A128546 -id:A128546 - cp's OEIS frontend

A128548 Primes p such that pq-p-q and pq+p+q are prime where q=nextprime(p).

3, 5, 13, 43, 89, 163, 479, 643, 683, 773, 811, 953, 1109, 1399, 1471, 2213, 2741, 3253, 4583, 5153, 5923, 6427, 7649, 9059, 10151, 10531, 12301, 12373, 13553, 13903, 13921, 14723, 14869, 14929, 16183, 17123, 17681, 21149, 21377, 21569, 21587

Offset: 1

Zak Seidov, Mar 10 2007

Intersection of A126148 and A128546.

			3*5-3-5=7 and 3*5+3+5=23 are prime, 5*7-5-23=7 and 5*7+5+7=47 are primes.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A096342, A126148, A128547.

Select[Partition[Prime[Range[2500]],2,1],AllTrue[Times@@#+{Total[#],-Total[#]},PrimeQ]&][[;;,1]] (* Harvey P. Dale, Aug 30 2025 *)

isok(p) = isprime(p) && (q = nextprime(p+1)) && isprime(p*q-p-q) && isprime(p*q+p+q); \\ Michel Marcus, Oct 11 2013

A097060 Revrepfigits (reverse replicating Fibonacci-like digits): Numbers k whose reversal occurs in a sequence generated by starting with the k digits of a number and then continuing the sequence with a number that is the sum of the previous k terms.

Original entry on oeis.org

12, 24, 36, 48, 52, 71, 341, 682, 1285, 5532, 8166, 17593, 28421, 74733, 90711, 759664, 901921, 1593583, 4808691, 6615651, 6738984, 8366363, 8422611, 26435142, 54734431, 57133931, 79112422, 89681171, 351247542, 428899438, 489044741, 578989902

Offset: 1

Jason Earls, Sep 15 2004

Numbers ending in zero are not permitted since the zeros are dropped upon reversal. However, terms with internal zeros such as 90711 are permitted. Conjectures: 1. Sequence is infinite. 2. Revrepfigits are more rare than repfigits.

There are no 12-digit revrepfigits.

			8166 is in the sequence since the sequence 8,1,6,6,21,34,67,128,250, 479,924,1781,3434,6618,..., contains the reversal of 8166.

J. Earls, Mathematical Bliss, Pleroma Publications, 2009, pages 11-13. ASIN: B002ACVZ6O [From Jason Earls, Nov 21 2009]

Bernardo Boncompagni and Anton Vrba, Table of n, a(n) for n = 1..59
Carlos Rivera, Primepuzzles.net Puzzle 384
Eric Weisstein's World of Mathematics, Keith Number

Cf. A007629.

Cf. A128546 (reverse of these numbers).

rKeithQ[n_Integer] := Module[{b = IntegerDigits[n], r, s, k = 0}, If[Mod[n, 10] == 0, False, r = FromDigits[Reverse[b]]; s = Total[b]; While[s < r, AppendTo[b, s]; k++; s = 2*s - b[[k]]]; s == r]]; Select[Range[10, 100000], rKeithQ] (* T. D. Noe, Mar 15 2011 *)

More terms from Bernardo Boncompagni and Anton Vrba (antonvrba(AT)yahoo.com), Jan 05 2007

A307863 Numbers x = concat(a,b) such that b and a are the first two terms for a Fibonacci-like sequence containing x itself.

Original entry on oeis.org

17, 21, 25, 42, 63, 84, 105, 123, 126, 147, 168, 189, 197, 246, 295, 369, 492, 787, 1033, 1115, 1141, 1248, 1279, 1997, 2066, 2230, 2282, 2496, 2995, 3099, 3345, 3423, 3744, 4460, 4564, 4992, 5411, 5575, 5705, 6690, 6846, 7987, 10112, 10483, 10822, 11059, 11107

Offset: 1

Paolo P. Lava, May 02 2019

Similar to A130792 but here the sums start b + a = c, a + c = d, etc.
First six terms are also the first six Inrepfigit numbers (A128546).
Being x = concat(a,b), the problem is to find an index y such that x = b*F(y) + a*F(y+1), where F(y) is a Fibonacci number (see file with values of x, b, a, y, for 1< x <10^6, in Links). All the listed numbers admit only one unique concatenation that, through the addition process, leads to themselves. Is there any number that admits more than one single concatenation?

			123 can be split into 1 and 23 and the Fibonacci-like sequence: 23, 1, 24, 25, 49, 74, 123, ... contains 123 itself.

Giovanni Resta, Table of n, a(n) for n = 1..900
Paolo P. Lava, Values of x, b, a, y, for 1< x <10^6

Cf. A128546, A130792, A289868.

P:=proc(n) local j,t,v; v:=array(1..100);
for j from 1 to length(n)-1 do v[1]:=n mod 10^j; v[2]:=trunc(n/10^j);
v[3]:=v[1]+v[2]; t:=3; while v[t]Paolo P. Lava, May 02 2019

cp's OEIS Frontend

A128548 Primes p such that pq-p-q and pq+p+q are prime where q=nextprime(p).

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A097060 Revrepfigits (reverse replicating Fibonacci-like digits): Numbers k whose reversal occurs in a sequence generated by starting with the k digits of a number and then continuing the sequence with a number that is the sum of the previous k terms.

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A307863 Numbers x = concat(a,b) such that b and a are the first two terms for a Fibonacci-like sequence containing x itself.

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cp's OEIS Frontend

A128548 Primes p such that p*q-p-q and p*q+p+q are prime where q=nextprime(p).

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A097060 Revrepfigits (reverse replicating Fibonacci-like digits): Numbers k whose reversal occurs in a sequence generated by starting with the k digits of a number and then continuing the sequence with a number that is the sum of the previous k terms.

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A307863 Numbers x = concat(a,b) such that b and a are the first two terms for a Fibonacci-like sequence containing x itself.

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Author

Keywords

Comments

Examples

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Programs

Maple

A128548 Primes p such that pq-p-q and pq+p+q are prime where q=nextprime(p).