A125312 Moessner triangle based on primes.

2, 3, 5, 10, 21, 13, 48, 105, 80, 29, 264, 628, 553, 232, 47, 1730, 4378, 4235, 2059, 543, 73, 13024, 34620, 36078, 19553, 6063, 1095, 107, 110542, 306362, 339554, 200769, 70350, 15166, 2000, 151, 1044900, 3003012, 3507070, 2228398, 861305, 212514

Offset: 1

Author

Gary W. Adamson, Dec 10 2006

Keywords

• nonn
• tabl

Comments

Row sums are 2, 8, 44, 262, 1724, 13024, ... Conjecture: log row n-th sum tends to (2n-3) + some unknown fractional part. E.g., log 1724 = 7.45... while log 13024 = 9.43... Right border = A011756.

Examples

			First few rows of the triangle are:
     2;
     3,    5;
    10,   21,   13;
    48,  105,   80,   29;
   164,  628,  553,  232,  47;
  1736, 4378, 4235, 2059, 543, 73;
  ...
		

References

• J. H. Conway and R. K. Guy, "The Book of Numbers", Springer-Verlag, 1996, p. 64.

Links

• Joshua Zucker, Table of n, a(n) for n = 1..55
• G. S. Kazandzidis, On a conjecture of Moessner and a general problem, Bull. Soc. Math. Grèce (N.S.) 2 (1961), 23-30.
• Dexter Kozen and Alexandra Silva, On Moessner's theorem, Amer. Math. Monthly 120(2) (2013), 131-139.
• R. Krebbers, L. Parlant, and A. Silva, Moessner's theorem: an exercise in coinductive reasoning in Coq, Theory and practice of formal methods, 309-324, Lecture Notes in Comput. Sci., 9660, Springer, 2016.
• Calvin T. Long, Strike it out--add it up, Math. Gaz. 66 (438) (1982), 273-277.
• Alfred Moessner, Eine Bemerkung über die Potenzen der natürlichen Zahlen, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 29, 1951.
• Ivan Paasche, Ein neuer Beweis des Moessnerschen Satzes S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 1952 (1952), 1-5 (1953). [Two years are listed at the beginning of the journal issue.]
• Ivan Paasche, Beweis des Moessnerschen Satzes mittels linearer Transformationen, Arch. Math. (Basel) 6 (1955), 194-199.
• Ivan Paasche, Eine Verallgemeinerung des Moessnerschen Satzes, Compositio Math. 12 (1956), 263-270.
• Hans Salié, Bemerkung zu einem Satz von A. Moessner, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 1952 (1952), 7-11 (1953). [Two years are listed at the beginning of the journal issue.]
• Oskar Perron, Beweis des Moessnerschen Satzes, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 31-34, 1951.

Crossrefs

Cf. A125714, A125777, A011756.

Formula

Begin with the primes and circle every (n*(n+1)/2)-th prime: 1, 5, 13, 29, 47, ... = A011756. Following the instructions in A125714, take partial sums of the uncircled terms, making this row 2. Circle the terms in row 2 one place to the left of row 1 terms. Take partial sums of the uncircled terms, continuing with analogous procedures for subsequent rows.

Extensions

Corrected and extended by Joshua Zucker, Jun 17 2007