A037074 -id:A037074 - cp's OEIS frontend

A344148 Primes which are two greater than A191746 terms.

17, 6779, 293617, 2992417, 24101863, 423722581, 625997497, 929306267, 3377032037, 3825265007, 6458885659, 7150892197, 13075407803, 13860035251, 19434399319, 32531231209, 47475445333, 50281049527, 53207636077, 62607479491, 85780812151, 106014038789, 109384656937, 121991823731, 125813698531

Offset: 1

Hartmut F. W. Hoft, May 10 2021

nonn

Among numbers a(1..564) are 38 twin primes of which 14 are twins to numbers in A344147 with the first of the latter pairs being A344147(16)=85780812149 and a(21)=85780812151. In contrast it appears that A097490, A097491, A097493 and A343778 contain only twin prime numbers from the set {5, 7, 17}.

			a(1)=17=A191746(1)+2 is the first prime and a(2)=6779=A191746(7)+2 is the second of the form A191746(k)+2; both are twin primes while a(3)=293617 is not.

Cf. A037074, A074040, A077800, A079164, A097490, A097491, A097493, A191746, A344147.

(* function a191746[ ] is defined in A344147 *)
a344148[n_] := Select[a191746[n] + 2, PrimeQ]
a344148[500]

A357928 a(n) is the smallest c for which (s+c)^2-n is a square, where s = floor(sqrt(n)), or -1 if no such c exists.

Original entry on oeis.org

0, 0, -1, 1, 0, 1, -1, 2, 1, 0, -1, 3, 1, 4, -1, 1, 0, 5, -1, 6, 2, 1, -1, 8, 1, 0, -1, 1, 3, 10, -1, 11, 1, 2, -1, 1, 0, 13, -1, 2, 1, 15, -1, 16, 6, 1, -1, 18, 1, 0, -1, 3, 7, 20, -1, 1, 2, 4, -1, 23, 1, 24, -1, 1, 0, 1, -1, 26, 10, 5, -1, 28, 1, 29, -1, 2, 12, 1, -1, 32

Offset: 0

Darío Clavijo, Oct 20 2022

sign

c exists iff n != 2 (mod 4), and it allows n to be written as the difference of two perfect squares.
This gives a factorization n = x*y where x and y may or may not be primes: let s = floor(sqrt(n)), u = a(n) + s and v = u^2 - n; then w = sqrt(v), x = u - w, y = u + w and x*y == n.
The Fermat factorization algorithm seeks such a form, starting from s, so that a(n) is the number of steps it must take for n != 2 (mod 4).
a(n) >= 1 if n is not square and is writable as a difference of squares.
a(n) = 0 if n is square.
a(n) = -1 if n is not writable as a difference of squares.

			n   prime  square  n == 2 (mod 4)   c   s  v=(s+c)^2-n   u   w   x   y  x*y
--  -----  ------  --------------  --  --  -----------  --  --  --  --  ---
76      F       F               F  12   8          324  20  68   2  38   76
13      T       F               F   4   3           36   7   6   1  13   13
25      F       T               F   0   0            0   5   0   5   5   25
7       T       F               T  -1   -            -   -   -   -   -    -

Wikipedia, Fermat's factorization method

Cf. A177713, A037074.

a(n) = if ((n%4)==2, -1, my(s=sqrtint(n), c=0); while (!issquare((s+c)^2-n), c++); c); \\ Michel Marcus, Oct 24 2022

from gmpy2 import *
def fermat(n):
    a, rem = isqrt_rem(n)
    b2 = -rem
    c0 = (a << 1) + 1
    c = c0
    while not is_square(b2):
        b2 += c
        c += 2
    return (c-c0) >> 1
def A357928(n):
  if is_square(n):
      return 0
  elif ((n-2) % 4) != 0:
      return fermat(n)
  else:
      return -1

from math import isqrt
from itertools import takewhile
from sympy import divisors
def A357928(n): return -1 if n&3==2 else min((m>>1 for d in takewhile(lambda d:d**2<=n,divisors(n)) if not((m:=n//d+d) & 1)),default=0) - isqrt(n) # Chai Wah Wu, Oct 26 2022

A135285 Sum of staircase twin primes according to the rule: top * bottom - next top.

Original entry on oeis.org

10, 24, 126, 294, 858, 1704, 3528, 5082, 10296, 11526, 18894, 22320, 32208, 36666, 38976, 51744, 57330, 72618, 79212, 96996, 120684, 175968, 186162, 212922, 271914, 324300, 359382, 381282, 411504, 434790, 655278, 674856, 684726, 735282, 776904

Offset: 1

Cino Hilliard, Dec 03 2007

nonn

While there is multiplication and subtraction in the generation of this sequence, it is still called a sum because the arithmetic processes -,*,/ are derived from addition.

g(n) = for(x=1,n,y=twinu(x) * twinl(x) - twinl(x+1);print1(y",")) twinl(n) = / *The n-th lower twin prime. */ { local(c,x); c=0; x=1; while(c

We list the twin primes in staircase fashion as in A135283. Then a(n) = tl(n) * tu(n) + (-tl(n+1)).

a(n) = A037074(n) -A001359(n+1). - R. J. Mathar, Sep 10 2016

A135286 Sum of staircase twin primes according to the rule: top * bottom + next top.

