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.

Previous Showing 21-27 of 27 results.

A177222 Numbers k that are the products of two distinct primes, such that 2*k + 1 and 4*k + 3 are also products of two distinct primes.

Original entry on oeis.org

38, 46, 106, 129, 133, 145, 201, 203, 235, 291, 298, 334, 335, 381, 407, 417, 458, 489, 497, 538, 579, 583, 597, 623, 626, 649, 685, 689, 694, 707, 758, 767, 781, 815, 898, 899, 921, 926, 959, 995, 1073, 1079, 1082, 1094, 1099, 1139, 1142, 1157, 1214, 1226
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 05 2010

Keywords

Examples

			38 is a term because 38 = 2*19, 2*38 + 1 = 77 = 7*11, and 4*38 + 1 = 155 = 5*31.

Crossrefs

Cf. A006881, A111153, A177210, A177211, A177212, A177213, A177214, A177215, A177216, A177217, A177220, A177221.
Cf. A133123 (allows squares also).

Programs

  • Mathematica
    f[n_] := Last/@FactorInteger[n] == {1,1};  lst = {}; Do[If[f[n] && f[2*n+1] && f[4*n+3], AppendTo[lst, n]], {n, 1000}]; lst

A177223 Numbers k that are the products of two distinct primes such that 2*k+1, 4*k+3 and 8*k+7 are also products of two distinct primes.

Original entry on oeis.org

145, 203, 291, 298, 407, 497, 649, 707, 758, 815, 899, 926, 959, 995, 1079, 1094, 1139, 1142, 1157, 1313, 1403, 1415, 1461, 1497, 1538, 1639, 1658, 1691, 1857, 1934, 1945, 1991, 2123, 2159, 2217, 2234, 2315, 2603, 2629, 2807, 2991, 3215, 3254, 3279, 3305
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 05 2010

Keywords

Comments

A number k is the product of two distinct primes iff k = p*q where p and q are distinct primes. - N. J. A. Sloane, Jan 11 2025

Examples

			145 is a term because 145 = 5*29, 2*145 + 1 = 291 = 3*97, 4*145 + 1 = 583 = 11*53, and 8*145 + 1 = 1167 = 3*389.

Links

Crossrefs

Cf. A006881, A111153, A177210, A177211, A177212, A177213, A177214, A177215, A177216, A177217, A177220, A177221, A177222.

Programs

  • Mathematica
    f[n_]:=Last/@FactorInteger[n]=={1,1}; lst={};Do[If[f[n]&&f[2*n+1]&&f[4*n+3]&&f[8*n+7],AppendTo[lst,n]],{n,0,2*7!}];lst
tdpQ[n_]:=With[{c={n, 2n+1, 4n+3,8n+7}},PrimeNu[c]==PrimeOmega[c]=={2,2,2,2}]; Select[Range[3500],tdpQ] (* Harvey P. Dale, Jan 11 2025 *)

A211169 The least n-almost Sophie Germain prime.

Original entry on oeis.org

2, 4, 52, 40, 688, 4900, 63112, 178240, 38272, 5357056, 1997824, 247221760, 586504192, 707436544, 15582115840, 47145459712, 77620412416, 1871289057280, 17787921498112, 10891875057664, 146305150615552, 535618317844480, 15921951753109504, 39754688251297792
Offset: 1

Views

Author

Don Reble, Paolo P. Lava, and Robert G. Wilson v, Jan 31 2013

Keywords

Examples

			a(1)=2 because 2 and 5 are primes (A000040),
a(2)=4 because 4 and 9 are semiprimes (A001358),
a(3)=52 because the pair, 52 and 105, are 3-almost primes (A014612) and they are the least such pair,
a(4)=40 because the pair, 40 and 81, are 4-almost primes (A014613) and they are the least such pair, etc.

Crossrefs

Cf. A005384 (Sophie Germain primes), A111153 (Sophie Germain semiprimes), A111173 (Sophie Germain 3-almost primes), A111176 (Sophie Germain 4-almost primes), A211162 (Sophie Germain 5-almost primes).

Programs

  • Maple
    with(numtheory);
A211169:=proc(q)
local a,b,c,d,g,f,i,j,n;
for j from 1 to q do for n from 1 to q do
    a:=ifactors(n)[2]; b:=nops(a); c:=ifactors(2*n+1)[2]; d:=nops(c); g:=0; f:=0;
    for i from 1 to b do g:=g+a[i][2]; od; for i from 1 to d do f:=f+c[i][2]; od;
    if g=f and g=j then print(n); break;
fi; od; od; end:
A211169(1000000000000);
  • Mathematica
    t = Table[0, {20}]; k = 2; While[k < 2700000001, x = PrimeOmega[k]; If[ t[[x]] == 0 && PrimeOmega[ 2k + 1] == x, t[[x]] = k; Print[{x, k}]]; k++]; t

Extensions

a(15)-a(24) from Giovanni Resta, Jan 31 2013

A238257 Numbers n such that n and 2n+1 use only odd decimal digits.

