A100570 -id:A100570 - cp's OEIS frontend

A241922 Smallest k^2>=0 such that n-k^2 is semiprime, or a(n)=2 if there is no such k^2.

2, 2, 2, 0, 1, 0, 1, 4, 0, 0, 1, 2, 4, 0, 0, 1, 2, 4, 4, 16, 0, 0, 1, 9, 0, 0, 1, 2, 4, 4, 9, 2, 0, 0, 0, 1, 4, 0, 0, 1, 16, 4, 4, 9, 36, 0, 1, 9, 0, 1, 0, 1, 4, 16, 0, 1, 0, 0, 1, 9, 4, 0, 1, 9, 0, 1, 9, 64, 0, 1, 9, 2, 4, 0, 1, 25, 0, 1, 64, 25, 4, 0, 1, 49, 0, 0, 0, 1, 4, 4, 0, 1, 0, 0, 0, 1, 4, 4, 4, 9, 16

Offset: 1

Vladimir Shevelev, May 01 2014

nonn

If n = m^2, m>=2, then the condition {a(n) differs from 2} is equivalent to the Goldbach binary conjecture. Indeed, if m^2 - k^2 is semiprime, then (m-k)*(m+k) = p*q, where p<=q are primes. Here we consider two possible cases. 1) m-k=1, m+k=p*q and 2) m-k=p, m+k=q. But in the first case k=m-1>m-p, i.e., more than k in the second case. In view of the minimality k, we only have to consider case 2). In this case we have m-/+k both are primes p<=q (with equality in case k=0) and thus 2*m = p + q. Conversely, let the Goldbach conjecture be true. Then for a perfect square n>=4, we have 2*sqrt(n)=p+q (p<=q are both primes). Thus n=((p+q)/2)^2 and n-((p-q)/2)^2=p*q is semiprime. Hence a(n) is a square not exceeding ((p-q)/2)^2.
Note that a(n)=2 for 1,2,3,12,17,28,32,72,...
All these numbers are in A100570. Thus the Goldbach binary conjecture is true if and only if A100570 does not contain perfect squares.
The largest term found in the first 2^28 terms is a(106956964) = 369^2 = 136161. This further encourages one to believe that Goldbach's binary conjecture holds true. - Daniel Mikhail, Nov 23 2020

Peter J. C. Moses, Table of n, a(n) for n = 1..1000
Daniel Mikhail, Lists of up to the first 15 integers that are a squared distance, k^2, away from a semiprime for all k's found between [5..2^28]

Cf. A000290, A001358, A100570, A152522, A152451, A156537.

a(n) = {my(lim = if (issquare(n), sqrtint(n)-1, sqrtint(n))); for (k=0, lim, if (bigomega(n-k^2) == 2, return (k^2));); return (2);} \\ Michel Marcus, Nov 26 2020

a(A001358(n)) = 0.

A100592 Least positive integer that can be represented as the sum of exactly two semiprimes in exactly n ways.

Original entry on oeis.org

1, 8, 18, 30, 43, 48, 60, 72, 91, 108, 132, 155, 120, 144, 192, 168, 216, 236, 227, 180, 320, 340, 240, 252, 348, 300, 324, 336, 488, 484, 456, 396, 614, 360, 524, 548, 706, 468, 536, 656, 628, 420, 624, 576, 612, 588, 540, 600, 648, 768, 732, 800, 832, 660

Offset: 0

Jonathan Vos Post, Nov 30 2004

A072931(a(n)) = n and A072931(m) < n for m < a(n). [From Reinhard Zumkeller, Jan 21 2010]

			a(0) = 1 because 1 is the smallest positive integer that cannot be represented as sum of two semiprimes (since 4 is the smallest semiprime). a(1) = 8 because 8 is the smallest such sum of two semiprimes: 4 + 4. Similarly a(2) = 18 because 18 = 14 + 4 = 9 + 9 where {4,9,14} are semiprimes and there is no third such sum for 18.

Reinhard Zumkeller and Zak Seidov, Table of n, a(n) for n = 0..1000 (Terms 0-250 from Reinhard Zumkeller)

Cf. A001358, A076768, A100570, A072966.

a(n) = min{i such that i = A001358(j) + A001358(k) in n ways}.

