A064097 -id:A064097 - cp's OEIS frontend

A105017 Positions of records in A064097.

1, 2, 3, 5, 7, 11, 19, 23, 43, 47, 94, 139, 235, 283, 517, 659, 1081, 1319, 2209, 2879, 5758, 8637, 13301, 20147, 30337, 49727, 61993, 103823, 135313, 247439, 366683, 606743, 811879, 1266767, 1739761, 2913671, 3797401, 5827343, 8288641, 16577282, 22784407, 37346483, 58003213, 81768767

Offset: 1

Hugo Pfoertner, Feb 17 2006

nonn

With a(1) = 1, a(n) is the smallest number m such that the number of iterations of k -> k - k/p, p being any prime factor of k, needed to reach 1 starting at k = m is equal to n-1. (See Example section.) - Jaroslav Krizek, Feb 15 2010

a(n) =~ sqrt(e^(5n/6)). - Robert G. Wilson v, Aug 11 2022

			a(6)=11 because m=11 requires 6-1 = 5 iterations of r -> r - (largest divisor d < r) to reach 1 (the 5 iterations are 11-1=10, 10-5=5, 5-1=4, 4-2=2, and 2-1=1) and 11 is the smallest such number m. - _Jaroslav Krizek_, Feb 15 2010

Robert G. Wilson v, Table of n, a(n) for n = 1..200
Hugo Pfoertner, Program

Cf. A000079, A064097, A067513, A175125.

A105017 := proc()
    local maxa,a ;
    maxa := -999 ;
    for n from 1 do
        a := A064097(n) ;
        if a > maxa then
            printf("%d\n",n) ;
            maxa :=a ;
        end if;
    end do:
end proc:
A105017() ; # R. J. Mathar, Aug 07 2022

g[n_] := Block[{p = Select[1 + Divisors@n, PrimeQ]}, n*p/(p - 1)]; f[n_] := f[n] = Block[{lst = Union@Flatten[g@# & /@ f[n - 1]]}, If[ Length@ lst > 325, lst = Take[lst, 325 (* This limit must be increased for greater n's from the start. *) ]]; lst]; f[1] = {1}; f[0] = {0}; lst = {}; Do[ AppendTo[lst, Min[ f[n]]]; f[n - 1] =., {n, 44}]; lst (* Robert G. Wilson v, Aug 11 2022 *)

a=vectorsmall(10^7); a[1]=0;
for(n=2,#a,if(isprime(n),a[n]=1+a[n-1],f=factor(n);a[n]=a[f[1,1]]+a[n/f[1,1]])); \\ computes A064097
r=-oo; for(k=1,#a,if(a[k]>r,print1(k,", ");r=a[k])); \\ Hugo Pfoertner, Mar 16 2020

a(1)=1 inserted by Robert G. Wilson v, Mar 16 2020

A334111 Irregular triangle where row n gives all terms k for which A064097(k) = n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 12, 16, 11, 13, 14, 15, 17, 18, 20, 24, 32, 19, 21, 22, 25, 26, 27, 28, 30, 34, 36, 40, 48, 64, 23, 29, 31, 33, 35, 37, 38, 39, 41, 42, 44, 45, 50, 51, 52, 54, 56, 60, 68, 72, 80, 96, 128, 43, 46, 49, 53, 55, 57, 58, 61, 62, 63, 65, 66, 70, 73, 74, 75, 76, 78, 81, 82, 84

Offset: 0

Antti Karttunen, Michael De Vlieger and Robert G. Wilson v, Apr 14 2020

Applying map k -> (p-1)*(k/p) to any term k on any row n > 1, where p is any prime factor of k, gives one of the terms on preceding row n-1.
Any prime that appears on row n is 1 + {some term on row n-1}.
The e-th powers of the terms on row n form a subset of terms on row (e*n). More generally, a product of terms that occur on rows i_1, i_2, ..., i_k can be found at row (i_1 + i_2 + ... + i_k), because A064097 is completely additive.
A001221(k) gives the number of terms on the row above that are immediate descendants of k.
A067513(k) gives the number of terms on the row below that lead to k.

			Rows 0-6 of the irregular table:
0 |   1;
1 |   2;
2 |   3, 4;
3 |   5, 6, 8;
4 |   7, 9, 10, 12, 16;
5 |  11, 13, 14, 15, 17, 18, 20, 24, 32;
6 |  19, 21, 22, 25, 26, 27, 28, 30, 34, 36, 40, 48, 64;

Michael De Vlieger, Table of n, a(n) for n = 0..14422 (rows 0 <= n <= 17, flattened)
Index entries for sequences that are permutations of the natural numbers

Cf. A001221, A064097, A067513, A333123, A334144.
Cf. A105017 (left edge), A000079 (right edge), A175125 (row lengths).
Cf. also A058812, A334100.

f[n_] := Length@ NestWhileList[# - #/FactorInteger[#][[1, 1]] &, n, # != 1 &]; SortBy[ Range@70, f]
(* Second program *)
With[{nn = 8}, Values@ Take[KeySort@ PositionIndex@ Array[-1 + Length@ NestWhileList[# - #/FactorInteger[#][[1, 1]] &, #, # > 1 &] &, 2^nn], nn + 1]] // Flatten (* Michael De Vlieger, Apr 18 2020 *)

A060681(n) = (n-if(1==n,n,n/vecmin(factor(n)[,1])));
A064097(n) = if(1==n,0,1+A064097(A060681(n)));
for(n=0, 10, for(k=1,2^n,if(A064097(k)==n, print1(k,", "))));

A076091 Numbers n such that A064097(n) - A003313(n) = 1.

