A078425 Primes in "Ulam's Prime sequence". A prime is in the sequence iff p+1 can be expressed in exactly 1 way as the sum of 2 previous distinct primes.

3, 5, 7, 11, 13, 19, 29, 41, 43, 59, 83, 89, 107, 109, 127, 139, 157, 163, 173, 199, 211, 223, 257, 271, 277, 293, 307, 331, 347, 367, 397, 421, 443, 457, 491, 541, 557, 587, 601, 631, 691, 761, 769, 821, 911, 941, 971, 991, 1009, 1033, 1103, 1129, 1153, 1201

Offset: 1

Author

Jon Perry, Dec 29 2002

Keywords

• nonn

Comments

a(1) = 3, a(2) = 5; for n >= 3, a(n) is smallest prime which is uniquely a(j) + a(k) - 1, with 1<= j < k < n.

Is the (3,5) sequence finite or infinite? Note that (3,7) as a starting sequence has only 2 terms and (7,11) yields 7, 11, 17, 23, 29 only. Equally using -1 as a rule creates more variants.

The sequence continues at least up to a(2227) = 400031.

After about 500 terms, the graph of this sequences appears almost linear. - T. D. Noe, Jan 20 2008

Examples

			a(3)=7 as 8=3+5. a(4)=11 as 12=5+7 (and nothing else).
		

Links

• T. D. Noe, Table of n, a(n) for n=1..10000
• Index entries for Ulam numbers

Crossrefs

Cf. A002858 (Ulam numbers), A002859, A003666, A003667, A001857, A048951, A007300.

Programs

PARI

v=vector(1220);vc=2;v[1]=3;v[2]=5; forprime (p=7,1220,p1=p+1;pc=0;fl=0;for (i=1,vc-1, for (j=i+1,vc,if (v[i]+v[j]==p1,pc++);if (pc>1,fl=1);if (fl,break));if (fl,break));if (pc==0,fl=1);if (!fl,vc++;v[vc]=p));print(vecextract(v,concat("1..",vc)))

Extensions

Edited and extended by Klaus Brockhaus, Apr 14 2005