A092506 -id:A092506 - cp's OEIS frontend

A263581 Prime powers (p^k, p prime, k >= 1) such that k*p^k - 1 is also a power of a prime.

2, 3, 4, 5, 8, 9, 17, 25, 49, 64, 121, 169, 257, 289, 729, 841, 1681, 1849, 3481, 5329, 11881, 12769, 16129, 18769, 24649, 32041, 32761, 38809, 39601, 44521, 59049, 63001, 65537, 69169, 76729, 85849, 96721, 124609, 134689, 143641, 167281, 175561, 187489

Offset: 1

Juri-Stepan Gerasimov, Apr 09 2016

nonn

Of course 1 = p^0 for any prime p, so 1 is definitely the power of a prime (comment in A000961).

Only primes of the form 2^m + 1 (2 and Fermat primes) are terms.

			8 is in this sequence because both 8 = 2^3 and 3*2^3 - 1 = 23 is prime power.

Cf. A000961, A019434 (Fermat primes), A092506 (primes of the form 2^m + 1).

ispp(n) = if ((n==1) || isprime(n), return (1), isprimepower(n));
isok(n) = ((k=ispp(n)) && ispp(k*n-1)); \\ Michel Marcus, Apr 11 2016

A272608 Number of positive integers k such that both n/(k + 2^x) and n/(n/k - 2^y) are integers for some nonnegative x, y.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 0, 3, 1, 2, 0, 4, 0, 0, 1, 4, 1, 2, 0, 3, 1, 0, 0, 6, 0, 0, 0, 0, 0, 3, 0, 5, 2, 2, 1, 4, 0, 0, 0, 4, 0, 2, 0, 0, 1, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 5, 0, 0, 1, 6, 2, 4, 0, 2, 0, 1, 0, 6, 0, 0, 0, 0, 1, 0, 0, 5, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2

Offset: 1

Juri-Stepan Gerasimov, Nov 09 2016

nonn

Where k, k + 2^x, n/k, n/k - 2^y, n/(k + 2^x) and n/(n/k - 2^y) are divisors of n.

			a(9) = 1 because both 9/(1 + 2^1) = 3 and 9/(9/1 - 2^4) = 1 are integers.
a(68) = 3 because (1) 68/(1 + 2^0) = 34 and 68/(68 - 2^6) = 17, (2) 68/(2 + 2^1) = 17 and 68/(34 - 2^5) = 34, and (3) 68/(4 + 2^6) = 1 and 68/(17 - 2^4) = 68 are all integers.

Cf. A027750, A092506, A277994.

t1(n,k)=for(x=0,logint(n,2), if(n%(k+2^x)==0, return(1))); 0
t2(n,d)=for(y=0,logint(d-1,2), if(n%(d-2^y)==0, return(1))); 0
a(n)=sumdiv(n,k, kCharles R Greathouse IV, Nov 09 2016

a(2^n) = n, a(A092506(n)) = 1.

a(68), a(70), a(90) corrected by Charles R Greathouse IV, Nov 09 2016

A275159 Primes p such that p-1 is the value of totient function of a product of distinct Fermat numbers (A000215).

Original entry on oeis.org

2, 3, 5, 17, 257, 65537, 548898078721, 1151122703583805441, 77370970260794891965562881, 632834090662785970268956262401, 1327149278901642923121482163604684801, 2787593149816327845958662202634335514787841, 91343852333181430856373443055921906148567941121

Offset: 1

Jaroslav Krizek, Nov 13 2016

nonn

Primes p such that p-1 = phi(A001317(x)) has solution.

			Prime 548898078721 is in the sequence because 548898078720 = phi(1095216660735) = phi(3*5*17*4294967297); all numbers 3, 5, 17 and 4294967297 are terms of A000215 (Fermat numbers).

Supersequence of A019434 (Fermat primes) and A092506 (primes of the form 2^n+1).

Cf. A000010, A000215, A001317, A019434, A092506.

Set(Sort([EulerPhi(k)+1: k in [A001317(n)] | IsPrime(EulerPhi(k)+1)]));

a(13) from Jinyuan Wang, Nov 01 2020

A275584 Primes p such that S_e(p-1)/S_o(p-1) is an integer, where S_e(x) is the sum of the even numbers and S_o(x) is the sum of the odd numbers in the Collatz iteration of x.

