login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A212173 First integer with same second signature as n (cf. A212172). 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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.

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)

FORMULA

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

EXAMPLE

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.

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 *)

PROG

(MAGMA) A212173 := func<n| &*[Integers()| NthPrime(j)^s[j]:j in[1..#s]] where s is Reverse(Sort([pe[2]:pe in Factorisation(n)| pe[2]gt 1]))>; [A212173(n):n in[1..85]]; // Jason Kimberley, Jun 14 2012

(Python)

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 map(a, xrange(1, 151)) # Indranil Ghosh, 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

CROSSREFS

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

Sequence in context: A074058 A088440 A300253 * A274006 A203025 A057521

Adjacent sequences:  A212170 A212171 A212172 * A212174 A212175 A212176

KEYWORD

nonn,easy

AUTHOR

Matthew Vandermast, Jun 03 2012

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 25 23:48 EDT 2019. Contains 322465 sequences. (Running on oeis4.)