A098983 -id:A098983 - cp's OEIS frontend

A304081 Number of ways to write n as p + 2^k + (1+(n mod 2))5^m, where p is an odd prime, and k and m are nonnegative integers with 2^k + (1+(n mod 2))5^m squarefree.

0, 0, 0, 0, 0, 1, 0, 2, 1, 3, 1, 2, 2, 2, 1, 3, 3, 3, 2, 4, 2, 3, 2, 5, 2, 4, 2, 3, 3, 3, 2, 4, 3, 5, 1, 7, 4, 4, 3, 7, 2, 4, 3, 8, 4, 7, 4, 6, 3, 7, 3, 6, 4, 5, 3, 5, 4, 5, 2, 7, 3, 5, 4, 8, 4, 5, 3, 5, 5, 8, 6, 6, 6, 9, 3, 9, 7, 6, 6, 8, 5, 6, 4, 6, 8, 7, 6, 8, 7, 4

Offset: 1

Zhi-Wei Sun, May 06 2018

nonn

Conjecture: a(n) > 0 for all n > 7.
This has been verified for n up to 2*10^10.
See also A303821, A303934, A303949, A304031 and A304122 for related information, and A304034 for a similar conjecture.
The author would like to offer 2500 US dollars as the prize to the first proof of the conjecture, and 250 US dollars as the prize to the first explicit counterexample. - Zhi-Wei Sun, May 08 2018

			a(6) = 1 since 6 = 3 + 2^1 + 5^0 with 3 an odd prime and 2^1 + 5^0 = 3 squarefree.
a(15) = 1 since 15 = 5 + 2^3 + 2*5^0 with 5 an odd prime and 2^3 + 2*5^0 = 2*5 squarefree.
a(35) = 1 since 35 = 29 + 2^2 + 2*5^0 with 29 an odd prime and 2^2 + 2*5^0 = 2*3 squarefree.
a(91) = 1 since 91 = 17 + 2^6 + 2*5^1 with 17 an odd prime and 2^6 + 2*5^1 = 2*37 squarefree.
a(9574899) = 1 since 9574899 = 9050609 + 2^19 + 2*5^0 with 9050609 an odd prime and 2^19 + 2*5^0 = 2*5*13*37*109 squarefree.
a(6447154629) = 2 since 6447154629 = 6447121859 + 2^15 + 2*5^0 with 6447121859 prime and 2^15 + 2*5^0 = 2*5*29*113 squarefree, and 6447154629 = 5958840611 + 2^15 + 2*5^12 with 5958840611 prime and 2^15 + 2*5^12 = 2*17*41*433*809 squarefree.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Mixed sums of primes and other terms, in: Additive Number Theory (edited by D. Chudnovsky and G. Chudnovsky), pp. 341-353, Springer, New York, 2010.
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, arXiv:1211.1588 [math.NT], 2012-2017.)

Cf. A000040, A000079, A000351, A005117, A098983, A118955, A156695, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303601, A303637, A303639, A303656, A303660, A303702, A303821, A303932, A303934, A303949, A304031, A304032, A304034, A304122.

PQ[n_]:=n>2&&PrimeQ[n];
tab={};Do[r=0;Do[If[SquareFreeQ[2^k+(1+Mod[n,2])*5^m]&&PQ[n-2^k-(1+Mod[n,2])*5^m],r=r+1],{k,0,Log[2,n]},{m,0,If[2^k==n,-1,Log[5,(n-2^k)/(1+Mod[n,2])]]}];tab=Append[tab,r],{n,1,90}];Print[tab]