Original entry on oeis.org

2, 3, 5, 17, 257, 59393, 65537, 331777, 534529, 1299457

Offset: 1

Jaroslav Krizek, Aug 04 2016

Primes p such that A213909(p-1)/A213916(p-1) is an integer.
Primes of the form A274796 + 1.
Fermat primes (A019434) are terms. Also supersequence of A092506 (primes of the form 2^n+1).
Corresponding values of S_e/o(a(n)-1): 0, 2, 6, 30, 510, 1567, 131070, ...

			Prime 59393 is a term because S_e/o(59392) = A213909(59392)/A213916(59392) = 119092/76 = 1567.

Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture

Cf. A033493, A213909, A213916, A274796.

[n+1: n in [A274796(m)] | IsPrime(n+1)]

e:= [&+[not IsOdd(h) select h else 0: h in [k eq 1 select n else IsOdd(Self(k-1)) and not IsOne(Self(k-1)) select 3*Self(k-1)+1 else Self(k-1) div 2: k in [1..5*n]]]: n in [1..1000]]; o:= [&+[IsOdd(h) select h else 0: h in [k eq 1 select n else IsOdd(Self(k-1)) and not IsOne(Self(k-1)) select 3*Self(k-1)+1 else Self(k-1) div 2: k in [1..5*n]]]: n in [1..1000]]; [n+1: n in [1..1000] | IsPrime(n+1) and e[n] mod o[n] eq 0]

Select[Prime@ Range[10^5], IntegerQ[Divide @@ Map[Total, TakeDrop[#, LengthWhile[#, EvenQ]]]] &@ SortBy[#, OddQ] &@ NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, # - 1, # > 1 &] &] (* Michael De Vlieger, Oct 15 2018 *)

S_e/o(A092506(n)-1) = A033493(A092506(n))-1.

More terms from Michael De Vlieger, Oct 15 2018.

A303435 Numbers n such that uphi(n) (the unitary totient function A047994) is a power of the number of unitary divisors of n (A034444).

Original entry on oeis.org

1, 2, 3, 5, 9, 10, 17, 30, 34, 85, 170, 257, 514, 765, 1285, 1542, 4369, 8738, 39321, 65537, 131070, 131074, 327685, 655370, 1114129, 2949165, 3342387, 16843009, 33686018, 100271610, 151587081, 572662306, 2863311530

Offset: 1

Amiram Eldar, Apr 24 2018

nonn

The unitary version of A289276.

Since A034444(n)=2^omega(n) is a power of 2, all the terms are products of 2 and the Fermat primes (A019434), each with multiplicity < 2, except for 3 that may be of multiplicity of 2 (since 3^2 = 2^3 + 1). If there is no 6th Fermat prime, then this sequence is finite with 33 terms.

			2863311530 = 2 * 5 * 17 * 257 * 65537 is in the sequence since it has 2^5 unitary divisors, and its uphi value is 2^30 = (2^5)^6.

Cf. A019434, A034444, A047994, A092506, A289276.

uphi[n_]:=If[n == 1,1,(Times @@ (Table[#[[1]]^#[[2]] - 1, {1}] & /@ FactorInteger [n]))[[1]]]; aQ[n_] := If[n == 1, True, IntegerQ[Log[2, uphi[n]]/PrimeNu[n]]]; v = Union[Times @@@ Rest[Subsets[{1, 2, 3, 5, 17, 257, 65537}]]]; w = Union[v, 3*v]; s = {}; Do[w1 = w[[k]]; If[aQ[w1], AppendTo[s, w1]], {k, 1, Length[w]}]; s

A363464 Numbers k in A052294 with arithmetic derivative k' (A003415) in A052294.