Original entry on oeis.org

20, 46, 160, 352, 940, 1822, 3670, 5284, 10510, 11800, 19192, 22678, 32590, 37060, 39430, 52222, 57868, 73180, 79834, 97690, 121522, 176830, 187084, 213964, 273052, 325498, 360616, 382564, 412822, 436408, 656920, 676510, 686440, 737044, 778942, 1041430, 1066072, 1103560, 1128934, 1193614, 1328332, 1514176, 1634572, 1665400, 1696522, 1743826, 2040634, 2109784, 2197810, 2215750

Offset: 1

Cino Hilliard, Dec 03 2007

nonn

While there is multiplication in the generation of this sequence, it is still called a sum because the arithmetic processes -,*,/ are derived from addition.

g(n) = for(x=1,n,y=twinu(x) * twinl(x) + twinl(x+1);print1(y",")) twinl(n) = / *The n-th lower twin prime. */ { local(c,x); c=0; x=1; while(c

We list the twin primes in staircase fashion as in A135283. Then a(n) = tl(n) * tu(n) + tl(n+1).

a(n) = A037074(n)+A001359(n+1). - R. J. Mathar, Sep 10 2016

All the entries were wrong. They have been corrected by Franklin T. Adams-Watters, Apr 29 2008

A167283 Products of distinct isolated primes.

Original entry on oeis.org

46, 74, 94, 106, 134, 158, 166, 178, 194, 226, 254, 262, 314, 326, 334, 346, 422, 446, 466, 502, 514, 526, 554, 586, 614, 634, 662, 674, 706, 718, 734, 746, 758, 766, 778, 794, 802, 818, 851, 878, 886, 898, 914, 934, 958, 974, 982, 998, 1006, 1018, 1081, 1082

Offset: 1

Juri-Stepan Gerasimov, Nov 01 2009, Nov 02 2009

nonn

Numbers of the form A007510(i)*A007510(j), i <> j.

			a(50) = 1081 because 23*37 = 1081, where 23 is the 2nd non-twin prime and 37 is the 3rd non-twin prime.

Cf. A007510, A037074, A085434.

Definition clarified and 851 inserted by R. J. Mathar, May 21 2010

A342870 a(n) is the number of twin primes between A001359(n)^2 and A001359(n)*(A001359(n)+1).

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 2, 1, 2, 2, 0, 2, 4, 1, 1, 1, 5, 1, 3, 4, 2, 5, 4, 2, 4, 3, 6, 6, 3, 5, 6, 6, 4, 6, 4, 7, 9, 6, 8, 9, 6, 8, 10, 6, 11, 9, 13, 8, 12, 6, 14, 4, 7, 11, 11, 15, 9, 10, 12, 11, 10, 12, 13, 8, 15, 14, 11, 12, 9, 11, 15, 14, 18, 16, 11, 18, 10

Offset: 1

Zhandos Mambetaliyev, Mar 28 2021

nonn

Cf. A171727, A342552, A001359, A037074, A006094.

{for(k=1, 400, if(prime(k+1)-prime(k)==2, my(c=0); forprime(m=prime(k)^2, prime(k)*(prime(k)+1), c+=isprime(m+2)); print1(c, ", ")))}

A358043 Numbers k such that phi(k) is a multiple of 8.

Original entry on oeis.org

15, 16, 17, 20, 24, 30, 32, 34, 35, 39, 40, 41, 45, 48, 51, 52, 55, 56, 60, 64, 65, 68, 70, 72, 73, 75, 78, 80, 82, 84, 85, 87, 88, 89, 90, 91, 95, 96, 97, 100, 102, 104, 105, 110, 111, 112, 113, 115, 116, 117, 119, 120, 123, 128, 130, 132, 135, 136, 137, 140, 143

Offset: 1

Darío Clavijo, Oct 26 2022

nonn

Cf. A000010 (phi), A053574 (its 2-adic valuation), A037074 (a subsequence).

Totient multiples: A066498 (3), A172019 (4), A066500 (5), A066502 (7), A332512 (12).

Select[Range[150], Divisible[EulerPhi[#], 8] &] (* Amiram Eldar, Oct 27 2022 *)

isok(k) = Mod(eulerphi(k), 8) == 0; \\ Michel Marcus, Oct 27 2022

from sympy.ntheory import totient
def isok(n): return totient(n) % 8 == 0

A000010(a(n)) == 0 (mod 8).

A373899 Semiprimes q*p such that q^p == p (mod (q - p)), where q > p.

Original entry on oeis.org

6, 15, 21, 33, 35, 55, 65, 77, 85, 91, 133, 143, 145, 155, 161, 187, 209, 217, 221, 247, 253, 265, 299, 301, 323, 341, 377, 391, 403, 415, 437, 451, 481, 493, 533, 545, 551, 553, 559, 581, 589, 611, 629, 667, 671, 689, 697, 703, 713, 781, 793, 799, 817, 893, 899, 901

Offset: 1

Juri-Stepan Gerasimov, Jun 22 2024

nonn

			15 = 3*5 is a term because 5^3 == 3 (mod 2).

Subsequence of A001358.

Cf. A037074 (subsequence), A046388, A371811.

seqQ[n_] := Module[{f = FactorInteger[n], p, q}, If[f[[;; , 2]] == {1, 1}, p = f[[1, 1]]; q = f[[2, 1]]; PowerMod[q, p, q - p] == Mod[p, q - p], False]]; Select[Range[1000], seqQ] (* Amiram Eldar, Jun 26 2024 *)

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A344148 Primes which are two greater than A191746 terms.

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A357928 a(n) is the smallest c for which (s+c)^2-n is a square, where s = floor(sqrt(n)), or -1 if no such c exists.

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A135285 Sum of staircase twin primes according to the rule: top * bottom - next top.

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A135286 Sum of staircase twin primes according to the rule: top * bottom + next top.

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A167283 Products of distinct isolated primes.

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A342870 a(n) is the number of twin primes between A001359(n)^2 and A001359(n)*(A001359(n)+1).

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A358043 Numbers k such that phi(k) is a multiple of 8.

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A373899 Semiprimes q*p such that q^p == p (mod (q - p)), where q > p.

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