Original entry on oeis.org

23, 33, 43, 46, 47, 49, 59, 65, 66, 67, 69, 77, 83, 86, 92, 94, 98, 99, 107, 115, 118, 121, 130, 131, 132, 133, 134, 138, 139, 141, 145, 147, 149, 154, 163, 165, 166, 167, 172, 173, 177, 179, 184, 188, 195, 196, 197, 198, 199, 201, 203, 207, 209

Offset: 1

Benoit Cloitre, Oct 31 2002

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000 (also based on the b-file of A003313 provided by D. W. Wilson and Antoine Mathys)

Cf. A003313, A064097, A076142.

Corrected and extended by Hugo Pfoertner, Feb 17 2006

A076142 a(n) = A064097(n) - A003313(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 1

Benoit Cloitre, Oct 31 2002

nonn

The positions where k = 0, 1, 2, ... occur for the first time are 1, 23, 129, 517, 2049, 4613, 33097, 33793, ... factorized as: 1, 23, 3*43, 11*47, 3*683, 7*659, 23*1439, 47*719, ... - Antti Karttunen, Aug 18 2017

Antti Karttunen, Table of n, a(n) for n = 1..10000 (also based on the b-file of A003313)

Cf. A003313, A064097, A076091.

It seems that sum(k = 1, n, a(k)) * log(n)/n^2 -> c (0.006 < c < 0.01).

Extended to 129 terms by Antti Karttunen, Aug 18 2017

A334090 a(1) = 0, and then after the first differences of A064097.

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 1, -1, 1, 0, 1, -2, 1, 0, 0, 0, 1, -1, 1, -2, 2, -1, 1, -1, 1, 0, 0, -1, 1, 0, 1, -1, 0, 1, 1, -3, 2, -1, 0, 0, 1, -1, 1, -1, 1, 0, 1, -2, 1, 0, 0, -2, 2, 0, 1, -2, 2, -1, 1, -2, 1, 0, 0, 0, 1, -1, 1, -2, 1, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, 0, 0, 1, -1, -2, 1, 1, 0, -1, 1, -1, 1, -1, 1

Offset: 1

Antti Karttunen, Apr 17 2020

sign

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

Cf. A064097, A334091, A334195.
Cf. A334197 (positions of records).
Cf. also A332903.

A064097(n) = if(1==n,0,1+A064097(n-(n/vecmin(factor(n)[,1]))));
A334090(n) = if(1==n,0,A064097(n)-A064097(n-1));

a(1) = 0, and for n > 1, a(n) = A064097(n) - A064097(n-1).

a(n) = A334091(n) + A334195(n).

A307742 Quasi-logarithm A064097(n) of von Mangoldt's exponential function A014963(n).

Original entry on oeis.org

0, 1, 2, 1, 3, 0, 4, 1, 2, 0, 5, 0, 5, 0, 0, 1, 5, 0, 6, 0, 0, 0, 7, 0, 3, 0, 2, 0, 7, 0, 7, 1, 0, 0, 0, 0, 7, 0, 0, 0, 7, 0, 8, 0, 0, 0, 9, 0, 4, 0, 0, 0, 8, 0, 0, 0, 0, 0, 9, 0, 8, 0, 0, 1, 0, 0, 9, 0, 0, 0, 9, 0, 8, 0, 0, 0, 0, 0, 9, 0, 2, 0, 9, 0, 0, 0, 0, 0, 9

Offset: 1

I. V. Serov, Apr 26 2019

nonn

I. V. Serov, Table of n, a(n) for n = 1..10000

Cf. A064097, A014963, A008683, A307743.

qLog[n_] := qLog[n] = Module[{p, e}, If[n == 1, 0, Sum[{p, e} = pe; (1 + qLog[p-1])e, {pe, FactorInteger[n]}]]];
a[n_] := qLog[Exp[MangoldtLambda[n]]];
Array[a, 100] (* Jean-François Alcover, May 07 2019 *)

mang(n) = ispower(n, , &n); if(isprime(n), n, 1); \\ A014963
ql(n) = if (n==1, 0, if(isprime(n),1+ql(n-1), sumdiv(n,p, if(isprime(p),ql(p)*valuation(n,p))))); \\ A064097
a(n) = ql(mang(n)); \\ Michel Marcus, Apr 26 2019

a(n) = A064097(A014963(n)).
a(n) = 1 + A064097(n-1) if n is prime.
a(n) = a(p) if n=p^k with k > 1.
a(n) = 0 if n is not a prime power or n = 1.
a(n) = -Sum_{d|n} A064097(d)*A008683(d) by Mobius inversion.

