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 1-6 of 6 results.

A354525 Numbers k such that A354512(k) = A001221(k).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 11, 13, 14, 15, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 55, 59, 61, 62, 67, 69, 71, 73, 77, 79, 83, 85, 89, 91, 93, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 127, 131, 133, 137, 139, 141, 143, 145, 149, 151, 155, 157
Offset: 1

Views

Author

Jianing Song, Aug 16 2022

Keywords

Comments

Numbers k such that for every prime factor p of k we have gpf(k+p) = p, gpf = A006530.
Numbers k such that for every prime factor p of k, k+p is p-smooth.
If k is an even term, then k+2 is a power of 2, so k is of the form 2*(2^m-1). Those m for which 2*(2^m-1) is a term are listed in A354531.

Examples

			15 is a term since the prime factors of 15 are 3,5, and we have gpf(15+3) = 3 and gpf(15+5) = 5.

Links

Crossrefs

Cf. A001221, A354512, A006530, A354531, A354532, A354533, A354534.
Indices of 0 in A354527. Complement of A354526.

Programs

  • PARI
    gpf(n) = vecmax(factor(n)[, 1]);
isA354525(n) = my(f=factor(n)[, 1]); for(i=1, #f, if(gpf(n+f[i])!=f[i], return(0))); 1

A354531 Numbers k such that 2*(2^k-1) is in A354525.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 13, 17, 19, 31, 61, 67, 89, 107, 127, 137, 521, 607, 727
Offset: 1

Views

Author

Jianing Song, Aug 16 2022

Keywords

Comments

Numbers k such that for every prime factor p of 2^k-1 we have gpf(2*(2^k-1)+p) = p.
Numbers k such that for every prime factor p of 2^k-1, 2*(2^k-1)+p is p-smooth.
All terms except 2 are odd: if k is even, then 3 is a factor of 2^k-1, so 3^m = 2*(2^k-1)+3 = 2^(k+1) + 1 => k+1 >= 3^(m-1). The only possible case is (k,m) = (2,2).
Clearly A000043 is a subsequence. The exceptional terms (1, 9, 67, 137, ...) are listed in A354532.
The next term is >= 349. The next composite term, if it exists, is >= 7921 = 89^2.

Examples

			See A354532.

Crossrefs

Cf. A000043, A354525, A354532, A354533, A354536.

Programs

  • PARI
    gpf(n) = vecmax(factor(n)[, 1]);
ispsmooth(n,p,{lim=1<<256}) = if(n<=lim, n==1 || gpf(n)<=p, my(N=n/p^valuation(n,p)); forprime(q=2, p, N=N/q^valuation(N,q); if((N<=lim && isprime(N)) || N==1, return(N<=p))); 0); \\ check if n is p-smooth, using brute force if n is too large
isA354531(n,{lim=256},{p_lim=1<<32}) = {
  my(N=2^n-1);
  if(isprime(N), return(1));
  if(n>lim, forprime(p=3, p_lim, if(N%p==0 && !ispsmooth(2*N+p,p), return(0)))); \\ first check if there is a prime factor p <= p_lim of 2^n-1 such that 2*(2^n-1)+p is not p-smooth (for large n)
  my(d=divisors(n));
  for(i=1, #d, my(f=factor(2^d[i]-1)[, 1]); for(j=1, #f, if(!ispsmooth(2*N+f[j],f[j],1<

Extensions

a(17)-a(19) from Jinyuan Wang, Jan 21 2025

A354533 Even terms in A354525.

Original entry on oeis.org

2, 6, 14, 62, 254, 1022, 16382, 262142, 1048574, 4294967294, 4611686018427387902, 295147905179352825854, 1237940039285380274899124222, 324518553658426726783156020576254, 340282366920938463463374607431768211454, 348449143727040986586495598010130648530942
Offset: 1

Views

Author

Jianing Song, Aug 16 2022

Keywords

Comments

Even numbers k such that for every prime factor p of k we have gpf(k+p) = p, gpf = A006530.
Even numbers k such that for every prime factor p of k, k+p is p-smooth.

Examples

			See A354532.

Crossrefs

Cf. A006530, A354525, A354531, A354532, A354534, A354536.

Programs

  • PARI
    lista(nn,{lim=256},{lim_p=1<<32}) = for(n=1, nn, if(isA354531(n,lim,lim_p), print1(2*(2^n-1), ", "))) \\ See A354531 for the function isA354531

Formula

a(n) = 2*(2^A354531(n) - 1).

A354534 Even terms in A354525 that are not twice the Mersenne primes (A000668).

Original entry on oeis.org

2, 1022, 295147905179352825854, 348449143727040986586495598010130648530942
Offset: 1

Views

Author

Jianing Song, Aug 16 2022

Keywords

Comments

Terms in A354533 that are not twice the Mersenne primes. Note that all twice the Mersenne primes are in A354533.

Examples

			See A354532.

Crossrefs

Cf. A006530, A354525, A354532, A354533, A354537.

Programs

  • PARI
    lista(nn,{lim=256},{lim_p=1<<32}) = for(n=1, nn, if(isA354532(n,lim,lim_p), print1(2*(2^n-1), ", "))) \\ See A354532 for the function isA354532

Formula

a(n) = 2*(2^A354532(n) - 1) = 2*A354537(n).

A354536 Numbers k such that 2*k is in A354525.

Original entry on oeis.org

1, 3, 7, 31, 127, 511, 8191, 131071, 524287, 2147483647, 2305843009213693951, 147573952589676412927, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727, 174224571863520493293247799005065324265471
Offset: 1

Views

Author

Jianing Song, Aug 17 2022

Keywords

Comments

Numbers k such that for every prime factor p of k we have gpf(2*k+p) = p, gpf = A006530.
Numbers k such that for every prime factor p of k, 2*k+p is p-smooth.
a(17) = 2^521 - 1 is too large to include here. - Jinyuan Wang, Jan 21 2025

Examples

			See A354532.

Crossrefs

Cf. A006530, A354525, A354531, A354532, A354533, A354537.

Programs

  • PARI
    lista(nn,{lim=256},{lim_p=1<<32}) = for(n=1, nn, if(isA354531(n,lim,lim_p), print1(2^n-1, ", "))) \\ See A354531 for the function isA354531

Formula

a(n) = 2^A354531(n) - 1 = A354533(n)/2.

A354537 Numbers k that are not Mersenne primes (A000668) such that 2*k is in A354525.

Original entry on oeis.org

1, 511, 147573952589676412927, 174224571863520493293247799005065324265471
Offset: 1

Views

Author

Jianing Song, Aug 17 2022

Keywords

Comments

Terms in A354536 that are not Mersenne primes. Note that all Mersenne primes are in A354536.
a(5) = 2^727 - 1 is too large to include here. - Jinyuan Wang, Jan 21 2025

Examples

			See A354532.

Crossrefs

Cf. A000668, A354525, A354532, A354534, A354536.

Programs

  • PARI
    lista(nn,{lim=256},{lim_p=1<<32}) = for(n=1, nn, if(isA354532(n,lim,lim_p), print1(2^n-1, ", "))) \\ See A354532 for the function isA354532

Formula

By definition, equals A354536 \ A000668.
a(n) = 2^A354532(n) - 1 = A354534(n)/2.
Showing 1-6 of 6 results.

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