Chris Fry - cp's OEIS frontend

User: Chris Fry

Chris Fry has authored 4 sequences.

A232768 Numbers n with the property that n^2+(n+1)^2 and n^2+(n+1)^2+(n+2)^2 are both prime.

2, 12, 14, 24, 34, 122, 154, 164, 272, 342, 464, 612, 674, 734, 784, 794, 854, 1174, 1262, 1274, 1364, 1392, 1524, 1554, 1664, 1682, 1844, 1854, 1862, 1892, 1924, 1942, 1994, 2232, 2294, 2354, 2442, 2592, 2802, 2884, 3124, 3164, 3292, 3394, 3544, 3594, 3632, 3724, 3892, 3904, 3922

Offset: 1

Chris Fry, Nov 29 2013

See A027862 for primes of the form x^2+(x+1)^2 = 2x^2+2x+1.
See A027864 for primes of the form x^2+(x+1)^2+(x+2)^2 = 3x^2+6x+5.
It is an open question whether either of these polynomials produces an infinite number of primes.  This sequence lists the values of x that produce a prime in both polynomials. x must be congruent to 0 or 2 (mod 4) and all the generated primes are of the form 4k+1.

			When x=14, 2x^2+2x+1=421 and 3x^2+6x+5=677. 14 is the third value of x for which both these polynomials produce a prime number, so a(3)=14.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 2005, page 266.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Wikipedia, Hardy and Littlewood's Conjecture F.

Cf. A027862, A027864. Equals n common to A027861 and A027863.

lst = {}; Do[If[And[PrimeQ[n^2 + (n + 1)^2], PrimeQ[n^2 + (n + 1)^2 + (n + 2)^2]], Print[n]; AppendTo[lst, n]], {n, 10000}]
Select[Range[2,4000,2],AllTrue[{(#^2+(#+1)^2),(#^2+(#+1)^2+(#+2)^2)},PrimeQ]&] (* Harvey P. Dale, Jul 30 2023 *)

A193108 The tetrahedral numbers A000292 mod 10.

Original entry on oeis.org

1, 4, 0, 0, 5, 6, 4, 0, 5, 0, 6, 4, 5, 0, 0, 6, 9, 0, 0, 0, 1, 4, 0, 0, 5, 6, 4, 0, 5, 0, 6, 4, 5, 0, 0, 6, 9, 0, 0, 0, 1, 4, 0, 0, 5, 6, 4, 0, 5, 0, 6, 4, 5, 0, 0, 6, 9, 0, 0, 0, 1, 4, 0, 0, 5, 6, 4, 0, 5, 0, 6, 4, 5, 0, 0, 6, 9, 0, 0, 0, 1, 4, 0, 0, 5, 6, 4, 0, 5, 0, 6, 4, 5, 0, 0, 6, 9, 0, 0, 0

Offset: 1

Chris Fry, Jul 16 2011

Periodic with period 20.
The cycle is symmetric about index 9 in that a(8)+a(10), a(7)+a(11), etc are all congruent to 0 mod 10.
If the first diagonal of Pascal's triangle is given index 0 this sequence is the 3rd diagonal of Pascal's triangle modulo 10, or the binomial coefficients C(n+2,3)mod 10. Note that the last three terms in the cycle are 0.
The Pisano period lengths of A000292 (mod m) are 1,  4,  9,  8,  5, 36,  7, 16, 27, 20, 11, 72, 13, 28, 45, 32, 17,108, 19, 40.., for m>=1. This sequence describes the case m=10. - R. J. Mathar, Oct 25 2011

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A000292, A008954.

Mathematica
```
Table[Mod[Binomial[n+2,3],10],{n,1,21}]
```

a(n) = a(n-20).
G.f.  -x*(1+4*x+5*x^4+6*x^5+4*x^6+5*x^8+6*x^10+4*x^11+5*x^12+6*x^15+9*x^16)  / ( (x-1)*(1+x^4+x^3+x^2+x)*(1+x)*(1-x+x^2-x^3+x^4)*(1+x^2)*(x^8-x^6+x^4-x^2+1) ). - R. J. Mathar, Oct 25 2011
a(n) = 55 -a(n-1) -a(n-2) … -a(n-18) -a(n-19). - Ant King, Oct 19 2012

Edited by N. J. A. Sloane, Jul 16 2011

A191969 Numbers that are indices of deficient oblong numbers (A002378).