A282192 Number of ways of writing n as a sum of a prime and a squarefree semiprime.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 4, 1, 2, 1, 2, 1, 3, 2, 3, 2, 3, 3, 4, 0, 2, 2, 4, 2, 3, 2, 5, 4, 4, 4, 5, 2, 4, 4, 5, 4, 3, 2, 4, 3, 6, 5, 6, 2, 3, 4, 7, 6, 4, 3, 3, 7, 6, 6, 6, 2, 6, 7, 7, 5, 4, 4, 4, 7, 7, 8, 6, 3, 6, 7, 8, 8, 3, 4, 7, 6, 8, 10, 8, 3, 4, 8, 11, 10, 6, 8, 7, 11, 9, 9, 5, 6, 5, 9, 11, 9, 5, 8

Offset: 0

Ilya Gutkovskiy, Feb 15 2017

Conjecture: a(n) > 0 for all n > 30.

			a(17) = 4 because we have [15, 2], [14, 3], [11, 6] and [10, 7].

Robert Israel, Table of n, a(n) for n = 0..10000
Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Semiprime
Eric Weisstein's World of Mathematics, Squarefree

Cf. A000040, A006881, A098983, A100949, A258139, A282318, A282355.

N:= 200: # for a(0)..a(N)
P:= select(isprime, [2,seq(i,i=3..N,2)]): nP:= nops(P):
SFS:= NULL: j:= nP:
for i from 1 to nP while j > 0 do
  while P[i]*P[j] > N do j:= j-1; if j = 0 then break fi; od:
  SFS:= SFS, op(map(`*`,P[1..min(i-1,j)],P[i]))
od:
gS:= add(x^i,i=[SFS]):
gP:= add(x^P[i],i=1..nP):
g:= gP*gS:
[seq(coeff(g,x,i),i=0..N)]; # Robert Israel, Jun 15 2020

nmax = 108; CoefficientList[Series[Sum[x^Prime[k], {k, 1, nmax}] Sum[MoebiusMu[k]^2 Floor[2/PrimeOmega[k]] Floor[PrimeOmega[k]/2] x^k, {k, 2, nmax}], {x, 0, nmax}], x]

A282318 Number of ways of writing n as a sum of a prime and a nonprime squarefree number.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 1, 0, 2, 1, 0, 1, 2, 2, 1, 1, 1, 4, 2, 2, 2, 2, 1, 3, 3, 3, 2, 3, 3, 4, 1, 2, 4, 5, 2, 4, 2, 6, 5, 4, 4, 6, 3, 5, 6, 6, 4, 5, 3, 6, 3, 6, 5, 8, 3, 4, 4, 7, 6, 6, 4, 5, 8, 6, 6, 7, 2, 7, 9, 8, 5, 7, 6, 8, 8, 8, 8, 9, 3, 8, 9, 10, 8, 8, 5, 10, 6, 9, 10, 13, 4, 6, 8, 12, 10, 9, 8, 10, 12, 10, 9, 9, 7, 8, 11, 12, 9, 10

Offset: 0

Ilya Gutkovskiy, Feb 11 2017

Conjecture: a(n) > 0 for all n > 10.

			a(17) = 4 because we have [15, 2], [14, 3], [11, 6] and [10, 7].

Robert Israel, Table of n, a(n) for n = 0..10000
Ilya Gutkovskiy, Extended graphical example

Cf. A001221, A000469, A008683, A098983.

N = 200; % to get a(0) to a(N)
Primes = primes(N);
B = zeros(1,N);
B(Primes) = 1;
LPrimes = Primes(Primes .^ 2 < N);
SF = 1 - B;
for p = LPrimes
   SF(p^2:p^2:N) = 0;
end
C = conv(SF, B);
C = [0,0,C(1:N-1)] % Robert Israel, Feb 12 2017

nmax = 107; CoefficientList[Series[(Sum[x^Prime[i], {i, 1, nmax}]) (x + Sum[Sign[PrimeNu[j] - 1] MoebiusMu[j]^2 x^j, {j, 2, nmax}]), {x, 0, nmax}], x]

G.f.: (Sum_{i>=1} x^prime(i))*(x + Sum_{j>=2} sgn(omega(j)-1)*mu(j)^2*x^j), where omega(j) is the number of distinct primes dividing j (A001221) and mu(j) is the Moebius function (A008683).

