A024037 -id:A024037 - cp's OEIS frontend

A304720 Number of nonnegative integers k such that n - (4^k - k) is positive and squarefree.

0, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 3, 2, 2, 1, 3, 1, 2, 1, 3, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 1, 3, 1, 2, 2, 3, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 2, 3, 2, 2, 2, 2, 3, 3, 2, 2, 3, 4, 2, 3, 3, 3, 1, 2, 2, 4, 2, 2, 3, 3, 2, 2, 3, 3, 1, 3, 2, 4, 1, 3, 2, 4, 2, 3, 2, 3

Offset: 1

Zhi-Wei Sun, May 17 2018

nonn

Conjecture: a(n) > 0 for all n > 1.
This has been verified for n up to 2*10^10.
See A304721 for the values of n with a(n) = 1.
See A281192 for N such that none of N - 1 or N + 1 is squarefree: then n = N + 2 is such that n - 1 and n - 3 are not squarefree, i.e., one cannot take k = 0 or k = 1 in the present definition, and k > 1 is required to satisfy the conjecture. - M. F. Hasler, May 23 2018

			a(2) = 1 with 2 - (4^0 - 0) = 1 squarefree.
a(178) = 1 with 178 - (4^0 - 0) = 3*59 squarefree.
a(245) = 1 with 245 - (4^2 - 2) = 3*7*11 squarefree.
a(9196727) = 1 with 9196727 - (4^6 - 6) = 19*211*2293 squarefree.
a(16130577) = 1 with 16130577 - (4^9 - 9) = 2*7934221 squarefree.
a(38029402) = 1 with 38029402 - (4^1 - 1) = 1153*32983 squarefree.
a(180196927) = 1 with 180196927 - (4^11 - 11) = 2*139*227*2789 squarefree.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100000
Zhi-Wei Sun, Mixed sums of primes and other terms, in: D. Chudnovsky and G. Chudnovsky (eds.), Additive Number Theory, Springer, New York, 2010, pp. 341-353.
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)

Cf. A000302, A005117, A024037, A304034, A304081, A304331, A304333, A304522, A304523, A304689, A304721, A304943, A304945.

Cf. A281192.

f[n_]:=f[n]=4^n-n;
tab={};Do[r=0;k=0;Label[bb];If[f[k]>=n,Goto[aa]];If[SquareFreeQ[n-f[k]],r=r+1];k=k+1;Goto[bb];Label[aa];tab=Append[tab,r],{n,1,100}];Print[tab]

A304721 Numbers m with A304720(m) = 1.

Original entry on oeis.org

2, 3, 5, 7, 9, 10, 11, 12, 13, 19, 21, 23, 26, 28, 30, 39, 41, 46, 50, 51, 53, 55, 57, 59, 77, 89, 93, 101, 113, 129, 149, 151, 153, 161, 165, 178, 185, 189, 201, 221, 237, 245, 246, 297, 364, 377, 489, 553, 581, 639

Offset: 1

Zhi-Wei Sun, May 17 2018

nonn

Conjecture: The sequence only has 112 terms as listed in the b-file.

We have verified that there is no new term below 2*10^9.

			a(9) = 13 since 13 - (4^1 - 1) = 2*5 is squarefree,  13 - (4^0 - 0) = 2^2*3 is not squarefree, and 13 - (4^k -k ) < 0 for any integer k > 1.

Zhi-Wei Sun, Table of n, a(n) for n = 1..112
Zhi-Wei Sun, Mixed sums of primes and other terms, in: D. Chudnovsky and G. Chudnovsky (eds.), Additive Number Theory, Springer, New York, 2010, pp. 341-353.
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)

Cf. A000302, A005117, A024037, A304034, A304081, A304331, A304333, A304522, A304523, A304689, A304720.

f[n_]:=f[n]=4^n-n;
tab={};Do[r=0;k=0;Label[bb];If[f[k]>=m,Goto[aa]];If[SquareFreeQ[m-f[k]],r=r+1];If[r>1,Goto[cc]];k=k+1;Goto[bb];Label[aa];If[r==1,tab=Append[tab,m]];Label[cc],{m,1,640}];Print[tab]

A252657 Numbers m such that 4^m - m is a semiprime.

Original entry on oeis.org

2, 11, 17, 33, 55, 59, 63, 153, 315

Offset: 1

Vincenzo Librandi, Dec 20 2014

549, 721, and 755 are in the sequence, but not necessarily the next three terms. The other possibilities for a(9) are 483, 503, and 543. - Robert Israel, Feb 10 2019

			2 is in this sequence because 4^2-2 = 2*7 is semiprime.
17 is in this sequence because 4^17-17 = 6971*2464477 and these two factors are prime.

factordb.com, Status of 4^483-483.

Cf. A024037 (4^n - n).

Cf. similar sequences listed in A252656.

IsSemiprime:=func; [m: m in [2..120] | IsSemiprime(s) where s is 4^m-m];

Mathematica

Select[Range[120], PrimeOmega[4^# - #]==2 &]

Extensions

a(8)-a(9) from Luke March, Jul 08 2015

cp's OEIS Frontend

A304720 Number of nonnegative integers k such that n - (4^k - k) is positive and squarefree.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A304721 Numbers m with A304720(m) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A252657 Numbers m such that 4^m - m is a semiprime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Extensions

A069993 a(n) = 2^(2n+1)*Sum_{k=1..2*n} binomial(2n+1,k)*Bernoulli(k)/2^k.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A069993 a(n) = 2^(2n+1)Sum_{k=1..2n} binomial(2n+1,k)*Bernoulli(k)/2^k.