A098048
Numbers k such that A098037(k) sets a new record. A098037 is the number of prime divisors (counting multiplicity) of the sums of two consecutive primes.
Original entry on oeis.org
1, 2, 5, 16, 18, 43, 162, 190, 532, 564, 3314, 3908, 10499, 30789, 53828, 62946, 278737, 364195, 629686, 3768344, 7827416, 9496221, 23159959, 68035462, 92566977, 370110663, 648634305, 4032924162, 7748432257, 17324573653, 19879357650, 34988238805
Offset: 1
-
s = PrimeOmega[Total[#]] & /@ Partition[Prime[Range[10^5]], 2, 1]; Map[FirstPosition[s, #][[1]] &, Union@FoldList[Max, s]] (* Amiram Eldar, Feb 07 2020 *)
With[{nn=10^6},DeleteDuplicates[Thread[{Range[nn-1],PrimeOmega[Total[#]]&/@Partition[Prime[ Range[nn]],2,1]}],GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]]] (* The program generates the first 19 terms of the sequence. *) (* Harvey P. Dale, Sep 20 2024 *)
A251600
Least k such that prime(k) + prime(k+1) contains n prime divisors (with multiplicity), otherwise 0.
Original entry on oeis.org
1, 0, 2, 5, 16, 20, 18, 43, 162, 190, 532, 916, 564, 3314, 3908, 10499, 30789, 53828, 153384, 62946, 278737, 364195, 629686, 3768344, 7827416, 9496221, 23159959, 184328920, 68035462, 92566977, 457932094, 370110663, 648634305, 4032924162, 7841376455
Offset: 1
a(5) = 16 because prime(16) + prime(17) = 53 + 59 = 112 = 7*2^4 with 5 prime divisors.
-
A251600 = {1, 0}; Do[k = 1; While[PrimeOmega[Prime[k] + Prime[k + 1]] != n, k++]; AppendTo[A251600, k], {n, 3, 10}]; A251600
A105418
Smallest prime p such that the sum of it and the following prime has n prime factors including multiplicity, or 0 if no such prime exists.
Original entry on oeis.org
2, 0, 3, 11, 53, 71, 61, 191, 953, 1151, 3833, 7159, 4093, 30713, 36857, 110587, 360439, 663547, 2064379, 786431, 3932153, 5242877, 9437179, 63700991, 138412031, 169869311, 436207613, 3875536883, 1358954453, 1879048183, 10066329587, 8053063661, 14495514619
Offset: 1
a(5) = 53 because (53 + 59) = 112 = 2^4*7.
a(24) = 63700991 because (63700991 + 63700993) = 127401984 = 2^19*3^5.
a(28) = 3875536883 because (3875536883 + 3875536909) = 7751073792 = 2^25*3*7*11.
a(29) = 1358954453 because (1358954453 + 1358954539) = 2717908992 = 2^25*3^4.
a(30) = 1879048183 because (1879048183 + 1879048201) = 3758096384 = 2^29*7.
-
f[n_] := Plus @@ Flatten[ Table[ #[[2]], {1}] & /@ FactorInteger[n]]; t = Table[0, {40}]; Do[a = f[Prime[n] + Prime[n + 1]]; If[a < 41 && t[[a]] == 0, t[[a]] = Prime[n]; Print[{a, Prime[n]}]], {n, 111500000}]; t
-
almost_primes(A, B, n) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p, ceil(A/m)), B\m, listput(list, m*q)), forprime(q=p, sqrtnint(B\m, n), list=concat(list, f(m*q, q, n-1)))); list); vecsort(Vec(f(1, 2, n)));
a(n) = if(n==2, return(0)); my(x=2^n, y=2*x); while(1, my(v=almost_primes(x, y, n)); for(k=1, #v, my(p=precprime(max(v[k]>>1, 2)), q=nextprime(p+1)); if(p+q == v[k], return(p))); x=y+1; y=2*x); \\ Daniel Suteu, Aug 06 2024
A230518
Smallest prime p = a(n) such that the sum of p and the next prime has n distinct prime factors.
Original entry on oeis.org
2, 5, 13, 103, 1783, 15013, 285283, 9699667, 140645501, 4127218087, 100280245063, 5625398263453, 202666375276361, 11602324073775431, 438272504610946003, 21828587281891445047, 1156915125940246587913, 66595945348137856405747, 4632891063696575353839163
Offset: 1
30 = 13+17 is the earliest case with 3 prime divisors, so a(3) = 13.
-
Clear[a]; a[_] = 0; Do[p = Prime[k]; q = Prime[k+1]; n = PrimeNu[p+q]; If[a[n] == 0, a[n] = p; Print["a(", n, ") = p = ", p, ", q = ", q]], {k, 1, 10^9}]; Table[a[n], {n, 1, 10}]
-
a(n) = {p = 2; while (omega(p+nextprime(p+1)) != n, p = nextprime(p+1)); p;} \\ Michel Marcus, Oct 22 2013
-
step(Fvec)=my([n,f]=Fvec,v=List(),t);for(i=1,#f~,t=f;t[i,2]++;listput(v,[n*f[i,1],t]);t=f;t[i,1]=nextprime(t[i,1]+1);if(i==#f~||t[i,1]1,1,prime(i))),v=[[factorback(f),f]],t); if(!bad(v[1][1]),return(precprime(v[1][1]/2))); v=vecsort(step(v[1]),1); while(bad(v[1][1]), v=vecsort(concat(step(v[1]),v[2..#v]),1,8)); precprime(v[1][1]/2); \\ Charles R Greathouse IV, Oct 22 2013
Showing 1-4 of 4 results.
Comments