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A096333 Prime numbers that are 2 less than a prime-indexed odd triangular number or 1 more than a prime-indexed even triangular number. 2
7, 13, 29, 67, 89, 151, 191, 277, 433, 701, 859, 947, 1129, 1429, 1889, 2557, 2699, 4003, 4751, 5779, 8647, 11173, 12401, 13367, 14029, 16111, 18719, 19501, 22367, 24977, 27259, 31627, 33151, 36313, 36857, 38501, 39619, 47279, 49139, 56951 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

REFERENCES

David Wells, The Penguin Dictionary of Curious & Interesting Numbers. In the entry for 496 he remarks that 496 is the smallest counterexample to the conjecture that an even, prime-indexed triangular plus 1 equals a prime, since 497 is not prime.

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000

FORMULA

Given the numbers of A034953, triangular numbers with prime indices, subtract 2 from the odd numbers on the list and add 1 to the even numbers on the list, then remove from the list the composite numbers.

EXAMPLE

a(2) = 13 because 15 is the 5th triangular number and since it is odd and we take 2 away from it, it yields the prime 13.

a(3) = 29 because 28 is the 7th triangular number and since it is even and we add 1 to it, it yields the prime 29.

497 is not on the list because although 496 is the 31st triangular number, but 496 + 1 = 7 * 71.

That sequence continues: 1771, 2279, 3161, 3487, 5149, 5357, 5993, 6439, 8129, 9451, 9731, ....

MATHEMATICA

tri[n_] := n(n + 1)/2; tp = Table[ tri[ Prime[n]], {n, 2, 70}]; f[n_] := If[ OddQ[n], n - 2, n + 1]; Select[f /@ tp, PrimeQ[ # ] &] (* Robert G. Wilson v, Aug 12 2004 *)

Select[If[OddQ[#], #-2, #+1]&/@Table[(n(n+1))/2, {n, Prime[Range[ 100]]}], PrimeQ] (* Harvey P. Dale, Sep 19 2016 *)

CROSSREFS

Cf. A034953.

Sequence in context: A082221 A230460 A182624 * A133325 A336117 A283709

Adjacent sequences:  A096330 A096331 A096332 * A096334 A096335 A096336

KEYWORD

nonn

AUTHOR

Alonso del Arte, Aug 02 2004

EXTENSIONS

Edited and extended by Robert G. Wilson v, Aug 12 2004

STATUS

approved

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Last modified May 8 12:24 EDT 2021. Contains 343666 sequences. (Running on oeis4.)