A133364 Number of ways of writing n as a sum of a prime and a square-full number.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Oct 26 2007

This is to square-full numbers A001694 as A098983 is to squarefree numbers A005117 and as A002471 is to squares A000290. Asymptotics of this should relate to A098983.

			a(3) = 1 because 3=2+1 where 2 is prime and 1 is square-full.
a(4) = 1 because 4=3+1 where 3 is prime and 1 is square-full.
a(5) = 0 because there is no positive solution to 5 = prime+(square-full).
a(6) = 2 because 6=5+1=2+4.
a(7) = 1 because 7=3+4.
a(8) = 1 because 8=7+1.
a(9) = 1 because 9=5+4.
a(10) = 1 because 10=2+8.
a(11) = 3 because 11=2+9=3+8=7+4.
a(12) = 2 because 12=3+9=11+1.
a(13) = 1 because 13=5+8.
a(14) = 2 because 14=5+9=13+1.
a(15) = 2 because 15=7+8=11+4.
a(16) = 1 because 16=7+9.
a(17) = 1 because 17=13+4.
a(18) = 2 because 18=2+16=17+1.
a(19) = 2 because 19=3+16=11+8.
a(20) = 2 because 20=19+1=11+9.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A000040, A000290, A001694, A002471, A005117, A098983.

isA001694 := proc(n) local digs,i ; digs := ifactors(n)[2] ; for i in digs do if op(2,i) = 1 then RETURN(false) ; fi ; od: RETURN(true) ; end: A133364 := proc(n) local a,p ; a := 0 ; p := 2 ; while p < n do if isA001694(n-p) then a := a+1 ; fi ; p := nextprime(p) ; od: RETURN(a) ; end: seq(A133364(n),n=3..90) ; # R. J. Mathar, Nov 09 2007

a = {}; For[n = 3, n < 100, n++, c = 0; For[j = 1, Prime[j] < n, j++, d = 1; b = FactorInteger[n - Prime[j]]; For[m = 1, m < Length[b] + 1, m++, If[b[[m, 2]] < 2, d = 0]]; If[d == 1, c++ ]]; AppendTo[a, c]]; a (* Stefan Steinerberger, Oct 29 2007 *)

a(n) = Card{n = i + j where i is in A000040 and j is in A001694}.

Corrected and extended by Stefan Steinerberger, Oct 29 2007 and by R. J. Mathar, Nov 09 2007

A282290 Expansion of (Sum_{p prime, i>=2} x^(p^i))(Sum_{j>=2} mu(j)^2x^j), where mu() is the Moebius function (A008683).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 2, 3, 1, 1, 3, 3, 1, 1, 3, 4, 1, 3, 3, 4, 1, 2, 3, 4, 2, 3, 6, 4, 3, 3, 4, 5, 1, 5, 7, 6, 3, 3, 7, 4, 3, 4, 7, 6, 3, 4, 5, 7, 2, 3, 5, 7, 4, 3, 4, 5, 4, 4, 7, 6, 4, 4, 8, 6, 4, 6, 7, 7, 2, 5, 7, 7, 2, 4, 9, 5, 4, 4, 7, 8, 4, 5, 9, 9, 4, 4, 7, 7, 5, 6, 8, 8, 5, 5, 8, 6, 4, 6, 8, 7, 5, 6, 6, 6, 2, 5, 10

Offset: 0

Ilya Gutkovskiy, Feb 11 2017

nonn

Number of ways of writing n as a sum of a proper prime power (A246547) and a squarefree number > 1 (A144338).

Conjecture: a(n) > 0 for all n > 8.

			a(19) = 4 because we have [16, 3], [15, 4], [11, 8] and [10, 9].

Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Prime Power
Eric Weisstein's World of Mathematics, Squarefree

Cf. A005117, A008683, A098983, A144338, A246547.

nmax = 110; CoefficientList[Series[Sum[Sign[PrimeOmega[i] - 1] Floor[1/PrimeNu[i]] x^i, {i, 2, nmax}] Sum[MoebiusMu[j]^2 x^j, {j, 2, nmax}], {x, 0, nmax}], x]

G.f.: (Sum_{p prime, i>=2} x^(p^i))*(Sum_{j>=2} mu(j)^2*x^j).

A283929 Number of ways of writing n as a sum of a twin prime (A001097) and a squarefree semiprime (A006881).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3, 1, 2, 1, 2, 1, 2, 1, 3, 2, 3, 2, 3, 0, 2, 2, 3, 2, 2, 1, 3, 3, 4, 3, 4, 2, 3, 3, 4, 4, 2, 1, 3, 3, 5, 4, 4, 2, 3, 3, 4, 4, 1, 2, 1, 5, 4, 5, 6, 2, 4, 5, 5, 4, 2, 3, 2, 5, 5, 6, 5, 2, 4, 5, 5, 6, 2, 3, 4, 4, 6, 5, 4, 3, 3, 5, 6, 8, 3, 7, 4, 9, 6, 6, 3, 3, 3, 5, 6, 7, 4, 5, 3, 5

Offset: 0

Ilya Gutkovskiy, Mar 18 2017

nonn

Conjecture: a(n) > 0 for all n > 30.

			a(17) = 3 because we have [14, 3], [11, 6] and [10, 7].

Robert Israel, Table of n, a(n) for n = 0..10000
Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Semiprime
Eric Weisstein's World of Mathematics, Squarefree
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A001097, A006881, A098983, A100949, A282192, A282318.

N:= 200: # to get a(0) to a(N)
V:= Vector(N):
Primes:= select(isprime,[2,seq(i,i=3..N+2)]):
PS:= convert(Primes,set);
Twins:= PS intersect map(`-`,PS,2):
Twins:= Twins union map(`+`,Twins,2):
Twins:= sort(convert(Twins,list)):
for i from 1 to nops(Twins) do
  for j from 1 to nops(Primes) while Twins[i]+2*Primes[j] <= N do
    for k from 1 to j-1 do
      v:= Twins[i]+Primes[k]*Primes[j];
      if v > N then break fi;
      V[v]:= V[v]+1;
od od od:
0, seq(V[i],i=1..N); # Robert Israel, Mar 29 2017

nmax = 110; CoefficientList[Series[Sum[Boole[PrimeQ[k] && (PrimeQ[k - 2] || PrimeQ[k + 2])] x^k, {k, 1, nmax}] Sum[MoebiusMu[k]^2 Floor[2/PrimeOmega[k]] Floor[PrimeOmega[k]/2] x^k, {k, 2, nmax}], {x, 0, nmax}], x]

concat([0, 0, 0, 0, 0, 0, 0, 0, 0], Vec(sum(k=1, 110, (isprime(k) && (isprime(k - 2) || isprime(k + 2)))* x^k) * sum(k=2, 110, moebius(k)^2 * floor(2/bigomega(k)) * floor(bigomega(k)/2) * x^k) + O(x^111))) \\ Indranil Ghosh, Mar 18 2017

G.f.: (Sum_{k>=1} x^A001097(k))*(Sum_{k>=1} x^A006881(k)).

