Morgan L. Owens - cp's OEIS frontend

A272627 Numbers n = pq where p, q are primes congruent to 3 and 7 mod 8, respectively.

21, 69, 77, 93, 133, 141, 213, 237, 253, 301, 309, 341, 381, 413, 437, 453, 469, 501, 517, 573, 581, 589, 597, 669, 717, 749, 781, 789, 813, 869, 893, 917, 933, 973, 989, 1077, 1101, 1133, 1141, 1149, 1253, 1293, 1317, 1333, 1349, 1357, 1389, 1397, 1437, 1461

Offset: 1

Morgan L. Owens, May 03 2016

nonn

Candidate moduli for Rabin cryptosystem using Williams padding to ensure sufficient redundancy that the decryption is unique.

Steven D. Galbraith, Mathematics of Public Key Cryptography, Cambridge University Press, 2012, page 493.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
H. C. Williams, A Modification of the RSA public encryption procedure, IEEE Trans. Inf. Theory 26(6) (1980), 726-729.

Cf. A016105.

With[{upto = 1000},
With[{primes = Prime@Range@PrimePi@NextPrime[upto/3]},
  With[{p = Pick[primes, Mod[primes, 8], 3], q = Pick[primes, Mod[primes, 8], 7]},
   Select[Union[Flatten@Outer[Times, p, q]], # <= upto &]] ]] (* after Harvey P. Dale at A016105 *)

ok(n)={n%8==5 && bigomega(n)==2 && factor(n)[1,1] % 4 == 3} \\ Andrew Howroyd, Dec 23 2019

Terms a(36) and beyond from Andrew Howroyd, Dec 23 2019

A250397 Differences between A250396 and A104080.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 12, 2, 30, 0, 0, 10, 10, 0, 2, 112, 0, 0, 6, 42, 52, 2, 0, 0, 54, 16, 0, 96, 0, 0, 46, 12, 60, 0, 84, 260, 6, 22, 126, 0, 60, 112, 14, 16, 124, 136, 40, 0, 150

Offset: 1

Morgan L. Owens, Nov 21 2014

nonn

Cf. A250396, A104080.

A250396 a(n) is the smallest prime greater than 2^n such that 2 is a primitive root modulo a(n).

Original entry on oeis.org

3, 3, 5, 11, 19, 37, 67, 131, 269, 523, 1061, 2053, 4099, 8219, 16421, 32771, 65539, 131213, 262147, 524309, 1048589, 2097211, 4194371, 8388619, 16777259, 33554467, 67108933, 134217773, 268435459, 536871019, 1073741827, 2147483659, 4294967357, 8589934621, 17179869269, 34359738421, 68719476851, 137438953741

Offset: 0

Morgan L. Owens, Nov 21 2014

nonn

Henri Cohen, A Course in Computational Algebraic Number Theory, Springer Verlag, (1993)

Amiram Eldar, Table of n, a(n) for n = 0..300
Joerg Arndt, Matters Computational; Ideas, Algorithms, Source Code, (§1.5.1, p.13).

Cf. A104080 (smallest prime >= 2^n).

With[{n = 20},
Module[{p = NextPrime[2^n]},
  While[FreeQ[PrimitiveRootList[p], 2], p = NextPrime[p]]; p]]

a(n)=forprime(p=2^n+1,,if(znorder(Mod(2,p))==p-1,return(p))); \\ Joerg Arndt, Nov 21 2014

A247209 Number of terms in generalized Swinnerton-Dyer polynomials.

Original entry on oeis.org

1, 2, 6, 35, 495, 20349, 2760681, 1329890705, 2353351951665, 15481400876017505, 379554034822178909121, 34676179189150610052785025, 11806724418359403847522843860225, 14998128029851443976142151169687652865, 71221988684076361563783957084457295633613825

Offset: 0

Morgan L. Owens, Nov 25 2014

If the sequence of primes used in the construction of Swinnerton-Dyer polynomials is replaced by the generic sequence a_1, a_2, ..., a_n, this sequence gives the number of terms in the resulting multivariate polynomial (treating the a_n as variables).

a(n-1) is the number of monomials obtained when multiplying all the possible cases Sum_{k=1..n} e_k*sqrt(x_k) where e_1 is 1 and all other e_i are +1 or -1; so that 1/(Sum_{k=1..n} sqrt(x_k)) is transformed into a fraction whose denominator has no radicals. See the French link. - Michel Marcus, Jun 12 2022

			a(3) = 35. For the three numbers a, b, c, the general Swinnerton-Dyer polynomial is
(sqrt(a)+sqrt(b)+sqrt(c)-z)(-sqrt(a)+sqrt(b)+sqrt(c)-z)(sqrt(a)-sqrt(b)+sqrt(c)-z)(-sqrt(a)-sqrt(b)+sqrt(c)-z)(sqrt(a)+sqrt(b)-sqrt(c)-z)(-sqrt(a)+sqrt(b)-sqrt(c)-z)(sqrt(a)-sqrt(b)-sqrt(c)-z)(-sqrt(a)-sqrt(b)-sqrt(c)-z)
which expands to
a^4-4a^3b+6a^2b^2-4ab^3+b^4-4a^3c+4a^2bc+4ab^2c-4b^3c+6a^2c^2+4abc^2+6b^2c^2-4ac^3-4bc^3+c^4- 4a^3z^2+4a^2bz^2+4ab^2z^2-4b^3z^2+4a^2cz^2-40abcz^2+ 4b^2cz^2+4ac^2z^2+4bc^2z^2-4c^3z^2+6a^2z^4+4abz^4+ 6b^2z^4+4acz^4+4bcz^4+6c^2z^4-4az^6-4bz^6-4cz^6+z^8
with 35 terms.

