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.

Showing 1-5 of 5 results.

A104210 Positive integers divisible by at least 2 consecutive primes.

Original entry on oeis.org

6, 12, 15, 18, 24, 30, 35, 36, 42, 45, 48, 54, 60, 66, 70, 72, 75, 77, 78, 84, 90, 96, 102, 105, 108, 114, 120, 126, 132, 135, 138, 140, 143, 144, 150, 154, 156, 162, 165, 168, 174, 175, 180, 186, 192, 195, 198, 204, 210, 216, 221, 222, 225, 228, 231, 234, 240
Offset: 1

Views

Author

Leroy Quet, Mar 13 2005

Keywords

Comments

If a perfect square is in this sequence, then so is its square root (e.g., 144 and 12). - Alonso del Arte, May 07 2012
The numbers of terms not exceeding 10^k, for k=1,2,..., are 1, 22, 242, 2456, 24632, 246414, 2464272, 24643281, 246433426, ... Apparently, the asymptotic density of this sequence is 0.24643... - Amiram Eldar, Apr 10 2021

Examples

			35 is divisible by both 5 and 7, and 5 and 7 are consecutive primes.
77 is divisible by both 7 and 11, and 7 and 11 are consecutive primes.
110 is not in the sequence because, although it is divisible by 2, 5 and 11, it is not divisible by 3 or 7.

Links

Crossrefs

Cf. A003961, A296210 (characteristic function), A319630 (complement), A379230 [= A252748(a(n))].
Positions of terms larger than 1 in A300820 and in A322361.
Subsequences: A006094, A349169 (conjectured, after its initial 1), A349176, A355527 (squarefree terms), A372566, A378884, A379232.

Programs

  • Maple
    N:= 1000: # for terms <= N
R:= {}:
p:= 2:
do
  q:= p; p:= nextprime(p);
  if p*q > N then break fi;
  R:= R union {seq(i,i=p*q..N,p*q)}
od:
sort(convert(R,list)); # Robert Israel, Apr 13 2020
  • Mathematica
    fQ[n_] := Block[{lst = PrimePi /@ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]}, Count[ Drop[lst, 1] - Drop[lst, -1], 1] > 0]; Select[ Range[244], fQ[ # ] &] (* Robert G. Wilson v, Mar 16 2005 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
is_A104210(n) = (gcd(n,A003961(n))>1); \\ Antti Karttunen, Dec 24 2024

Formula

{k such that gcd(k, A003961(k)) > 1}. - Antti Karttunen, Dec 24 2024

Extensions

More terms from Robert G. Wilson v, Mar 16 2005

A349174 Odd numbers k for which gcd(k, A003961(k)) is equal to gcd(sigma(k), A003961(k)), where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 55, 59, 61, 63, 67, 69, 71, 73, 79, 81, 83, 85, 89, 91, 93, 95, 97, 101, 103, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 167, 169
Offset: 1

Views

Author

Antti Karttunen, Nov 10 2021

Keywords

Comments

Odd numbers k for which A322361(k) = A342671(k).
Odd numbers k for which A348994(k) = A349161(k).
Odd numbers k such that A319626(k) = A349164(k).
Odd terms of A336702 form a subsequence of this sequence. See also A349169.
Ratio of odd numbers residing in this sequence, vs. in A349175 seems to slowly decrease, but still apparently stays > 2 for a long time. E.g., for range 2 .. 2^28, it is 95302074/38915653 = 2.4489...

Links

Crossrefs

Cf. A000203, A003961, A319626, A322361, A336702, A342671, A348994, A349161, A349164, A349169.
Cf. A349175 (complement among the odd numbers).
Union of A349176 and A349177.

Programs

  • Mathematica
    Select[Range[1, 169, 2], GCD[#1, #3] == GCD[#2, #3] & @@ {#, DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &] (* Michael De Vlieger, Nov 11 2021 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349174(n) = if(!(n%2),0,my(u=A003961(n)); gcd(u,sigma(n))==gcd(u,n));

A372567 Odd numbers k such that k, sigma(k) and A003961(k) have a common divisor larger than 1, where A003961(n) is fully multiplicative function with a(prime(i)) = prime(i+1).

