cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 11 results. Next

A131834 Indices of records in A100949.

Original entry on oeis.org

6, 9, 11, 17, 38, 51, 62, 88, 93, 98, 122, 148, 152, 188, 222, 232, 248, 266, 272, 296, 308, 326, 388, 398, 458, 488, 500, 518, 572, 602, 686, 692, 708, 860, 912, 972, 992, 1068, 1112, 1128, 1146, 1152, 1270, 1272, 1340, 1356, 1422, 1536, 1542, 1578
Offset: 1

Views

Author

Jonathan Vos Post, Oct 04 2007

Keywords

Examples

			a(15) = 222 because there are 22 partitions of n into a prime and a semiprime and that 22 is a record.
For n = 6, 9, 11, 17, 38, 51, 62, 88, 93, 98, 122, 148, 152, 188, 222, A100949(n) = 1, 2, 3, 5, 6, 8, 10, 11, 12, 13, 16, 17, 19, 21, 22.

Crossrefs

Cf. A000040, A001358, A061358, A100949.

Programs

  • Mathematica
    nPar[n_] := Length@ Select[Prime@ Range[ PrimePi@ n], PrimeOmega[n - #] == 2 &]; r = 0; L = {}; n = 2; While[Length[L] < 50, p = nPar[++n]; If[p > r, r = p; AppendTo[L, n]]]; L (* Giovanni Resta, Jun 19 2016 *)
DeleteDuplicates[Table[{n,Count[Sort/@(PrimeOmega/@IntegerPartitions[n,{2}]),{1,2}]},{n,1600}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]]//Rest (* Harvey P. Dale, Jun 14 2024 *)

Formula

Numbers n such that the number of partitions of n into a prime and a semiprime is a record.

Extensions

Data corrected by Giovanni Resta, Jun 19 2016

A152164 Erroneous version of A100949.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 3, 2, 2, 1, 3, 2, 5, 1, 2, 2, 3, 2, 4, 2, 3, 3, 5, 5, 4, 1, 2, 4, 5, 2, 4, 3, 5
Offset: 1

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Author

Keywords

References

  • Found on the internet.

A152197 Erroneous version of A100949.

Original entry on oeis.org

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

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Author

Keywords

References

  • Found on the internet.

A100951 Number of ways to write n = p*q + r with three distinct primes p, q and r.

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, Nov 23 2004

Keywords

Comments

a(n)<=A100950(n)<=A100949(n)<=Min{A000720(n), A072000(n)}.
a(n) = A000040(i) + A006881(j) for some i and j, such that A000040(i) is not a factor of A006881(j);
a(A100952(n)) = 0.
a(A160373(n)) = n and a(m) <> n for m < A160373(n). [From Reinhard Zumkeller, May 11 2009]

Examples

			A100949(21) = #{7+2*7, 11+2*5, 17+2*2} = 3,
a(21) = #{11+2*5} = 1.

Links

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

Views

Author

Ilya Gutkovskiy, Feb 15 2017

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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]

A100950 Number of partitions of n into a coprime pair of a prime and a semiprime.

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, Nov 23 2004

Keywords

Comments

A100951(n)<=a(n)<=A100949(n)<=Min{A000720(n), A072000(n)}.
a(n) = A000040(i) + A001358(j) for some i and j, such that A000040(i) is not a factor of A001358(j).

Examples

			A100949(21) = #{7+2*7, 11+2*5, 17+2*2} = 3,
a(21) = #{11+2*5, 17+2*2} = 2.

A100952 Numbers that cannot be written as p*q+r with three distinct primes p, q and r.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 30, 36, 42, 60
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 23 2004

Keywords

Comments

A100951(a(n)) = 0;
Conjecture: the sequence is complete.
A weaker conjecture: every integer greater than 60 (or some larger value based on further search) may be partitioned into a prime p and a semiprime qr, where the prime p is bounded by log(min(q,r)). Chen (1978) showed that all sufficiently large even numbers are the sum of a prime and the product of at most two primes. Zumkeller's conjecture effectively extends this from "even" to both even and odd integers. - Jonathan Vos Post, Nov 25 2004
Conjecture: Every positive integer can be represented as p*q-r with distinct primes p, q, r. - Zak Seidov, Aug 28 2012

Examples

			A100949(60) = #{11+7*7, 5+5*11, 3+3*19, 2+2*29} = 4, but A100951(60) = 0 as in each partition only 2 primes are used, therefore 60 is a term.

References

  • Chen, J.-R. "On the Representation of a Large Even Number as the Sum of a Prime and the Product of at Most Two Primes, II." Sci. Sinica 21, 421-430, 1978.

Crossrefs

Cf. A100949, A100951.

A282355 Expansion of (Sum_{i>=1} x^prime(prime(i)))*(Sum_{j = p*q, p prime, q prime} x^j).

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Feb 13 2017

Keywords

Comments

Number of ways of writing n as a sum of a prime with prime subscript (A006450) and a semiprime (A001358).
Every sufficiently large even number can be written as the sum of two primes, or a prime and a semiprime (Chen's theorem).
Conjecture: a(n) > 0 for all n > 527 (addition: only 18 positive integers cannot be represented as a sum of a prime number with prime subscript and a semiprime).

Examples

			a(9) = 2 because we have [6, 3] and [5, 4].

Links

Crossrefs

Cf. A001358, A006450, A100949.

Programs

  • Mathematica
    nmax = 110; CoefficientList[Series[Sum[x^Prime[Prime[k]], {k, 1, nmax}] Sum[Floor[PrimeOmega[k]/2] Floor[2/PrimeOmega[k]] x^k, {k, 2, nmax}], {x, 0, nmax}], x]

Formula

G.f.: (Sum_{i>=1} x^prime(prime(i)))*(Sum_{j = p*q, p prime, q prime} 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

Views

Author

Ilya Gutkovskiy, Mar 18 2017

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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]
  • PARI
    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

Formula

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

A152165 Largest number which is not the sum of an n-almost prime and a prime.

Original entry on oeis.org

10, 300, 60060, 3573570, 446185740
Offset: 2

Views

Author

Jonathan Vos Post, Mar 30 2009

Keywords

Comments

All the values are conjectural and untrustworthy. - N. J. A. Sloane, Oct 05 2009
For n=1 see A061358. - N. J. A. Sloane, Oct 05 2009

Crossrefs

Cf. A130588, A146295, A146296, A146297, A100949.
Showing 1-10 of 11 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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