A373365 a(n) = gcd(A001414(n), A064097(n)), where A001414 is the sum of prime factors with repetition, and A064097 is a quasi-logarithm defined inductively by a(1) = 0 and a(p) = 1 + a(p-1) if p is prime and a(n*m) = a(n) + a(m) if m,n > 1.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 3, 1, 1, 2, 1, 5, 7, 1, 1, 2, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 8, 1, 2, 1, 1, 1, 1, 1, 1, 6, 2, 8, 1, 7, 1, 2, 1, 1, 1, 1, 1, 1, 9, 2, 1, 1, 4, 1, 1, 2, 2, 9, 1, 1, 1, 1, 1, 9, 1, 1, 3, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3

Offset: 1

Antti Karttunen, Jun 02 2024

nonn

As A001414 and A064097 are both fully additive sequences, all sequences that give the positions of multiples of some k > 1 in this sequence are closed under multiplication.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A001414, A064097.

Cf. also A082299, A373362, A373363, A373364, A373366.

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
A064097(n) = if(1==n,0,1+A064097(n-(n/vecmin(factor(n)[,1]))));
A373365(n) = gcd(A001414(n), A064097(n));

A373366 a(n) = gcd(A064097(n), A083345(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 2, 1, 1, 1, 5, 7, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 7, 1, 1, 8, 1, 2, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 9, 1, 1, 1, 4, 1, 1, 1, 2, 9, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Jun 02 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A064097, A083345.

Cf. also A373363, A373365.

A064097(n) = if(1==n,0,1+A064097(n-(n/vecmin(factor(n)[,1]))));
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A373366(n) = gcd(A064097(n), A083345(n));

A334202 a(n) = A064097(n) - A323077(n).

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 2, 5, 4, 3, 1, 5, 2, 6, 3, 4, 5, 7, 2, 3, 5, 2, 4, 7, 3, 7, 1, 5, 5, 4, 2, 7, 6, 5, 3, 7, 4, 8, 5, 3, 7, 9, 2, 4, 3, 5, 5, 8, 2, 5, 4, 6, 7, 9, 3, 8, 7, 4, 1, 5, 5, 9, 5, 7, 4, 9, 2, 8, 7, 3, 6, 5, 5, 9, 3, 2, 7, 9, 4, 5, 8, 7, 5, 9, 3, 5, 7, 7, 9, 6, 2, 8, 4, 5, 3, 9, 5, 9, 5, 4

Offset: 1

Antti Karttunen, May 11 2020

nonn

a(A122111(n)) differs from A001222(n) for the first time at n=64.

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

Cf. A006530, A060681, A064097, A323077.

A060681(n) = (n-if(1==n,n,n/vecmin(factor(n)[,1])));
A064097(n) = if(1==n,0,1+A064097(A060681(n)));
A323077(n) = if(1>=bigomega(n),0,1+A323077(A060681(n)));
A334202(n) = (A064097(n)-A323077(n));

A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A334202(n) = A064097(A006530(n));

a(n) = A064097(n) - A323077(n).

a(n) = A064097(A006530(n)).

A334203 a(n) = A064097(A032742(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 2, 2, 3, 0, 3, 0, 4, 3, 3, 0, 4, 0, 4, 4, 5, 0, 4, 3, 5, 4, 5, 0, 5, 0, 4, 5, 5, 4, 5, 0, 6, 5, 5, 0, 6, 0, 6, 5, 7, 0, 5, 4, 6, 5, 6, 0, 6, 5, 6, 6, 7, 0, 6, 0, 7, 6, 5, 5, 7, 0, 6, 7, 7, 0, 6, 0, 7, 6, 7, 5, 7, 0, 6, 6, 7, 0, 7, 5, 8, 7, 7, 0, 7, 5, 8, 7, 9, 6, 6, 0, 8, 7, 7, 0, 7, 0, 7, 7

Offset: 1

Antti Karttunen, May 13 2020

nonn

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

Cf. A020639, A032742, A060681, A064097, A323077.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A060681(n) = (n-if(1==n,n,n/vecmin(factor(n)[,1])));
A064097(n) = if(1==n,0,1+A064097(A060681(n)));
A334203(n) = A064097(A032742(n));

a(n) = A064097(A032742(n)) = A064097(n/A020639(n)).

cp's OEIS Frontend

A105017 Positions of records in A064097.

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A334111 Irregular triangle where row n gives all terms k for which A064097(k) = n.

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A076091 Numbers n such that A064097(n) - A003313(n) = 1.

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A076142 a(n) = A064097(n) - A003313(n).

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A334090 a(1) = 0, and then after the first differences of A064097.

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A307742 Quasi-logarithm A064097(n) of von Mangoldt's exponential function A014963(n).

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A373365 a(n) = gcd(A001414(n), A064097(n)), where A001414 is the sum of prime factors with repetition, and A064097 is a quasi-logarithm defined inductively by a(1) = 0 and a(p) = 1 + a(p-1) if p is prime and a(n*m) = a(n) + a(m) if m,n > 1.

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A373366 a(n) = gcd(A064097(n), A083345(n)).

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A334202 a(n) = A064097(n) - A323077(n).

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