Group theory and computation, 59-90, Indian Stat. Inst. Ser., Springer, Singapore, 2018.

GAP
```
Product(List(G, Order))
```

A337705 Possible sums of orders of elements of finite groups.

Original entry on oeis.org

1, 3, 7, 11, 13, 15, 19, 21, 23, 25, 27, 31, 33, 39, 43, 45, 47, 49, 55, 57, 59, 61, 63, 67, 71, 73, 75, 77, 79, 83, 85, 87, 95, 97, 99, 101, 103, 105, 111, 113, 115, 119, 121, 125, 127, 129, 133, 135, 143, 147, 151, 153, 157, 159, 161, 163

Offset: 1

M. Farrokhi D. G., Sep 16 2020

nonn

The sum of the orders of all elements of a finite group G is denoted by psi(G).
psi(A X B) = psi(A)*psi(B) for finite groups A and B of coprime orders.
psi(G) <= 7/11 psi(C_n) < psi(C_n) for every noncyclic finite group G of order n.
psi(G) < 1/(p - 1) psi(C_n) for every noncyclic finite group G of order n, where p the smallest prime divisor of n.
Conjecture: If S is a simple group and G is a soluble group satisfying |S|=|G|, then psi(S) < psi(G).

			psi(C_6) = 1 + 2 + 3 + 3 + 6 + 6 = 21.

M. Farrokhi D. G., Table of n, a(n) for n = 1..1027
H. Amiri, S. M. Jafarian Amiri, and I. M. Isaacs, Sums of element orders in finite groups, Comm. Algebra 37(9) (2009), 2978-2980.
M. Herzog, P. Longobardi, and M. Maj, Properties of finite and periodic groups determined by their element of orders (a survey), Group theory and computation, 59-90, Indian Stat. Inst. Ser., Springer, Singapore, 2018.

GAP
```
Sum(List(G, Order));
```

A336526 a(n) is the least k > 1 such that A031121(n) = F(k*m)/F(m) for some m.

Original entry on oeis.org

2, 3, 2, 2, 5, 2, 3, 2, 7, 3, 2, 4, 2, 9, 2, 3, 5, 4, 2, 11, 3, 2, 6, 2, 13, 5, 2, 3, 4, 7, 2, 15, 3, 2, 8, 6, 4, 2, 17, 2, 3, 5, 9, 2, 19, 7, 3, 2, 4, 10, 2, 21, 5, 2, 3, 6, 11, 8, 4, 2, 23, 3, 2, 12, 2, 25, 9, 2, 3, 4, 5, 7, 13, 6, 2, 27, 3, 2

Offset: 1

M. Farrokhi D. G., Jul 25 2020

M. Farrokhi D. G., Table of n, a(n) for n = 1..999
M. Farrokhi D. G., Some remarks on the equation F_n=kF_m in Fibonacci numbers, J. Integer Seq. 10 (2007), no. 5, Article 07.5.7, 9 pp.

Cf. A000045, A031121, A031122.

A333357 Integers that cannot be expressed as the ratio of 2 Fibonacci numbers.

Original entry on oeis.org

6, 9, 10, 12, 14, 15, 16, 19, 20, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 77, 78, 79, 80

Offset: 1

M. Farrokhi D. G., Jul 24 2020

Any product of m > 1 Fibonacci numbers > 1 belongs to the sequence except for 4, 18, 48, and 72.

M. Farrokhi D. G., Table of n, a(n) for n = 1..10000
M. Farrokhi D. G., Some remarks on the equation F_n=kF_m in Fibonacci numbers, J. Integer Seq. 10 (2007), no. 5, Article 07.5.7, 9 pp.

Cf. A000045.

Complement of A031121.

l := Filtered(Set(List(Cartesian([1..21], [1..21]), x -> Fibonacci(x[1] * x[2])/Fibonacci(x[1]))), x -> x < 10000);;
Filtered([1..10000], x ->  not x in l);

A336191 Numbers k of the form k = ab (the decimal concatenation of a and b) such that phi(ab) = a*b + 1.

Original entry on oeis.org

57, 195, 319, 5595, 11709, 77097, 114765, 1313667, 1348559, 4752465, 10219099, 11031119, 185573199, 2918945715, 3165616929, 12233666703, 16996664613, 18052811909, 20650199699, 38081370319, 58943659521, 195823876095, 236323024041, 242687655369, 342764528277, 677924155713

Offset: 1

M. Farrokhi D. G., Jul 11 2020

Is the sequence infinite?

			phi(57) = 5 * 7 + 1
phi(195) = 1 * 95 + 1 = 19 * 5 + 1
phi(319) = 31 * 9 + 1
phi(5595) = 5 * 595 + 1
phi(11709) = 11 * 709 + 1
phi(77097) = 7 * 7097 + 1
phi(114765) = 11 * 4765 + 1
phi(1313667) = 1313 * 667 + 1
phi(1348559) = 134855 * 9 + 1
phi(4752465) = 47 * 52465 + 1
phi(10219099) = 1021 * 9099 + 1
phi(11031119) = 1103111 * 9 + 1
phi(185573199) = 185 * 573199 + 1

Cf. A000010, A109606, A336192.

seqQ[n_] := Module[{d = IntegerDigits[n]}, MemberQ[Times @@@ Table[FromDigits /@ {Take[d, k], Take[d, -Length[d] + k]}, {k, 1, Length[d] - 1}], EulerPhi[n] - 1]]; Select[Range[10, 10^5], seqQ] (* Amiram Eldar, Jul 11 2020 *)

isok(m) = {my(tm=eulerphi(m)-1, d=digits(m)); for (i=1, #d-1, if (fromdigits(vector(i, k, d[k]))*fromdigits(vector(#d-i, k, d[i+k])) == tm, return(1)););} \\ Michel Marcus, Jul 11 2020

Missing terms 1348559 & 4752465, and a(12) from Amiram Eldar, Jul 11 2020

More terms from Giovanni Resta, Jul 13 2020

cp's OEIS Frontend

User: M. Farrokhi D. G.

A367195 Primes p such that (2^(p-1)*(p+1) - 1) / p is prime.

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A366219 Smallest positive integer whose smallest coprime divisor shift is n.

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A366330 Minimal numbers (with respect to division) with no coprime divisor shift.

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A366251 Numbers with a coprime divisor shift.

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A348358 Primes which are not the concatenation of smaller primes (in base 10 and allowing leading 0's).

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A337706 Possible values for the product of the orders of all elements of a finite group.

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GAP

A337705 Possible sums of orders of elements of finite groups.

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GAP

A336526 a(n) is the least k > 1 such that A031121(n) = F(k*m)/F(m) for some m.

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A333357 Integers that cannot be expressed as the ratio of 2 Fibonacci numbers.

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