A210444
a(n) = |{0
0, 0, 1, 2, 0, 4, 1, 0, 2, 2, 0, 4, 0, 1, 4, 2, 0, 6, 1, 3, 2, 2, 0, 5, 2, 1, 3, 1, 2, 11, 0, 1, 4, 1, 2, 6, 0, 2, 4, 3, 1, 9, 2, 3, 4, 2, 0, 7, 1, 4, 4, 5, 0, 8, 4, 1, 3, 3, 0, 15, 0, 3, 4, 4, 4, 13, 2, 4, 2, 5, 2, 10, 0, 2, 11, 2, 3, 12, 0, 6, 6, 2, 2, 13, 3, 5, 7, 5, 1, 16, 4, 4, 6, 3, 2, 11, 0, 8, 6, 7
Offset: 1
Keywords
A230514 Number of ways to write n = a + b + c (0 < a <= b <= c) such that all the three numbers a*(a+1)-1, b*(b+1)-1, c*(c+1)-1 are Sophie Germain primes.
0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 3, 4, 3, 4, 4, 4, 3, 5, 4, 4, 4, 5, 4, 4, 2, 4, 4, 4, 2, 3, 2, 3, 2, 1, 2, 2, 3, 3, 3, 4, 5, 3, 2, 5, 6, 5, 5, 6, 5, 7, 9, 6, 7, 9, 9, 8, 10, 8, 8, 10, 7, 8, 10, 6, 9, 8, 6, 5, 8, 4, 7, 4, 4, 8, 7, 5, 3, 5, 3, 7, 3, 3, 5, 7, 5, 4, 6, 5, 6, 7, 5, 6, 10, 9, 6
Offset: 1
Keywords
Comments
Conjecture: a(n) > 0 for all n > 5.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Sep 22 2023
This implies that there are infinitely many Sophie Germain primes of the form x^2 + x - 1.
See also A230516 for a similar conjecture.
Examples
a(10) = 2 since 10 = 2 + 2 + 6 = 2 + 3 + 5, and 2*3 - 1 = 5, 6*7 - 1 = 41, 3*4 - 1 = 11, 5*6 - 1 = 29 are all Sophie Germain primes. a(39) = 1 since 39 = 9 + 15 + 15, and both 9*10 - 1 = 89 and 15*16 - 1 = 239 are Sophie Germain primes.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
- Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 2012-2017.
Programs
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Mathematica
pp[n_]:=PrimeQ[n(n+1)-1]&&PrimeQ[2n(n+1)-1] a[n_]:=Sum[If[pp[i]&&pp[j]&&pp[n-i-j],1,0],{i,1,n/3},{j,i,(n-i)/2}] Table[a[n],{n,1,100}]
A210452
Number of integers k
0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 1, 2, 2, 3, 3, 3, 1, 3, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 4, 2, 5, 5, 4, 5, 5, 2, 4, 5, 5, 1, 5, 2, 6, 6, 5, 2, 6, 6, 6, 6, 6, 2, 6, 6, 6, 6, 5, 2, 6, 3, 5, 7, 7, 7, 7, 3, 7, 7, 7, 3, 7, 4, 6, 8, 8, 8, 8, 3, 8, 8, 6, 3, 8, 8, 6, 8, 8, 3, 8, 8, 8, 7, 6, 8, 8, 3, 8, 8, 8
Offset: 1
Keywords
Comments
Conjecture: a(n)>0 for all n>4.
This implies the twin prime conjecture since k*p is not practical for any prime p>sigma(k)+1.
Zhi-Wei Sun also made the following conjectures:
(1) For each integer n>197, there is a practical number k
(2) For every n=9,10,... there is a practical number k
(3) For any integer n>26863, the interval [1,n] contains five consecutive integers m-2, m-1, m, m+1, m+2 with m-1 and m+1 both prime, and m-2, m, m+2, m*n all practical.
Examples
a(11)=1 since 5 and 7 are twin primes, and 6 and 6*11 are both practical.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
- G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
- Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.
- Zhi-Wei Sun, Sandwiches with primes and practical numbers, a message to Number Theory List, Jan. 13, 2013.
Crossrefs
Programs
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Mathematica
f[n_]:=f[n]=FactorInteger[n] Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2]) Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}] pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0) a[n_]:=a[n]=Sum[If[PrimeQ[k-1]==True&&PrimeQ[k+1]==True&&pr[k]==True&&pr[k*n]==True,1,0],{k,1,n-1}] Do[Print[n," ",a[n]],{n,1,100}]
A210722 Number of ways to write n = (2-(n mod 2))p+q+2^k with p, q-1, q+1 all prime, and p-1, p+1, q all practical.
0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 1, 4, 2, 4, 2, 4, 3, 4, 4, 3, 4, 5, 3, 8, 5, 8, 4, 5, 4, 5, 3, 5, 4, 4, 3, 9, 2, 12, 4, 9, 5, 7, 6, 7, 5, 5, 7, 10, 5, 13, 6, 10, 6, 8, 6, 7, 5, 1, 7, 7, 1, 10, 5, 8, 4, 9, 7, 8, 6, 3, 10, 6, 6, 10, 7, 7, 9, 11, 7, 10, 10, 5, 10, 7, 5, 10, 7, 4, 8, 8, 5, 11, 5, 8, 10, 7, 5, 12, 5
Offset: 1
Keywords
Comments
Conjecture: a(n)>0 except for n = 1,...,8, 10, 520, 689, 740.
