A226770 -id:A226770 - cp's OEIS frontend

A226856 Numbers A226770((prime(n))^2-1).

1, 1, 1, 3, 1, 1, 3, 1, 3, 5, 1, 46, 1, 5, 66, 1, 1, 3, 3, 3, 10, 3, 1, 1, 172, 1, 1, 398, 5, 132, 255, 1, 336, 41, 1, 3, 615, 1, 462, 918, 283, 1, 1, 1, 453, 1, 11, 713, 1503, 311, 1, 112, 1, 308, 2272, 1, 788, 481, 2301, 1, 2686, 1358, 3, 3, 2651, 3346, 1

Offset: 1

Vladimir Shevelev, Jun 19 2013

nonn

a(n) = 1 iff (prime(n))^2 + prime(n) - 1 is prime (A053184).

Peter J. C. Moses, Table of n, a(n) for n = 1..400

Cf. A226770, A053184.

Table[(div=Most[Divisors[n+1]]; Length[FixedPoint[Union[Flatten[AppendTo[div, Map[Most[Rest[Divisors[n+#]]]&, #]]]]&, div]]-1), {n, Prime[Range[40]]^2-1}] (* Peter J. C. Moses, Jun 20 2013 *)

More terms from Peter J. C. Moses, Jun 20 2013

A226859 Number of prime sums in the process described in A226770.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 3, 1, 3, 2, 3, 1, 4, 1, 4, 2, 5, 1, 5, 1, 6, 3, 7, 1, 6, 1, 7, 4, 7, 3, 8, 1, 9, 4, 9, 1, 9, 1, 9, 4, 10, 1, 9, 2, 10, 2, 11, 1, 11, 2, 13, 5, 14, 1, 13, 1, 12, 5, 12, 5, 13, 1, 13, 6, 14, 1, 14, 1, 13, 6, 14, 7, 15, 1, 15, 3, 15

Offset: 1

Vladimir Shevelev, Jun 20 2013

nonn

			Let n=76. We have 77; d=7,11; 76+7=83 (prime), 76+11=87; d=3,29; 76+3=79(prime), 76+29=105; d=5,15,21,35; 76+5=81, 76+15=91, 76+21=97(prime), 76+35=111; d=9,27,13,37, 76+9=85,76+27=103(prime),76+13=89(prime), 76+37=113(prime), d=17, 76+17=93; d=31, 76+31=107(prime). Thus the set of prime sums is {83,79,97,103,89,113,107} and therefore a(76)=7.

Peter J. C. Moses, Table of n, a(n) for n = 1..2000

Cf. A226770, A226856, A053184.

Table[(div=Most[Divisors[n+1]]; Count[n+FixedPoint[Union[Flatten[AppendTo[div, Map[Most[Divisors[n+#]]&, #]]]]&, div],?PrimeQ]),{n,50}] (* _Peter J. C. Moses, Jun 20 2013 *)

a(n) = 1 iff either n = 5 or n + 1 = p or n + 1 = q^2, where p,q and q^2+q-1 are primes.

More terms from Peter J. C. Moses, Jun 20 2013

A227129 Semiprimes n = p*q, p

Original entry on oeis.org

15, 51, 85, 91, 133, 145, 235, 249, 265, 427, 451, 493, 519, 559, 565, 589, 591, 681, 721, 871, 879, 1003, 1149, 1177, 1189, 1207, 1411, 1441, 1509, 1561, 1603, 1651, 1837, 1945, 2059, 2071, 2119, 2227, 2335, 2391, 2419, 2599, 2661, 2827, 2869, 2965, 2995

Offset: 1

Vladimir Shevelev, Jul 02 2013

nonn

Subsequence of A006881.

Peter J. C. Moses, Table of n, a(n) for n = 1..5000
Since 591 = 3*197 and numbers 591 + 3 - 1 = 593, 591 + 197 - 1 = 787 are both primes, then 591 is in the sequence.

Cf. A006881, A226770.

Select[Range[10000],(Last[#2]=={1,1}&&And@@PrimeQ[#1+First[#2]-1]&)[#1,Transpose[FactorInteger[#1]]]&] (* Peter J. C. Moses, Jul 03 2013 *)
spQ[n_]:=Module[{fi=Transpose[FactorInteger[n]]},fi[[2]]=={1,1}&&AllTrue[ n-1+fi[[1]],PrimeQ]]; Select[Range[3000],spQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 24 2015 *)

A226770(a(n)-1) = 2.

A227276 Primes p for which p^2 + p - 1 = q*r (q

Original entry on oeis.org

7, 17, 23, 61, 67, 71, 79, 151, 307, 311, 383, 389, 409, 439, 613, 677, 1559, 1627, 1637, 2377, 2719, 2801, 3407, 3821, 4229, 4799, 4919, 5557, 5641, 5743, 5779, 5851, 5867, 6133, 6733, 7121, 7723, 8009, 8527, 8573, 10163, 10729, 11317, 11789, 11987, 14107, 14629, 14653, 14669, 17189, 17401, 18077

Offset: 1

Vladimir Shevelev, Jul 04 2013

nonn

Peter J. C. Moses, Table of n, a(n) for n = 1..5000

Cf. A227129, A226770.

Select[Prime[Range[3000]],And@@PrimeQ[#1^2-1+First[#2]]&&Last[#2]=={1,1}&[#1,Transpose[FactorInteger[#^2+#-1]]]&] (* Peter J. C. Moses, Jul 05 2013 *)

A226770(a^2(n) - 1) = 3.

More terms from Peter J. C. Moses.

A226938 Primes p such that p^4 + p - 1, p^4 + p^2 - 1, p^4 + p^3 - 1 are also prime.

Original entry on oeis.org

2, 3, 13, 43, 4909, 21283, 47417, 57301, 59951, 72647, 98713, 132623, 135841, 149101, 153371, 285463, 343489, 355519, 360823, 375101, 396997, 405901, 447197, 452377, 458797, 501173, 532379, 557153, 605947, 610199, 614071, 616079, 627901, 644051, 656141, 668417

Offset: 1

Vladimir Shevelev, Jul 05 2013

nonn

Peter J. C. Moses, Table of n, a(n) for n = 1..2000

Cf. A226770, A227129, A227276.

Select[Prime[Range[10000]],And@@PrimeQ[#1+{#2,#2^2,#2^3}]&[#^4-1,#]&] (* Peter J. C. Moses, Jul 05 2013 *)
Select[Prime[Range[60000]],AllTrue[#^4-1+#^Range[3],PrimeQ]&] (* Harvey P. Dale, Mar 26 2023 *)

A226770(a^4(n) - 1) = 3.

More terms from Peter J. C. Moses

cp's OEIS Frontend

A226856 Numbers A226770((prime(n))^2-1).

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A226859 Number of prime sums in the process described in A226770.

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A227129 Semiprimes n = p*q, p

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A227276 Primes p for which p^2 + p - 1 = q*r (q

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A226938 Primes p such that p^4 + p - 1, p^4 + p^2 - 1, p^4 + p^3 - 1 are also prime.

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