Original entry on oeis.org

135, 285, 435, 455, 675, 855, 885, 945, 1185, 1287, 1305, 1335, 1365, 1425, 1435, 1485, 1635, 1755, 1995, 2085, 2175, 2235, 2275, 2295, 2565, 2655, 2685, 2905, 2985, 3045, 3105, 3135, 3185, 3311, 3375, 3395, 3435, 3555, 3585, 3705, 3915, 4005, 4035, 4095, 4185, 4235, 4275, 4305, 4425, 4725, 4785, 4845, 4865, 4905
Offset: 1

Views

Author

Antti Karttunen, May 19 2024

Keywords

Comments

Most seem to be multiples of 5.

Examples

			135 = 3^3 * 5, sigma(135) = 240 = 2^4 * 3 * 5, and A003961(135) = 875 = 5^3 * 7 have 5 as their common divisor, therefore 135 is present in this sequence.

Links

Crossrefs

Odd terms in A372566.
Cf. A000203, A003961, A349176 (subsequence).
Cf. also conjecture 1 in A349753.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA372567(n) = ((n%2) && (1A003961(n)])));

A387166 Numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)) > 1, and that satisfy Euler's condition for odd perfect numbers (A228058).

Original entry on oeis.org

14157, 33525, 101025, 118825, 129605, 281025, 300713, 301725, 335405, 348525, 358925, 438525, 573525, 618525, 686025, 688205, 696725, 742577, 776025, 838125, 909225, 911025, 978525, 1046025, 1079225, 1099805, 1226025, 1293525, 1316025, 1322893, 1428889, 1451025, 1529045, 1563525, 1698525, 1721025, 1788525, 1991025, 2036025
Offset: 1

Views

Author

Antti Karttunen, Aug 28 2025

Keywords

Links

Crossrefs

Intersection of A228058 and A349176.
Intersection of A387164 and A104210, or equally, intersection of A387164 and A349166.
Setwise difference A387164 \ A387167.
Cf. A000203, A003961.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA228058(n) = if(!(n%2)||(omega(n)<2), 0, my(f=factor(n), y=0); for(i=1, #f~, if(1==(f[i, 2]%4), if((1==y)||(1!=(f[i, 1]%4)), return(0), y=1), if(f[i, 2]%2, return(0)))); (y));
isA349176(n) = if(!(n%2),0,my(u=A003961(n),t=gcd(u,n)); (t>1)&&(gcd(u,sigma(n))==t));
isA387166(n) = (isA228058(n) && isA349176(n));

A349177 Odd numbers k for which gcd(k, A003961(k)) = gcd(sigma(k), A003961(k)) = 1, where A003961(n) is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 55, 59, 61, 63, 67, 69, 71, 73, 79, 81, 83, 85, 89, 91, 93, 95, 97, 101, 103, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 137, 139, 141, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 167, 169, 173
Offset: 1

Views

Author

Antti Karttunen, Nov 11 2021

Keywords

Comments

Odd numbers k for which k and A003961(k) are relatively prime, and also sigma(k) and A003961(k) are coprime.

Links

Crossrefs

Subsequence of A349174 from this first differs by not having term 135 (see A349176).
Intersection of A319630 and A349174, or equally, intersection of A349165 and A349174.
Cf. A000203, A003961, A319626, A348994, A349161, A349164, A349169.

Programs

  • Mathematica
    Select[Range[1, 173, 2], GCD[#1, #3] == GCD[#2, #3] == 1 & @@ {#, DivisorSigma[1, #], Times @@ Map[NextPrime[#1]^#2 & @@ # &, FactorInteger[#]]} &] (* Michael De Vlieger, Nov 11 2021 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA349177(n) = if(!(n%2),0,my(u=A003961(n),t=gcd(u,n)); (1==t)&&(gcd(u,sigma(n))==t));
Showing 1-5 of 5 results.

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