Original entry on oeis.org

6, 9, 10, 14, 18, 20, 21, 22, 24, 25, 33, 34, 35, 38, 40, 42, 44, 48, 49, 52, 62, 65, 66, 68, 69, 70, 76, 80, 84, 88, 91, 93, 94, 96, 100, 104, 110, 115, 117, 118, 121, 132, 133, 134, 138, 140, 143, 144, 145, 148, 152, 155, 158, 164, 174, 182, 185, 186, 188, 192

Offset: 1

Marius A. Burtea, Jul 08 2023

If p > 2 is in A092506 then m = 2*p and u = 4*p are terms. Indeed, if p = 2^k + 1, k >= 1, m = 2*(2^k + 1) = 2^(k+1) + 2^1 has two 1's in its binary expansion, and m' = p+2 = 2^k + 3 = 2^k + 2^1 + 1 has three 1's in its binary expansion. Similarly u = 4*(2^k + 1) = 2^(k+2) + 2^2 and u' = 4*p + 4 = 2^(k+2) + 2^3.
If p is in A057733 then the number m = 2*p is a term. Indeed, if p = 2^k + 3, k >= 1, m = 2*(2^k + 3) = 2^(k+1) + 2^2 + 2 has three 1's in its binary expansion, and m' = p+2 = 2^k + 5 = 2^k + 2^2 + 1 has three 1's in its binary expansion.
If p > 7 is in A057733 then the number m = 4*p is a term. Indeed, if p = 2^k + 3, k >= 3, m = 4*(2^k + 3) = 2^(k+2) + 2^3 + 2 has three 1's in its binary expansion, and m' = 4*(p + 1) = 4*(2^k + 4) = 2^(k+2) + 2^4 has two 1's in its binary expansion.
If p is in A123250 then the number m = 4*p is a term. Indeed, if p = 2^k + 5, k >= 1, m = 4*(2^k + 5) = 2^(k+2) + 2^4 + 2^2 has three 1's its binary expansion, and m' = 4*(p+1) = 4*(2^k + 6) = 2^(k+2) + 2^4 + 2^2 has three 1's in its binary expansion.
If p is in A104070 then the number m = 4*p is a term. Indeed, if p = 2^k + 9, k >= 1, m = 4*(2^k + 9) = 2^(k+2) + 2^5 + 2^2 has three 1's its binary expansion, and m' = 4*(p+1) = 4*(2^k + 10) = 2^(k+2) + 2^5 + 2^3 has three 1's in its binary expansion.

			6 = 110_2 has two 1's, 6' = 5 = 101_2 has two 1's, so 6 is a term.
9 = 101_2 has two 1's, 9' = 6 = 110_2 has two 1's, so 9 is a term.
10 = 1010_2 has two 1's, 10' = 7 = 111_2 has three 1's, so 10 is a term.
18 = 10010_2 has two 1's, 18' = 21 = 10101_2 has three 1's, so 18 is a term.

Cf. A000120, A003415, A052294, A057733, A081092, A092506, A104070, A123250.

fp:=func; f:=func; [n:n in [1..200]| fp(n) and fp(Floor(f(n)))];

pernQ[n_] := PrimeQ[DigitCount[n, 2, 1]]; d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[200], And @@ pernQ[{#, d[#]}] &] (* Amiram Eldar, Jul 10 2023 *)

A365640 Prime powers of the form 2^k + 1.

Original entry on oeis.org

2, 3, 5, 9, 17, 257, 65537

Offset: 1

Juri-Stepan Gerasimov, Nov 10 2023

nonn

By Mihăilescu's theorem, this is just the union of {2, 9} and the Fermat primes A019434. - Robert Israel, Nov 10 2023

Subsequence of A000961. Supersequence of A019434 and A092506.

[2^k+1: k in [0..1000] | IsPrimePower(2^k+1)];

cp's OEIS Frontend

A263581 Prime powers (p^k, p prime, k >= 1) such that k*p^k - 1 is also a power of a prime.

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A272608 Number of positive integers k such that both n/(k + 2^x) and n/(n/k - 2^y) are integers for some nonnegative x, y.

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A275159 Primes p such that p-1 is the value of totient function of a product of distinct Fermat numbers (A000215).

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A275584 Primes p such that S_e(p-1)/S_o(p-1) is an integer, where S_e(x) is the sum of the even numbers and S_o(x) is the sum of the odd numbers in the Collatz iteration of x.

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A303435 Numbers n such that uphi(n) (the unitary totient function A047994) is a power of the number of unitary divisors of n (A034444).

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A363464 Numbers k in A052294 with arithmetic derivative k' (A003415) in A052294.

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A365640 Prime powers of the form 2^k + 1.

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