Original entry on oeis.org

1, 3, 5, 7, 9, 15, 17, 19, 35, 37, 39, 55, 57, 59, 75, 77, 79, 95, 97, 99, 155, 157, 159, 175, 177, 179, 195, 197, 199, 355, 357, 359, 375, 377, 379, 395, 397, 399, 555, 557, 559, 575, 577, 579, 595, 597, 599, 755, 757, 759, 775, 777, 779, 795, 797, 799, 955, 957, 959, 975, 977, 979, 995, 997, 999, 1555, 1557, 1559, 1575, 1577, 1579
Offset: 1

Views

Author

Zak Seidov, Feb 21 2014

Keywords

Comments

Sophie-Germain-analog for numbers with odd digits.
Also numbers with the first digit an odd digit and the other digits in {5, 7, 9}. - David A. Corneth, May 15 2018

Links

Crossrefs

Cf. A005384, A111153, A211162, A111173, A111176.

Programs

  • Mathematica
    Select[Range[1600],AllTrue[Join[IntegerDigits[#],IntegerDigits[ 2#+1]], OddQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 15 2018 *)
  • PARI
    is(n)=#setminus(setunion(Set(digits(n)),Set(digits(2*n+1))), [1,3,5,7,9])==0 \\ Charles R Greathouse IV, May 15 2018

Formula

a(5(3^k-1)/2) = 10^k-1. - Giovanni Resta, Feb 22 2014

A367496 Semiprimes k such that 2*k - 1 and 2*k + 1 are also semiprimes.

Original entry on oeis.org

25, 46, 93, 118, 133, 145, 161, 206, 226, 235, 259, 267, 291, 295, 334, 335, 361, 377, 395, 407, 447, 497, 529, 573, 579, 583, 669, 674, 685, 694, 695, 781, 843, 898, 899, 921, 926, 961, 979, 1059, 1079, 1114, 1115, 1142, 1159, 1214, 1227, 1241, 1257, 1285, 1286, 1294, 1315, 1379, 1393, 1405
Offset: 1

Views

Author

Zak Seidov and Robert Israel, Nov 20 2023

Keywords

Comments

One of the three semiprimes k, 2*k - 1 and 2*k + 1 is 3 times a prime.

Examples

			a(3) = 93 is a term because 93 = 3 * 31, 2 * 93 - 1 = 185 = 5 * 37 and 2 * 93 + 1 = 187 = 11 * 17 are semiprimes.

Links

Crossrefs

Cf. A001358. Intersection of A111153 and A111168.

Programs

  • Maple
    select(t -> numtheory:-bigomega(t) = 2 and numtheory:-bigomega(2*t-1) = 2 and numtheory:-bigomega(2*t+1) = 2, [$1..2000]);
  • Mathematica
    Select[Range[1410],PrimeOmega[#]==PrimeOmega[2#+1]==PrimeOmega[2#-1]==2 &] (* Stefano Spezia, Nov 20 2023 *)

A381465 Semiprimes k such that 6*k + 1 is also a semiprime.

Original entry on oeis.org

4, 9, 14, 15, 22, 34, 39, 49, 65, 69, 74, 82, 85, 86, 93, 94, 111, 133, 145, 158, 159, 183, 185, 194, 201, 203, 209, 214, 219, 226, 235, 259, 265, 267, 289, 299, 301, 303, 319, 321, 323, 326, 327, 334, 341, 346, 358, 361, 362, 365, 371, 377, 386, 393, 403, 407, 413, 415, 422, 427, 437, 469, 471
Offset: 1

Views

Author

Zak Seidov and Robert Israel, Feb 24 2025

Keywords

Examples

			a(3) = 14 is a term because 14 = 2 * 7 is a semiprime and 6 * 14 + 1 = 85 = 5 * 17 is also a semiprime.

Links

Crossrefs

Cf. A001358, A007693, A111153, A111170.

Programs

  • Maple
    select(t -> numtheory:-bigomega(t) = 2 and numtheory:-bigomega(6*t+1)=2, [$4..1000]);
  • Mathematica
    s = {}; Do[If[{2, 2} == PrimeOmega[{k, 6*k + 1}], AppendTo[s, k]], {k, 1000}] ; s
  • PARI
    isok(k) = (bigomega(k)==2) && (bigomega(6*k+1)==2); \\ Michel Marcus, Feb 26 2025

A085649 Brilliant numbers k such that 2k+1 is also brilliant.

Original entry on oeis.org

4, 10, 533, 671, 869, 1139, 2279, 14873, 17819, 20651, 22523, 23129, 23381, 25199, 28013, 28943, 31439, 31571, 33899, 35369, 35621, 37241, 37403, 39059, 39449, 41291, 42401, 42431, 42881, 43259, 45173, 45803, 49583, 49649, 49949, 50621
Offset: 1

Views

Author

Jason Earls, Amarnath Murthy, Jul 11 2003

Keywords

Examples

			1139 is a member because 1139 = 17*67 and 1139*2+1 = 2279 = 43*53.

Crossrefs

Subsequence of A111153.
Cf. A078972.

Programs

  • PARI
    isA078972(k)={my(f=factor(k));bigomega(f)==2 && (f[1,2]==2 || logint(f[1,1],10)==logint(f[2,1],10))}
isok(k)={isA078972(k) && isA078972(2*k+1)} \\ Andrew Howroyd, Sep 18 2024

Extensions

Offset changed by Andrew Howroyd, Sep 18 2024
Previous Showing 21-27 of 27 results.

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

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