Original entry on oeis.org

1, 10, 13, 22, 37, 43, 46, 52, 58, 61, 67, 73, 82, 85, 94, 97, 106, 109, 118, 121, 130, 133, 136, 142, 145, 148, 151, 157, 163, 166, 172, 178, 181, 190, 193, 202, 205, 211, 214, 217, 226, 229, 232, 238, 241, 250, 253, 262, 268, 277, 283, 289, 292, 298, 301, 310, 313, 316, 322, 331, 334, 337, 346, 358, 361, 373, 382, 388, 394, 397

Offset: 1

Chris Fry, Jun 22 2011

Numbers k such that A002378(k) = k*(k+1) is deficient.
"In 1700, Charles de Neuveglise claimed the product of two consecutive integers n(n+1) with n>=3 is abundant." - Tattersall, p. 144.  In other words, de Neuveglise claimed that all oblong numbers greater than 6 are abundant.  In fact, up to A002378(1100), 17.6% of the oblong numbers are deficient.  The per-100 count of deficient oblong numbers from A002378(1) to A002378(1100) is 16, 19, 19, 16, 17, 20, 18, 17, 17, 15, 20. For most deficient oblong numbers A002378(k) in this range, either k or k+1 is prime, but this is not always the case, explaining why the density of deficient oblong numbers does not decrease in line with the primes.
All the terms are congruent to 1 or 4 mod 6, and there are no terms that are congruent to 0, 4, 15, or 19 mod 20. Therefore, the asymptotic density of this sequence is less than 4/15. The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 2, 16, 174, 1831, 18237, 182432, 1824453, 18241059, 182414767, 1824169736, ... . Apparently, the asymptotic density of this sequence equals 0.18241... . - Amiram Eldar, Mar 15 2024

			The third deficient oblong number is A002378(13) = 13*14 = 182: sigma(182) = 336 < 364 = 2*182.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Second Edition, Cambridge University Press, 2005.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005101, A005100, A002378, A124672, A077804.

Select[Range[400], DivisorSigma[1, o = # (# + 1)] < 2 o &] (* Amiram Eldar, Jun 21 2019 *)

for(n=1, 400, o=n*(n+1); if(sigma(o)<2*o, print1(n, ", ")))

A002378(a(n)) = A077804(n). - Amiram Eldar, Mar 15 2024

A177927 3-Monica numbers.

Original entry on oeis.org

4, 9, 10, 22, 24, 25, 27, 34, 42, 46, 55, 58, 60, 72, 78, 81, 82, 85, 94, 105, 106, 114, 115, 118, 121, 126, 128, 132, 142, 145, 150, 166, 178, 180, 186, 187, 192, 195, 202, 204, 205, 214, 216, 222, 224, 226, 231, 234, 235, 243, 253, 256, 258, 262, 265, 274, 276, 285, 289, 295

Offset: 1

Chris Fry, Dec 26 2010

3-Monica numbers are composite positive integers k for which 3 divides S(k)-Sp(k), where S(k) denotes the sum of the digits of k and Sp(k) denotes the sum of the digits in an extended prime factorization of k.

			S(10)=1+0=1, 10=2*5, Sp(10)=2+5=7, S(10)-Sp(10)=-6 which is divisible by 3.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 2005, page 93.
E. W. Weisstein, The CRC Concise Encyclopedia of Mathematics, CRC Press, 1999, pages 1192-1193.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Michael Smith, Cousins of Smith Numbers: Monica and Suzanne Sets, Fibonacci Quarterly, Vol. 34, No. 2 (1996), pp. 102-104.
Eric Weisstein's World of Mathematics, Monica Set.

Cf. A006753 (Smith numbers are a subset of every n-Monica sequence).
Cf. A102217 (n-Suzanne numbers are a subset of n-Monica numbers).
Cf. A102219 (This list of '3-Monica' numbers is incorrect. It does not contain all the Smith numbers and appears to be based on S(n)+Sp(n) ==0 (mod 3), instead of S(n)-Sp(n) == 0 (mod 3)).
Cf. A007953, A118503.

s[n_] := Plus @@ IntegerDigits[n]; f[p_, e_] := e*s[p]; sp[n_] := Plus @@ f @@@ FactorInteger[n]; mon3Q[n_] := CompositeQ[n] && Divisible[s[n] - sp[n], 3]; Select[Range[300], mon3Q] (* Amiram Eldar, Apr 23 2021 *)

cp's OEIS Frontend

User: Chris Fry

A232768 Numbers n with the property that n^2+(n+1)^2 and n^2+(n+1)^2+(n+2)^2 are both prime.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

A193108 The tetrahedral numbers A000292 mod 10.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A191969 Numbers that are indices of deficient oblong numbers (A002378).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A177927 3-Monica numbers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica