A097272 -id:A097272 - cp's OEIS frontend

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

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

Ray Chandler, Aug 22 2004

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A097272, A097274, A097275.

Cf. A025487, A070826, A147516.

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 *)

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

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

Ray Chandler, Aug 22 2004

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.

			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,

Eric Weisstein's World of Mathematics, Pythagorean Triple.

Cf. A097272, A097273, A097275.

A097275 Least integer "mod 2 prime signatures" k ordered by number of primitive Pythagorean triples with leg = k.

Original entry on oeis.org

1, 2, 3, 6, 4, 12, 18, 8, 15, 60, 30, 9, 24, 105, 420, 54, 16, 36, 120, 840, 4620, 90, 27, 45, 180, 1155, 9240, 60060, 162, 32, 48, 240, 1260, 13860, 120120, 1021020, 210, 64, 72, 315, 1680, 15015, 180180, 2042040, 19399380, 270, 81, 96, 360, 2520, 18480, 240240

Offset: 0

Ray Chandler, Aug 22 2004

Row 0 of table represents "mod 2 prime signature" values k such that no PPTs have leg=k.
Row n of table, n>0, represents "mod 2 prime signature" values k such that 2^(n-1) PPTs have leg=k. Table read by antidiagonals.
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.

			Table begins:
0: 1,2,6,18,30,54,90,162,210,270,...
1: 3,4,8,9,16,27,32,64,81,128,...
2: 12,15,24,36,45,48,72,96,108,135,...
4: 60,105,120,180,240,315,360,480,540,720,...
8: 420,840,1155,1260,1680,2520,3360,3465,3780,5040,...
16: 4620,9240,13860,15015,18480,27720,36960,41580,45045,55440,...
32: 60060,120120,180180,240240,255255,360360,480480,540540,...
64: 1021020,2042040,3063060,4084080,4849845,6126120,8168160,...
128: 19399380,38798760,58198140,77597520,111546435,116396280,...
256: 446185740,892371480,1338557220,1784742960,2677114440,...

Eric Weisstein's World of Mathematics, Pythagorean Triple.

Cf. A097272, A097273, A097274.

Row 1 is A006899 except for starting point.

A322022 Lexicographically earliest such sequence a that a(i) = a(j) => A305891(i) = A305891(j) and A319697(i) = A319697(j), for all i, j.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 8, 3, 9, 3, 10, 11, 12, 3, 13, 3, 14, 11, 15, 3, 16, 7, 17, 18, 19, 3, 20, 3, 21, 11, 22, 11, 23, 3, 24, 11, 25, 3, 26, 3, 27, 28, 29, 3, 30, 7, 31, 11, 32, 3, 33, 11, 34, 11, 35, 3, 36, 3, 37, 28, 38, 11, 39, 3, 40, 11, 39, 3, 41, 3, 42, 28, 43, 11, 44, 3, 45, 46, 47, 3, 48, 11, 49, 11, 50, 3, 51, 11, 52, 11, 53, 11, 54, 3, 55, 28

Offset: 1

Antti Karttunen, Nov 29 2018

nonn

Restricted growth sequence transform of the ordered pair [A305891(n), A319697(n)], or equally, of the triple [A007814(n), A046523(n), A319697(n)].

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A007814, A046523, A097272, A305891, A319697, A319698, A322021.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007814(n) = valuation(n,2);
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A319697(n) = sumdiv(n, d, (!(d%2))*issquarefree(d)*d);
v322022 = rgs_transform(vector(up_to, n, [A007814(n), A046523(n), A319697(n)]));
A322022(n) = v322022[n];

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

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

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A097275 Least integer "mod 2 prime signatures" k ordered by number of primitive Pythagorean triples with leg = k.

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A322022 Lexicographically earliest such sequence a that a(i) = a(j) => A305891(i) = A305891(j) and A319697(i) = A319697(j), for all i, j.

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