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 1-3 of 3 results.

A097272 Least integer with same "mod 2 prime signature" as n.

Original entry on oeis.org

1, 2, 3, 4, 3, 6, 3, 8, 9, 6, 3, 12, 3, 6, 15, 16, 3, 18, 3, 12, 15, 6, 3, 24, 9, 6, 27, 12, 3, 30, 3, 32, 15, 6, 15, 36, 3, 6, 15, 24, 3, 30, 3, 12, 45, 6, 3, 48, 9, 18, 15, 12, 3, 54, 15, 24, 15, 6, 3, 60, 3, 6, 45, 64, 15, 30, 3, 12, 15, 30, 3, 72, 3, 6, 45, 12, 15, 30, 3, 48, 81, 6, 3, 60
Offset: 1

Views

Author

Ray Chandler, Aug 22 2004

Keywords

Comments

For n = 2^e_0 * p_1^e_1 * ... * p_n^e_n where p_i is odd prime and e_1 >= e_2 >= ... >= e_n, define "mod 2 prime signature" to be ordered prime exponents (e_0,e_1,...,e_n).
Least integer with a given "mod 2 prime signature" is obtained by replacing p_1 with 3, p_2 with 5,..., p_n with n-th odd prime.

Links

Crossrefs

Cf. A046523, A097273, A097274, A097275.

Programs

  • Mathematica
    a[n_] := Module[{m = IntegerExponent[n, 2], e}, 2^m * If[n == 2^m, 1, e = FactorInteger[n/2^m][[;; , 2]]; Times @@ (Prime[Range[2, Length[e] + 1]]^ReverseSort[e])]]; Array[a, 100] (* Amiram Eldar, Jul 23 2024 *)
  • PARI
    A000265(n) = (n/2^valuation(n, 2));
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A006519(n) = (1<A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A097272(n) = A006519(n)*A003961(A046523(A000265(n))); \\ Antti Karttunen, Sep 27 2018

Formula

a(n) = A006519(n)*A003961(A046523(A000265(n))). - Antti Karttunen, Sep 27 2018

Extensions

Offset corrected by Antti Karttunen, Sep 27 2018

A097273 Least integer with each "mod 2 prime signature".

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 15, 16, 18, 24, 27, 30, 32, 36, 45, 48, 54, 60, 64, 72, 81, 90, 96, 105, 108, 120, 128, 135, 144, 162, 180, 192, 210, 216, 225, 240, 243, 256, 270, 288, 315, 324, 360, 384, 405, 420, 432, 450, 480, 486, 512, 540, 576, 630, 648, 675, 720
Offset: 1

Views

Author

Ray Chandler, Aug 22 2004

Keywords

Comments

For n = 2^e_0 * p_1^e_1 * ... * p_n^e_n where p_i is odd prime and e_1 >= e_2 >= ... >= e_n, define "mod 2 prime signature" to be ordered prime exponents (e_0,e_1,...,e_n).
Least integer with a given mod 2 prime signature is obtained by replacing p_1 with 3, p_2 with 5,..., p_n with n-th odd prime.
A097272 sorted and duplicates removed.
Numbers k such that A097272(k) = k.
Verified up to a(68) = 972, 2*a(n) is also the order of a dihedral group D such that the lattice of normal subgroups of D is not isomorphic to the lattice of normal subgroups of any dihedral group of order less than 2*a(n). - Miles Englezou, May 18 2025

Links

Crossrefs

Cf. A097272, A097274, A097275.
Cf. A025487, A070826, A147516.

Programs

  • Mathematica
    lpsQ[n_] := n==1 || (Max@ Differences[(f = FactorInteger[n])[[;;,2]]] < 1 && f[[-1,1]] == Prime[Length[f] + 1]); Select[Range[1000], lpsQ[# / 2^IntegerExponent[#, 2]] &] (* Amiram Eldar, Jul 23 2024 *)

Formula

Sum_{n>=1} 1/a(n) = 2 * Product_{n>=2} 1/(1 - 1/A070826(n)) = 3.2482341898... . - Amiram Eldar, Jul 23 2024

Extensions

Offset corrected by Amiram Eldar, Jul 23 2024

A097274 Least integer "mod 2 prime signatures" k ordered by number of Pythagorean triples with leg = k.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 18, 16, 27, 54, 12, 15, 30, 32, 81, 162, 64, 243, 486, 128, 729, 1458, 24, 36, 45, 90, 256, 2187, 4374, 512, 6561, 13122, 1024, 19683, 39366, 48, 108, 135, 270, 2048, 59049, 118098, 4096, 177147, 354294, 72, 225, 450, 8192, 531441
Offset: 0

Views

Author

Ray Chandler, Aug 22 2004

Keywords

Comments

For n=2^a_0*p_1^a_1*...*p_n^a_n where p_i is odd prime and a_1>=a_2>=...>=a_n, define "mod 2 prime signature" to be ordered prime exponents (a_0,a_1,...,a_n).
Least integer with a given mod 2 prime signature is obtained by replacing p_1 with 3, p_2 with 5,..., p_n with n-th odd prime.

Examples

			Table begins:
0: 1,2,
1: 3,4,6,
2: 8,9,18,
3: 16,27,54,
4: 12,15,30,32,81,162,
5: 64,243,486,
6: 128,729,1458,
7: 24,36,45,90,256,2187,4374,
8: 512,6561,13122,
9: 1024,19683,39366,
10: 48,108,135,270,2048,59049,118098,
11: 4096,177147,354294,
12: 72,225,450,8192,531441,1062882,

Links

Crossrefs

Cf. A097272, A097273, A097275.
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.