A241927 Smallest k^2>=1 such that n-k^2 is semiprime p*q in Fermi-Dirac arithmetic (A176525) with additional requirement that, if n is a square, then p and q are of the same parity; or a(n)=2 if there is no such k^2.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 1, 2, 1, 4, 1, 4, 1, 4, 1, 1, 9, 4, 1, 2, 1, 1, 1, 4, 4, 4, 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 1, 4, 1, 1, 9, 4, 4, 9, 1, 1, 1, 4, 4, 4, 1, 1, 1, 4, 4, 1, 9, 1, 1, 9, 4, 4, 1, 1, 1, 1, 4, 4, 1, 1, 9, 4, 4, 9, 1, 1, 1, 1, 4, 4, 4, 25, 1, 4, 9, 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 1, 4, 4, 1, 1, 1, 4, 4, 4, 25

Offset: 1

Vladimir Shevelev, May 02 2014

nonn

A semiprime in Fermi-Dirac arithmetic is a product of two distinct terms of A050376, or, equivalently, an infinitary semiprime. The conjecture that every even number>=4 is a sum of two A050376 terms is a weaker form of the Goldbach conjecture; as such, it is natural to refer to it as a Goldbach conjecture in Fermi-Dirac arithmetic (FDGC).
Let us prove that the condition {a(m^2) differs from 2} is equivalent to the FDGC.
Indeed, from the FDGC for a perfect square n>=4, we have 2*sqrt(n)=p+q (pA050376 terms of the same parity). Thus n=((p+q)/2)^2 and n-((p-q)/2)^2=p*q is Fermi-Dirac semiprime. Hence, a(n)>=1 is a square not exceeding ((p-q)/2)^2. Thus the condition {a(m^2) differs from 2} is necessary for the truth of the FDGC.
Let us prove that the condition {a(m^2) differs from 2} is also sufficient. Indeed, a(m^2)-k^2 = p*q, where, say, pA050376, and p,q are of the same parity. If p,q are primes, then the proof repeats one in A241922. Let, e.g., p=s^2A050376). Consider two principal cases: 1) m-k = s, m+k = s*q; 2) m-k = s^2, m+k = q. In 1) k=m-s, in 2) k=m-s^2. In view of the minimality of k, we should accept 2) and thus m-k=p, m+k=q. So, 2*m=p+q as the FDGC requires.
The sequence of numbers n for which a(n)=2 begins 1, 2, 3, 4, 5, 6, 8, 20, ... (A241947).

			a(17)=9, since 9 is the smallest square such that 17-9 = 8 = 2*4 is a Fermi-Dirac semiprime.

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 (in Russian; MR 2000f: 11097, pp. 3912-3913).

Peter J. C. Moses, Table of n, a(n) for n = 1..10000

Cf. A000290, A001358, A050376, A176525, A100570, A152522, A152451, A156537, A241922.

A101181 Least positive integer that can be represented as sum of a semiprime and a square in exactly n ways.

Original entry on oeis.org

1, 4, 13, 10, 26, 50, 58, 74, 146, 159, 218, 302, 290, 458, 515, 647, 650, 794, 962, 986, 1178, 1403, 1322, 1418, 1658, 1898, 1802, 2126, 1970, 2210, 3062, 2930, 3143, 3263, 3482, 3527, 4142, 4667, 4010, 4562, 5123, 4955, 5018, 6242, 5330, 6695, 7178, 7103

Offset: 0

Jonathan Vos Post and Ray Chandler, Dec 14 2004

			a(0) = 1 because 1 is the smallest positive integer that cannot be represented as sum of a semiprime and a square (since 4 is the smallest semiprime).
a(1) = 4 = 4 + 0; a(2) = 13 = 4 + 9 = 9 + 4; a(3) = 10 = 6 + 4
= 9 + 1 = 10 + 0.

Donovan Johnson, Table of n, a(n) for n = 0..1000

Cf. A000290, A001358, A100570.

mx=4345802; v=vector(mx); sp=vector(856467); c=0; for(i=4, mx, if(bigomega(i)==2, c++; sp[c]=i)); for(i=0, 2084, sq=i^2; for(j=1, c, s=sq+sp[j]; if(s<=mx, v[s]++, next(2)))); n=vector(1049); for(i=4, mx, if(v[i]>0, if(n[v[i]]==0, n[v[i]]=i))); for(i=1, 1000, write("b101181.txt", i " " n[i])) /* Donovan Johnson, Feb 04 2013 */

a(n) = min(i such that i = A001358(j) + A000290(k) in n ways).

A241947 Numbers n for which A241927(n) = 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 20

Offset: 1

Vladimir Shevelev, May 03 2014

If the sequence contains no perfect squares>4, then the Goldbach conjecture in Fermi-Dirac arithmetic (FDGC) is true (see comment in A241927).
Essentially, the sequence is the Fermi-Dirac analog of A100570. Since A100570 is conjecturally finite, it is natural to suppose that this sequence is also finite.
There is not another term up to 10^6. - Peter J. C. Moses, May 05 2014
Thus, if 20 is the last term of the sequence, then the FDGC is true. - Vladimir Shevelev, May 05 2014

Cf. A241927, A241922, A100570.

Name corrected by Michel Marcus, Dec 14 2018

A111908 Numbers that are not the sum of a prime and a nonzero triangular number.

Original entry on oeis.org

1, 2, 7, 36, 61, 105, 171, 210, 211, 216, 325, 351, 406, 528, 561, 630, 741, 780, 990, 1081, 1176, 1275, 1596, 1711, 1830, 1953, 2016, 2145, 2346, 2628, 2775, 3003, 3081, 3240, 3321, 3655, 3741, 3916, 4278, 4371, 4465, 4560, 4851, 5253, 5460, 5565, 5886

Offset: 1

Stefan Steinerberger, Nov 25 2005

nonn

Can anybody prove or disprove a(n) = O(n^c) for some constant c?

Jonathan Vos Post has observed that every term in A076768 also occurs in this sequence.

			7 = 1+6 = 2+5 = 3+4; 7 is in the sequence because there is no sum where the first element is a prime and the second one a triangular number.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A064233, A000217, A076768, A020756, A072386, A014089, A014090, A000404, A001481, A002654, A100570, A101181.

lim=6000;plim=PrimePi[lim];tlim=Ceiling[Sqrt[2lim]];Complement[Range[lim],Union[Flatten[Table[Prime[i]+PolygonalNumber[j],{i,plim},{j,tlim}]]]] (* James C. McMahon, Jun 04 2024 *)

a(47) and offset corrected by Donovan Johnson, Feb 09 2013

A100591 Least positive integer that can be represented as sum of semiprime and a triangular number in exactly n ways. Triangular numbers include t(0)=0 and (1)=1.

Original entry on oeis.org

1, 4, 7, 10, 35, 25, 49, 61, 121, 140, 211, 268, 224, 392, 472, 517, 565, 529, 707, 1006, 1039, 994, 1213, 989, 1274, 1717, 1769, 1822, 2047, 2272, 2419, 2573, 2642, 3029, 3149, 3152, 3848, 3359, 4199, 4019, 4307, 4847, 5027, 4877, 5492, 6077

Offset: 0

Jonathan Vos Post, Nov 30 2004

Computed by Ray Chandler.

			a(0) = 1 because 1 is the smallest positive integer that cannot be represented as sum of semiprime and a triangular number (since 4 is the smallest semiprime). a(1) = 4 because 4 is the smallest such sum, namely semiprime(1)=4 + t(0)=0. Similarly a(2) = 7 because 7 = 4 + 3 and 7 = 6 + 1, where 4 and 6 are semiprimes, 3 and 1 are triangular.

Cf. A000217, A001358, A076768, A100570.

a(n) = min(i such that i = A001358(j) + A000217(k) in n ways).

A100593 Greatest positive integer that can be represented as the sum of exactly two semiprimes in exactly n ways.

Original entry on oeis.org

33, 62, 105, 122, 135, 174, 285, 214, 294, 315, 318, 366, 525, 405, 394, 474, 498, 585, 495, 529, 765, 645, 735, 693, 945, 705, 761, 825, 1155, 1109, 901, 989, 1049, 1123, 1365, 1063, 1121, 1181, 1243, 1129, 1231, 1169, 1349, 1485, 1399, 1577

Offset: 0

Jonathan Vos Post, Nov 30 2004

nonn

			a(0) = 33 is only a conjecture, based on computer searches by Lior Manor and by Ray Chandler. a(1) = 62 because 62 = 58 + 4 is the only way to partition 62 into two semiprimes.

Cf. A001358, A076768, A100570, A072966.

a(n) = max{i such that i = A001358(j) + A001358(k) in n ways}.

cp's OEIS Frontend

A241922 Smallest k^2>=0 such that n-k^2 is semiprime, or a(n)=2 if there is no such k^2.

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A100592 Least positive integer that can be represented as the sum of exactly two semiprimes in exactly n ways.

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A241927 Smallest k^2>=1 such that n-k^2 is semiprime p*q in Fermi-Dirac arithmetic (A176525) with additional requirement that, if n is a square, then p and q are of the same parity; or a(n)=2 if there is no such k^2.

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A101181 Least positive integer that can be represented as sum of a semiprime and a square in exactly n ways.

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A241947 Numbers n for which A241927(n) = 2.

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A111908 Numbers that are not the sum of a prime and a nonzero triangular number.

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A100591 Least positive integer that can be represented as sum of semiprime and a triangular number in exactly n ways. Triangular numbers include t(0)=0 and (1)=1.

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Formula

A100593 Greatest positive integer that can be represented as the sum of exactly two semiprimes in exactly n ways.

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Author

Keywords

Examples

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