A212173 First integer with same second signature as n (cf. A212172).

1, 1, 1, 4, 1, 1, 1, 8, 4, 1, 1, 4, 1, 1, 1, 16, 1, 4, 1, 4, 1, 1, 1, 8, 4, 1, 8, 4, 1, 1, 1, 32, 1, 1, 1, 36, 1, 1, 1, 8, 1, 1, 1, 4, 4, 1, 1, 16, 4, 4, 1, 4, 1, 8, 1, 8, 1, 1, 1, 4, 1, 1, 4, 64, 1, 1, 1, 4, 1, 1, 1, 72, 1, 1, 4, 4, 1, 1, 1, 16, 16, 1, 1, 4

Offset: 1

Author

Matthew Vandermast, Jun 03 2012

Keywords

• nonn
• easy

Comments

Two integers have the same second signature iff the same exponents >= 2 occur in the canonical prime factorization of each integer, regardless of the order in which they occur in each factorization.

Examples

			12 = 2^2*3 has 1 exponent >= 2 in its prime factorization, namely, 2. Hence, its second signature is {2}.  The smallest number with second signature {2} is 4; hence, a(12) = 4.
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Links

• Jason Kimberley, Table of n, a(n) for n = 1..10000
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• Primefan, The First 2500 Integers Factored (1st of 5 pages)

Crossrefs

Cf. A212172, A046523. All terms belong to A181800.

Programs

Magma

A212173 := func; [A212173(n):n in[1..85]]; // Jason Kimberley, Jun 14 2012

Maple

f:= proc(n) local E,i;
E:= sort(select(`>`, map(t -> t[2], ifactors(n)[2]), 1),`>`);
mul(ithprime(i)^E[i],i=1..nops(E))
end proc:
map(f, [$1..100]); # Robert Israel, Jul 19 2017

Mathematica

Function[s, Sort[Apply[Join, Map[Function[k, Map[{#, First@ k} &, k]], Values@ s]]][[All, -1]]]@ KeySort@ PositionIndex@ Table[Sort@ DeleteCases[FactorInteger[n][[All, -1]], e_ /; e < 2] /. {} -> {1}, {n, 84}] (* Michael De Vlieger, Jul 19 2017 *)

PARI

a(n) = {my(sn = vecsort(select(x->(x>=2), factor(n)[,2]))); for (i=1, n, if (vecsort(select(x->(x>=2),factor(i)[,2])) == sn, return(i)););} \\ Michel Marcus, Jul 19 2017

Python

from functools import reduce
from sympy import factorint
from operator import mul
def P(n): return sorted(factorint(n).values())
def a046523(n):
    x=1
    while True:
        if P(n)==P(x): return x
        else: x+=1
def a057521(n): return 1 if n==1 else reduce(mul, [1 if e==1 else p**e for p, e in factorint(n).items()])
def a(n): return a046523(a057521(n))
print([a(n) for n in range(1, 151)]) # Indranil Ghosh, Jul 19 2017

Formula

a(n) = A046523(A057521(n)) = A057521(A046523(n)).