A097621 -id:A097621 - cp's OEIS frontend

A344826 Integers k such that k/A097621(k) is an integer.

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60, 891, 1584, 1782, 3564, 4032, 4455, 4752, 7920, 8910, 17820, 20160, 22275, 23760, 44550, 49896, 86400, 89100, 100800, 118800, 249480, 349272, 399168, 694008, 1097712, 1746360, 1778400, 1995840, 2181168, 2774304, 2794176, 3470040

Offset: 1

Michel Marcus, May 29 2021, after a suggestion from Allan C. Wechsler

nonn

Allan C. Wechsler remarks that one can derive larger terms from existing terms. For instance, k = 5552064 has q = k/A097621(k) = 18. So multiplying 5552064 by 31 = A000961(18) will give a new term with q = 31.

More precisely, if k = a(n) has q = A343886(k) and m = A000961(q) such that gcd(k, m) = 1, then k*m is also a term. We could call "primitive" those terms not derived from a smaller term in this way. All the listed terms are primitive, but a({35, 36, 38, 42, 43}) allow the sequence to be extended to five larger non-primitive terms. The second and fourth one, having q = 17 resp. q = 23, both lead to a whole chain of many new terms. - M. F. Hasler, Jun 15 2021

Ray Chandler, Table of n, a(n) for n = 1..187 (terms 1..90 from Michel Marcus, terms 91..133 from Lars Blomberg)

Cf. A000961, A095874, A097621, A127724, A343886 (the ratios k/A097621(k)).

f(n) = if(isprimepower(n), sum(i=1, logint(n, 2), primepi(sqrtnint(n, i)))+1, n==1); \\ A095874
ff(n) = my(fr=factor(n)); for (k=1, #fr~, fr[k,1] = f(fr[k,1]^fr[k,2]); fr[k,2] = 1); factorback(fr); \\ A097621
isok(k) = denominator(k/ff(k)) == 1;

mappp(nn) = {my(map = Map()); mapput(map, 1, 1); my(nb=1); for (n=2, nn, if (isprimepower(n), nb++; mapput(map, n, nb));); map;}
ff(n, map) = my(fr=factor(n)); for (k=1, #fr~, fr[k, 1] = mapget(map, fr[k, 1]^fr[k, 2]); fr[k, 2] = 1); factorback(fr); \\ A097621
wa(na, nb) = {my(map = mappp(nb)); for (k=na, nb, if (denominator(k/ff(k, map)) == 1, print1(k, ", ")););}
wa(1, 10^8)

is_A344826(n)=!(n%A097621(n))
extend(n)=n*if(gcd(n, n=A000961(n/A097621(n)))==1,n) \\ Return the larger non-primitive term "derived" from a term n = a(k) with gcd(n,q) = 1, cf. COMMENTS, or zero if gcd(n,q) > 1, i.e., it cannot be "extended" that way. This allows the production of (infinitely?) many new terms from the existing ones. - M. F. Hasler, Jun 15 2021

A097622 A097621(A097621(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 6, 6, 7, 10, 8, 12, 10, 12, 15, 9, 12, 11, 10, 20, 16, 16, 12, 18, 15, 20, 11, 21, 12, 30, 16, 13, 16, 21, 30, 19, 20, 20, 30, 30, 18, 32, 18, 32, 35, 24, 14, 27, 21, 30, 32, 35, 15, 19, 40, 36, 30, 24, 20, 60, 16, 32, 33, 24, 30, 32, 17, 33, 36, 60, 30, 42, 18

Offset: 1

Reinhard Zumkeller, Aug 17 2004

nonn

a(A097621(n))=A097621(a(n))=A097623(n).

A097623 a(n) = A097621(A097621(A097621(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 6, 6, 6, 10, 7, 12, 10, 12, 15, 8, 12, 9, 10, 20, 11, 11, 12, 16, 15, 20, 9, 18, 12, 30, 11, 10, 11, 18, 30, 13, 20, 20, 30, 30, 16, 19, 16, 19, 30, 21, 12, 16, 18, 30, 19, 30, 15, 13, 35, 32, 30, 21, 20, 60, 11, 19, 27, 21, 30, 19, 12, 27, 32, 60, 30, 36, 16, 30

Offset: 1

Reinhard Zumkeller, Aug 17 2004

nonn

a(n) = A097622(A097621(n)) = A097621(A097622(n)).

Cf. A097621, A097622.