A287299 Number of ways of writing n as a sum of a proper prime power (A246547) and a nonprime squarefree number (A000469).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 0, 2, 1, 0, 1, 2, 2, 0, 0, 2, 2, 1, 1, 3, 0, 1, 1, 4, 3, 0, 2, 2, 2, 0, 3, 4, 3, 1, 2, 6, 3, 1, 0, 5, 4, 2, 2, 4, 3, 0, 2, 3, 5, 0, 1, 3, 4, 3, 2, 4, 3, 3, 4, 5, 4, 0, 2, 5, 5, 0, 4, 6, 2, 1, 1, 7, 3, 1, 2, 7, 4, 2, 4, 5, 5, 1, 3, 6, 5, 1, 3, 6, 6, 3, 4, 4, 4, 2, 4, 7, 6, 3, 1, 4, 4, 0, 4, 6, 5, 2, 2, 7, 5, 2, 1, 7, 8, 4

Offset: 0

Ilya Gutkovskiy, May 23 2017

nonn

Conjecture: a(n) > 0 for all n > 108.

			a(26) = 3 because we have [25, 1], [22, 4] and [16, 10].

Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Prime Power
Eric Weisstein's World of Mathematics, Squarefree

Cf. A000469, A098983, A246547, A282192, A282290, A282318, A282947.

nmax = 120; CoefficientList[Series[(Sum[Boole[SquareFreeQ[k] && ! PrimeQ[k]] x^k, {k, 1, nmax}]) (Sum[Boole[PrimePowerQ[k] && ! PrimeQ[k]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

x='x+O('x^120); concat([0, 0, 0, 0, 0], Vec(sum(k=1, 120, (issquarefree(k) && !isprime(k))*x^k) * sum(k=1, 120, (isprimepower(k) && !isprime(k))*x^k))) \\ Indranil Ghosh, May 23 2017

G.f.: (Sum_{k>=1} x^A246547(k))*(Sum_{k>=1} x^A000469(k)).

A280616 Smallest m such that the m - s is a prime for exactly n distinct squarefree numbers s.

Original entry on oeis.org

3, 4, 9, 8, 16, 18, 26, 32, 24, 36, 42, 44, 48, 66, 70, 60, 74, 72, 94, 106, 84, 90, 102, 112, 130, 108, 126, 114, 166, 160, 150, 144, 184, 218, 174, 208, 168, 220, 138, 222, 232, 204, 216, 262, 302, 268, 234, 252, 246, 240, 264, 276, 306, 270, 340, 318, 294, 312, 342, 336, 406, 330, 324

Offset: 1

Juri-Stepan Gerasimov, Jan 06 2017

nonn

			a(1) = 3 because 3 - 1 = 2 is prime where 1 is squarefree number.
a(2) = 4 because 4 - 1 = 3 and 4 - 2 = 2 are primes where 1 and 2 are squarefree numbers.
a(3) = 9 because 9 - 2 = 7, 9 - 6 = 3, 9 - 7 = 2 are primes where 2, 6, 7 are squarefree numbers.

Cf. A004767, A005117, A098983, A279027.

cp's OEIS Frontend

A304081 Number of ways to write n as p + 2^k + (1+(n mod 2))*5^m, where p is an odd prime, and k and m are nonnegative integers with 2^k + (1+(n mod 2))*5^m squarefree.

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A282192 Number of ways of writing n as a sum of a prime and a squarefree semiprime.

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A282318 Number of ways of writing n as a sum of a prime and a nonprime squarefree number.

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A133364 Number of ways of writing n as a sum of a prime and a square-full number.

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A282290 Expansion of (Sum_{p prime, i>=2} x^(p^i))*(Sum_{j>=2} mu(j)^2*x^j), where mu() is the Moebius function (A008683).

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A283929 Number of ways of writing n as a sum of a twin prime (A001097) and a squarefree semiprime (A006881).

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A287299 Number of ways of writing n as a sum of a proper prime power (A246547) and a nonprime squarefree number (A000469).

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A304081 Number of ways to write n as p + 2^k + (1+(n mod 2))5^m, where p is an odd prime, and k and m are nonnegative integers with 2^k + (1+(n mod 2))5^m squarefree.

A282290 Expansion of (Sum_{p prime, i>=2} x^(p^i))(Sum_{j>=2} mu(j)^2x^j), where mu() is the Moebius function (A008683).