cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A367576 Semiprimes p*q such that 2*p divides q-1.

Original entry on oeis.org

10, 21, 26, 34, 39, 55, 57, 58, 74, 82, 93, 106, 111, 122, 129, 146, 155, 178, 183, 194, 201, 202, 203, 205, 218, 219, 226, 237, 253, 274, 291, 298, 301, 305, 309, 314, 327, 346, 355, 362, 381, 386, 394, 417, 453, 458, 466, 471, 482, 489, 497, 505, 514, 538
Offset: 1

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Author

Jon E. Schoenfield, Nov 24 2023

Keywords

Comments

These terms and their squarefree proper multiples are the squarefree numbers that never appear as terms in A367566.

Examples

			10 is a term: 10 = 2*5, and 2*2 | 5 - 1.
33 is not a term: 33 = 3*11, but 2*3 does not divide 11 - 1.

Crossrefs

Cf. A156592, A231966, A367566.
Subsequence of A006881.

A251606 Let {d(i)}, i=1..q, be the set of the q prime distinct divisors of a number n, and consider the alternating sum s = Sum_{i=1..q} i*((-1)^(i+1))/d(i) and the product p = Product_{i=1..q} d(i) respectively. The sequence lists the composite numbers n such that s = 1/p.

Original entry on oeis.org

4, 8, 9, 10, 16, 20, 21, 25, 27, 32, 40, 49, 50, 55, 63, 64, 80, 81, 100, 102, 121, 125, 128, 147, 160, 169, 189, 200, 204, 243, 250, 253, 256, 275, 289, 306, 320, 343, 361, 400, 408, 441, 500, 512, 529, 567, 605, 612, 625, 640, 729, 800, 816, 841, 918, 961
Offset: 1

Views

Author

Michel Lagneau, Dec 13 2014

Keywords

Comments

The semiprimes p*q, p and q prime with q = 2*p + 1 (A156592) are in the sequence.
The trivial values of prime powers p^m (A025475) are in the sequence.
The primitive values of the sequence are 4, 9, 10, 21, 25, 49, 55, ... .

Examples

			21 is in the sequence because the prime factors of 21 are {3,7} => s = 1/3 - 2/7, 1/p = 1/21 and s = p = 1/21.

Crossrefs

Cf. A001248, A025475, A156592, A252044.

Programs

  • Maple
    with(numtheory):nn:=1000:
for n from 1 to nn do:
  if type(n,prime)=false
  then
  x:=factorset(n):n0:=nops(x):
   s:=sum('i*((-1)^(i+1))/x[i]','i'=1..n0):s0:=product('x[i]','i'=1..n0):
   s2:=s-1/s0:
   if s2=0
   then
   printf(`%d, `,n):
   else
   fi:
  fi:
od:
  • PARI
    lista(nn) = {forcomposite(n=2, nn, my(f = factor(n)[,1]); if (sum(i=1, #f, i*((-1)^(i+1))/f[i]) == prod(i=1, #f, 1/f[i]), print1(n, ", ")););} \\ Michel Marcus, Sep 13 2017

A365849 Triangular numbers that are the product of two distinct prime numbers of the form 4*k + 3.

Original entry on oeis.org

21, 253, 1081, 13861, 34453, 64261, 73153, 114481, 126253, 258121, 351541, 371953, 392941, 482653, 869221, 933661, 1034641, 1104841, 1660753, 2077741, 2126953, 2434321, 2992681, 4142881, 4212253, 4495501, 4567753, 4862521, 5013361, 6561253, 7459453, 8026021
Offset: 1

Views

Author

Marius A. Burtea, Oct 09 2023

Keywords

Comments

Intersection of A068443 and A016105.
Subsequence of A156592.
Apparently, a(n) == 1 (mod 36) for n >= 2. - Hugo Pfoertner, Nov 03 2023

Examples

			21 = A068443(4) and 21 = A016105(1), so 21 is a term.
253 = A068443(7) and 253 = A016105(18), so 253 is a term.

Crossrefs

Cf. A000217, A001358, A068443, A156592, A016105.

Programs

  • Magma
    pd:=PrimeDivisors; blum:=func; [n:n in [1..9000000]|IsSquare(8*n+1) and blum(n)];
  • Mathematica
    Select[Accumulate[Range[4500]], (f = FactorInteger[#])[[;; , 2]] == {1, 1} && Mod[f[[;; , 1]], 4] == {3, 3} &] (* Amiram Eldar, Oct 11 2023 *)
Previous Showing 11-13 of 13 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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