Alois P. Heinz, Table of n, a(n) for n = 0..60
Allan Berele and Stefan Catoiu, Rationalizing Denominators, Mathematics Magazine, Vol. 88, No. 2 (2015), pp. 121-136.
Les Tablettes du Chercheur, Problem 21, Solution to problem 21, Addition to problem 21, pp. 4, 30 and 64, 1892 (in French).
Eric Weisstein's World of Mathematics, Swinnerton-Dyer Polynomial.

Cf. A153731.

a:= n-> (t-> binomial(t+n, t))(2^(n-1)):
seq(a(n), n=0..14);  # Alois P. Heinz, Nov 28 2024

a[n_]:= Module[{a,x}, Length@Fold[Expand[(#1 /. x -> x + #2) (#1 /. x -> x - #2)] &, x, Sqrt[a /@ Range[n]]]]; a[0] = 1; Array[a, 5, 0] (* or *)
a[n_]:= Binomial[2^(n - 1) + n, 2^(n - 1)]; Array[a, 10, 0]

a(0) = 1 prepended by Michel Marcus, Jun 12 2022

A247146 As a binary numeral, the bit 2^(m-1) of a(n) is 1 iff m is a proper divisor of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 7, 1, 11, 5, 19, 1, 47, 1, 67, 21, 139, 1, 295, 1, 539, 69, 1027, 1, 2223, 17, 4099, 261, 8267, 1, 16951, 1, 32907, 1029, 65539, 81, 133423, 1, 262147, 4101, 524955, 1, 1056871, 1, 2098187, 16661, 4194307, 1, 8423599, 65, 16777747, 65541

Offset: 1

Morgan L. Owens, Nov 21 2014

nonn

a(n)==1 iff n is prime.
Apparently Moebius transform of A178472.
For n>1, the binary representation of a(n) is given by row (n-1) of A077049 (when read as a triangular array). - Tom Edgar, Nov 28 2014

Cf. A034729, A077049, A178472.

With[{n=Range[100]},(1/2) ((Total/@(2^Divisors[n])) - 2^n)]

a(n) = sumdiv(n, k, 2^(k-1)) - 2^(n-1); \\ Michel Marcus, Nov 25 2014

from sympy import divisors
def A247146(n): return sum(1<Chai Wah Wu, Jul 15 2022

a(n) = A034729(n) - 2^(n-1). - Michel Marcus, Nov 22 2014

A247123 Palindromes obtained after one iteration of Reverse and Add applied to the terms of A015976.

Original entry on oeis.org

2, 4, 6, 8, 11, 22, 33, 44, 55, 66, 77, 88, 99, 22, 33, 44, 55, 66, 77, 88, 99, 121, 33, 44, 55, 66, 77, 88, 99, 121, 44, 55, 66, 77, 88, 99, 121, 55, 66, 77, 88, 99, 121, 66, 77, 88, 99, 121, 77

Offset: 2

Morgan L. Owens, Nov 21 2014

Cf. A015976 (One iteration of Reverse and Add is needed to reach a palindrome).

Select[(FromDigits[#] + FromDigits[Reverse[#]]) & /@ IntegerDigits[Range[1000]], IntegerDigits[#] == Reverse[IntegerDigits[#]] &]

A247110 n + reversal of digits of n, when n is not palindromic.

Original entry on oeis.org

11, 33, 44, 55, 66, 77, 88, 99, 110, 22, 33, 55, 66, 77, 88, 99, 110, 121, 33, 44, 55, 77, 88, 99, 110, 121, 132, 44, 55, 66, 77, 99, 110, 121, 132, 143, 55, 66, 77, 88, 99, 121, 132, 143, 154, 66, 77, 88, 99, 110, 121, 143, 154, 165, 77, 88, 99, 110, 121, 132, 143, 165, 176, 88, 99, 110, 121, 132, 143, 154, 165, 187, 99, 110, 121

Offset: 10

Morgan L. Owens, Nov 21 2014

Cf. A056964 (n + reversal of digits of n)

With[{n=50}, (FromDigits[#] + FromDigits[Reverse[#]]) & /@ Select[IntegerDigits[Range[n]], # != Reverse[#] &]]
Table[If[n==IntegerReverse[n],Nothing,n+IntegerReverse[n]],{n,100}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 29 2016 *)

cp's OEIS Frontend

User: Morgan L. Owens

A272627 Numbers n = pq where p, q are primes congruent to 3 and 7 mod 8, respectively.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A250397 Differences between A250396 and A104080.

Views

Author

Keywords

Crossrefs

A250396 a(n) is the smallest prime greater than 2^n such that 2 is a primitive root modulo a(n).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

A247209 Number of terms in generalized Swinnerton-Dyer polynomials.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A247146 As a binary numeral, the bit 2^(m-1) of a(n) is 1 iff m is a proper divisor of n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A247123 Palindromes obtained after one iteration of Reverse and Add applied to the terms of A015976.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A247110 n + reversal of digits of n, when n is not palindromic.

Views

Author

Keywords

Crossrefs

Programs

Mathematica