Zhi-Wei Sun also guessed that any integer n>6 different from 407 can be written as p+q+F_k, where p is a prime with p-1 and p+1 practical, q is a practical number with q-1 and q+1 prime, and F_k (k>=0) is a Fibonacci number.
Examples
a(1832)=1 since 1832=2*881+6+2^6 with 5, 7, 881 all prime and 6, 880, 882 all practical. a(11969)=1 since 11969=127+11778+2^6 with 127, 11777, 11779 all prime and 126, 128, 11778 all practical.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
- G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
- Zhi-Wei Sun, Sandwiches with primes and practical numbers, a message to Number Theory List, Jan. 13, 2013.
- Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.
Crossrefs
Programs
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Mathematica
f[n_]:=f[n]=FactorInteger[n] Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2]) Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}] pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0) pp[k_]:=pp[k]=pr[Prime[k]-1]==True&&pr[Prime[k]+1]==True pq[n_]:=pq[n]=PrimeQ[n-1]==True&&PrimeQ[n+1]==True&&pr[n]==True a[n_]:=a[n]=Sum[If[pp[j]==True&&pq[n-2^k-(2-Mod[n,2])Prime[j]]==True,1,0],{k,0,Log[2,n]},{j,1,PrimePi[(n-2^k)/(2-Mod[n,2])]}] Do[Print[n," ",a[n]],{n,1,100}]
A211165 Number of ways to write n as the sum of a prime p with p-1 and p+1 both practical, a prime q with q+2 also prime, and a Fibonacci number.
0, 0, 0, 0, 0, 1, 1, 3, 3, 4, 5, 3, 5, 3, 4, 4, 3, 4, 4, 4, 6, 6, 8, 6, 8, 3, 7, 3, 6, 5, 5, 5, 7, 6, 11, 8, 12, 4, 8, 4, 7, 8, 6, 8, 8, 7, 11, 9, 13, 5, 8, 4, 7, 7, 6, 6, 6, 5, 7, 6, 10, 4, 9, 3, 9, 7, 8, 7, 6, 6, 7, 4, 7, 4, 7, 4, 8, 8, 11, 7, 6, 6, 8, 5, 6, 4, 7, 2, 9, 7, 12, 8, 7, 4, 10, 5, 9, 5, 8, 5
Offset: 1
Keywords
Comments
Conjecture: a(n)>0 for all n>5.
This has been verified for n up to 300000.
Note that for n=406 we cannot represent n in the given way with q+1 practical.
Examples
a(6)=a(7)=1 since 6=3+3+0 and 7=3+3+1 with 3 and 5 both prime, 2 and 4 both practical, 0 and 1 Fibonacci numbers.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
- G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210 [MR96i:11106].
- Zhi-Wei Sun, New Goldbach-type conjectures involving primes and practical numbers, a message to Number Theory List, Jan. 29, 2013.
- Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2017.
Crossrefs
Programs
-
Mathematica
f[n_]:=f[n]=FactorInteger[n] Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2]) Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}] pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0) pp[k_]:=pp[k]=pr[Prime[k]-1]==True&&pr[Prime[k]+1]==True q[n_]:=q[n]=PrimeQ[n]==True&&PrimeQ[n+2]==True a[n_]:=a[n]=Sum[If[k!=2&&Fibonacci[k]
A261963 Smallest number that can be written as the sum of a Sophie Germain prime (A005384) and a practical number (A005153) in exactly n ways.
2, 3, 6, 23, 31, 35, 65, 59, 95, 131, 173, 233, 203, 239, 257, 299, 317, 323, 473, 443, 635, 563, 671, 719, 809, 701, 779, 839, 803, 1109, 971, 1049, 1139, 1343, 1103, 1409, 1433, 1607, 1481, 1499, 1559, 1523, 1769, 1679, 1643, 2069, 2063, 2309, 2111, 2141
Offset: 0
Keywords
Examples
23 can be written as the sum of a Sophie Germain prime and a practical number in the following three ways: 3 + 20, 5 + 18, 11 + 12. Since 23 is the smallest number that can be expressed like that in exactly three ways, a(3) = 23.
Programs
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PARI
\\ First define the function is_a005153(n) as in A005153 is_a005384(n) = ispseudoprime(n) && ispseudoprime(2*n+1) count(n) = x=1; y=n-1; i=0; while(y > n/2, if((is_a005153(x) && is_a005384(y)) || (is_a005153(y) && is_a005384(x)), i++); x++; y--); i a(n) = k=2; while(count(k)!=n, k++); k
Comments
Examples
Links
Crossrefs
Programs
Mathematica
f[n_]:=f[n]=FactorInteger[n] Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2]) Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}] pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0) a[n_]:=a[n]=Sum[If[PrimeQ[k*n-1]==True&&PrimeQ[k*n+1]==True&&pr[k*n]==True,1,0],{k,1,n-1}] Do[Print[n," ",a[n]],{n,1,100}]