A100951 -id:A100951 - cp's OEIS frontend

A100949 Number of partitions of n into a prime and a semiprime.

0, 0, 0, 0, 0, 1, 1, 1, 2, 0, 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, 7, 13, 9, 9, 6

Offset: 1

Reinhard Zumkeller, Nov 23 2004

Marnell conjectures that a(n) > 0 for n > 10 after analyzing "many thousands of whole numbers". I find no exceptions below 100 million. - Charles R Greathouse IV, May 04 2010

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

Geoffrey R. Marnell, "Ten Prime Conjectures", Journal of Recreational Mathematics 33:3 (2004-2005), pp. 193-196.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A061358, A000040, A001358.

Cf. A010051.

a100949 n = sum $ map (a010051 . (n -)) $ takeWhile (< n) a001358_list
-- Reinhard Zumkeller, Jun 26 2013

Table[Count[Sort/@(PrimeOmega/@IntegerPartitions[n,{2}]),{1,2}],{n,110}] (* Harvey P. Dale, Mar 25 2018 *)

list(lim)=my(p=primes(primepi(lim)),sp=select(n->bigomega(n)==2, vector(lim\1,i,i)),x=O('x^(lim\1+1))+'x); concat([0,0,0,0,0], Vec(sum(i=1,#p,x^p[i])*sum(i=1,#sp,x^sp[i]))) \\ Charles R Greathouse IV, Jun 14 2013

A100951(n) <= A100950(n) <= a(n) <= min(A000720(n), A072000(n)).

a(n) = Sum_{i=1..floor(n/2)} A010051(i) * A064911(n-i) + A010051(n-i) * A064911(i). - Wesley Ivan Hurt, May 02 2019

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

Reinhard Zumkeller, Nov 23 2004

nonn

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).

			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

Reinhard Zumkeller, Nov 23 2004

nonn

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

			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.

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.

Cf. A100949, A100951.

A160373 Smallest number m such that exactly n triples (p,q,r) of distinct primes exist with m=p*q+r.

Original entry on oeis.org

1, 11, 13, 23, 17, 37, 53, 62, 81, 99, 93, 105, 118, 122, 148, 152, 165, 166, 208, 224, 214, 225, 232, 250, 284, 285, 308, 314, 332, 346, 326, 382, 388, 400, 448, 476, 458, 494, 454, 518, 520, 478, 525, 530, 578, 598, 640, 602, 632, 716, 634, 740, 710, 692

Offset: 0

Reinhard Zumkeller, May 11 2009

nonn

A100951(a(n)) = n and A100951(m) <> n for m < a(n);
a(42) = 525 seems to be the largest odd term.
Note that switching p and q does not make a different triple. - Robert Israel, Mar 09 2018

			A100951(37) = #{2*3+31,2*7+23,2*13+11,2*17+3,5*7+2} = 5.

Robert Israel, Table of n, a(n) for n = 0..6567 (n=0..500 from Reinhard Zumkeller)

N:= 10^4: # to get terms before the first term > N
Primes:= select(isprime, [2, seq(i,i=3..N,2)]):
V:= Vector(N):
for r in Primes do
  for j from 1 while Primes[j]^2 <= N do
    p:= Primes[j];
    if p = r then next fi;
    for k from j+1 while p*Primes[k]+r <= N do
      q:= Primes[k];
      if q = r then next fi;
      V[p*q+r]:= V[p*q+r]+1;
    od
  od
od:
mv:= max( V):
F:= Vector(mv):
for i from 1 to N do
  if V[i] > 0 and F[V[i]] = 0 then F[V[i]]:= i fi
od:
F0:= min(select(t -> F[t] = 0, [$1..max(V)])):
1, seq(F[i],i=1..F0-1); # Robert Israel, Mar 09 2018
N:= 10^4: # to get terms before the first term > N
Primes:= select(isprime, [2, seq(i,i=3..N,2)]):
V:= Vector(N):
for r in Primes do
  for j from 1 while Primes[j]^2 <= N do
    p:= Primes[j];
    if p = r then next fi;
    for k from j+1 to nops(Primes) while p*Primes[k]+r <= N do
      q:= Primes[k];
      if q = r then next fi;
      V[p*q+r]:= V[p*q+r]+1;
    od
  od
od:
mv:= max( V):
F:= Vector(mv):
for i from 1 to N do
  if V[i] > 0 and F[V[i]] = 0 then F[V[i]]:= i fi
od:
F0:= min(select(t -> F[t] = 0, [$1..max(V)])):
if F0 = infinity then F0:= mv fi:
1, seq(F[i],i=1..F0-1); # Robert Israel, Mar 09 2018

A347933 Positive integers that can't be written in the form a+b+c for some positive integers a, b, and c satisfying gcd(a,b)=1, gcd(a,c)>1, and gcd(b,c)>1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, 24, 30, 36, 42, 48, 60, 84, 90, 210

Offset: 1

José Hernández, Sep 20 2021

			1 and 2 are the first two terms of the sequence because they can't even be written as sums of three positive integers.
3 is the third term of the sequence because there is only one way to express it as a sum of three positive integers (1+1+1).
11 does not belong to the sequence because 11=2+3+6 (and gcd(2,3)=1, gcd(2,6)=2, and gcd(3,6)=3).

Freddy Barrera, Bernardo Recamán, and Stan Wagon, Sums of triples with one pair relatively prime. American Mathematical Monthly, 127 (2020), no. 1, pp. 89-90.

Cf. A224535, A100951, A100952.

Select[Range@210,Select[Flatten[Permutations/@IntegerPartitions[#,{3}],1],GCD[#[[1]],#[[2]]]==1&&GCD[#[[1]],#[[3]]]>1&&GCD[#[[2]],#[[3]]]>1&]=={}&] (* Giorgos Kalogeropoulos, Sep 20 2021 *)

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A100949 Number of partitions of n into a prime and a semiprime.

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

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

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A160373 Smallest number m such that exactly n triples (p,q,r) of distinct primes exist with m=p*q+r.

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A347933 Positive integers that can't be written in the form a+b+c for some positive integers a, b, and c satisfying gcd(a,b)=1, gcd(a,c)>1, and gcd(b,c)>1.

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