A343886 a(n) = k/A097621(k) for k=A344826(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 5, 3, 5, 4, 8, 5, 5, 5, 4, 7, 8, 9, 12, 7, 6, 8, 12, 13, 14, 9, 16, 9, 12, 18, 12, 11, 21, 18, 12, 24, 7, 13, 14, 14, 15, 27, 17, 27, 20, 18, 33, 12, 11, 21, 32, 21, 20, 11, 22, 23, 21, 14, 28, 27, 24, 17

Offset: 1

Michel Marcus, Jun 03 2021

nonn

A344826 is the sequence of numbers k such that k/A097621(k) is an integer. The current sequence lists the corresponding integer quotients.

			The first 12 terms of A344826 are (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60), fixed points of A097621, for which k/A097621(k) is 1.
A344826(13) is 891, so a(13) = 891/A097621(891) = 891/297 = 3, so a(13) = 3.

Ray Chandler, Table of n, a(n) for n = 1..187 (terms 1..90 from Michel Marcus, terms 91..133 from Lars Blomberg)

Cf. A097621, A344826.

A343886(n)=(n=A344826(n))/A097621(n) \\ M. F. Hasler, Jun 15 2021

A097624 Smallest m such that A097621(m) = n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 11, 10, 16, 12, 19, 23, 15, 18, 29, 21, 32, 20, 24, 43, 47, 28, 53, 38, 33, 46, 67, 30, 73, 36, 48, 58, 40, 42, 101, 103, 57, 45, 113, 56, 125, 86, 55, 94, 137, 63, 149, 65, 87, 76, 167, 66, 80, 72, 96, 134, 193, 60, 199, 146, 88, 108, 95, 112, 239, 116

Offset: 1

Reinhard Zumkeller, Aug 17 2004

nonn

A095874 a(n) = k if n = A000961(k) (powers of primes), a(n) = 0 if n is not in A000961.

Original entry on oeis.org

1, 2, 3, 4, 5, 0, 6, 7, 8, 0, 9, 0, 10, 0, 0, 11, 12, 0, 13, 0, 0, 0, 14, 0, 15, 0, 16, 0, 17, 0, 18, 19, 0, 0, 0, 0, 20, 0, 0, 0, 21, 0, 22, 0, 0, 0, 23, 0, 24, 0, 0, 0, 25, 0, 0, 0, 0, 0, 26, 0, 27, 0, 0, 28, 0, 0, 29, 0, 0, 0, 30, 0, 31, 0, 0, 0, 0, 0, 32, 0, 33, 0, 34, 0, 0, 0, 0, 0, 35, 0

Offset: 1

Reinhard Zumkeller, Jun 10 2004

nonn

The name has been edited to clarify that the indices k refer to A000961 ("powers of primes" = {1} U A246655) and not to the list A246655 of proper prime powers. - M. F. Hasler, Jun 16 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000961 (right inverse), A049084, A097621.

a095874 n | y == n    = length xs + 1
          | otherwise = 0
          where (xs, y:ys) = span (< n) a000961_list
-- Reinhard Zumkeller, Feb 16 2012, Jun 26 2011

Join[{1},Module[{k=2},Table[If[PrimePowerQ[n],k;k++,0],{n,2,100}]]] (* Harvey P. Dale, Aug 15 2020 *)

a(n)=if(isprimepower(n), sum(i=1,logint(n,2), primepi(sqrtnint(n,i)))+1, n==1) \\ Charles R Greathouse IV, Apr 29 2015

{M95874=Map(); A095874(n,k)=if(mapisdefined(M95874,n,&k),k, isprimepower(n), mapput(M95874,n, k=sum(i=1,exponent(n), primepi(sqrtnint(n,i)))+1); k,n==1)} \\ Variant with memoization, possibly useful to compute A097621, A344826 and related. One may omit "isprimepower(n)," (possibly requiring factorization) and ",n==1" if n is known to be a power of a prime, i.e., to get a left inverse for A000961. - M. F. Hasler, Jun 15 2021

from sympy import primepi, integer_nthroot, primefactors
def A095874(n): return 1+int(primepi(n)+sum(primepi(integer_nthroot(n,k)[0]) for k in range(2,n.bit_length()))) if n==1 or len(primefactors(n))==1 else 0 # Chai Wah Wu, Jan 19 2025

a(n) = Sum_{1 <= k <= n} A010055(k); [corrected by M. F. Hasler, Jun 15 2021]
a(n) = A065515(n)*(A065515(n)-A065515(n-1)).
a(n) = A065515(n)*A069513(n). - M. F. Hasler, Jun 16 2021

Edited by M. F. Hasler, Jun 15 2021

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A344826 Integers k such that k/A097621(k) is an integer.

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A097622 A097621(A097621(n)).

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A097623 a(n) = A097621(A097621(A097621(n))).

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A343886 a(n) = k/A097621(k) for k=A344826(n).

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A097624 Smallest m such that A097621(m) = n.

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A095874 a(n) = k if n = A000961(k) (powers of primes), a(n) = 0 if n is not in A000961.

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