A155043 -id:A155043 - cp's OEIS frontend

A261088 Number of steps needed to reach zero when starting from k = n^2 and repeatedly applying the map that replaces k with k - d(k), where d(k) is the number of divisors of k (A000005).

0, 1, 2, 3, 5, 6, 10, 10, 19, 15, 19, 21, 24, 28, 39, 33, 53, 44, 49, 53, 60, 61, 69, 72, 79, 82, 92, 93, 117, 108, 115, 115, 140, 121, 174, 146, 205, 155, 233, 217, 267, 192, 295, 209, 225, 222, 238, 249, 267, 270, 299, 290, 336, 313, 373, 328, 411, 347, 451, 380, 486, 400, 534, 422, 447, 441, 460, 460, 511, 479, 496, 504, 545, 529, 602, 553, 579, 577, 626, 612, 681, 632, 747, 665, 796, 695

Offset: 0

Antti Karttunen, Sep 23 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..5792

Cf. A000005, A000290, A155043, A049820, A259934, A261085.

Cf. also A260732, A261222.

f[n_]:=Length[NestWhileList[#-DivisorSigma[0,#]&,n^2,#!= 0&]]-1;f/@Range[0,85] (* Ivan N. Ianakiev, Sep 25 2015 *)

allocatemem((2^31)+(2^30));
uplim = 2^25;
v155043 = vector(uplim);
v155043[1] = 1; v155043[2] = 1;
for(i=3, uplim, v155043[i] = 1 + v155043[i-numdiv(i)]; if(!(i%65536),print1(i,", ")););
A155043 = n -> if(!n,n,v155043[n]);
A261088 = n -> A155043(n^2);
for(n=0, 5792, write("b261088.txt", n, " ", A261088(n)));

(define (A261088 n) (A155043 (A000290 n)))

a(n) = A155043(A000290(n)) = A155043(n^2).

A261085 Number of steps needed to reach zero when starting from the n-th prime [i.e., setting k to A000040(n)] and repeatedly applying the map that replaces k with k - d(k), where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 15, 17, 19, 20, 22, 23, 24, 26, 16, 18, 20, 21, 22, 22, 23, 24, 23, 24, 25, 26, 28, 29, 31, 33, 33, 34, 36, 37, 39, 40, 40, 41, 44, 47, 34, 35, 49, 51, 52, 54, 54, 55, 57, 58, 59, 58, 59, 62, 48, 49, 50, 66, 69, 71, 73, 74, 74, 76, 55, 57, 59, 60, 61, 63, 63, 65, 68, 69, 71, 72, 74, 62, 64, 65, 66, 67, 67, 70, 72, 73, 75, 76, 77, 80, 81, 75, 77, 79, 79, 81

Offset: 1

Antti Karttunen, Sep 23 2015

nonn

			For n=4 we have prime(4) = 7, from which we start subtracting the number of divisors, to get the following path: 7 - 2 = 5, 5 - 2 = 3, 3 - 2 = 1, 1 - 1 = 0, and we have reached zero in four steps, thus a(4) = 4.
For n=5 we have prime(5) = 11, for which the similar process results: 11 - 2 = 9, 9 - 3 = 6, 6 - 4 = 2, 2 - 2 = 0, and again we have reached zero in four steps, thus also a(5) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..6543

Cf. A000005, A000040, A049820, A155043, A259934, A261088.

Cf. A261086 (gives the positions of drops, i.e., where nonmonotonic) and A261087 (the corresponding primes).

mpr[p_]:=Length[NestWhileList[#-DivisorSigma[0,#]&,p,#>0&]]-1; mpr/@Prime[ Range[ 120]] (* Harvey P. Dale, Aug 18 2022 *)

uplim = 65537;
v155043 = vector(uplim);
v155043[1] = 1; v155043[2] = 1;
for(i=3, uplim, v155043[i] = 1 + v155043[i-numdiv(i)]);
A155043 = n -> if(!n,n,v155043[n]);
n=0; forprime(p=2, uplim, n++; write("b261085.txt", n, " ", A155043(p)));

(define (A261085 n) (A155043 (A000040 n)))

a(n) = A155043(A000040(n)).

A261103 a(n) = A259934(n) - A261089(n).

Original entry on oeis.org

0, 1, 3, 7, 11, 9, 13, 15, 19, 19, 25, 27, 27, 33, 37, 47, 43, 49, 49, 55, 57, 56, 58, 58, 71, 75, 83, 23, 25, 31, 35, 37, 49, 51, 49, 53, 57, 61, 65, 63, 77, 75, 79, 85, 87, 87, 91, 91, 99, 109, 109, 115, 117, 117, 123, 127, 127, 133, 139, 141, 141, 143, 149, 147, 159, 151, 157, 165, 167, 179, 187, 193, 205, 211, 213, 209, 215, 213, 75

Offset: 0

Antti Karttunen, Sep 23 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A155043, A259934, A261089.

(define (A261103 n) (- (A259934 n) (A261089 n)))

a(n) = A259934(n) - A261089(n).

A262677 Number of odd numbers encountered when iterating A049820 starting from n: a(0) = 0 and for n >= 1, a(n) = A000035(n) + a(A049820(n)).

Original entry on oeis.org

0, 1, 0, 2, 1, 3, 0, 4, 1, 1, 0, 2, 0, 3, 0, 3, 2, 4, 0, 5, 0, 5, 0, 6, 2, 1, 0, 7, 0, 8, 0, 9, 0, 9, 0, 10, 7, 11, 0, 11, 0, 12, 0, 13, 0, 12, 0, 13, 0, 1, 0, 14, 0, 15, 0, 15, 0, 16, 0, 17, 0, 18, 0, 17, 16, 19, 0, 20, 0, 20, 0, 21, 0, 22, 0, 21, 0, 23, 0, 24, 0, 1, 0, 2, 0, 2, 0, 3, 0, 4, 0, 4, 0, 5, 0, 5, 0, 6, 0, 6, 4, 7, 0, 8, 0, 7, 0, 8

Offset: 0

Antti Karttunen, Oct 03 2015

nonn

Number of odd numbers encountered before zero is reached when starting from k = n and repeatedly applying the map that replaces k by k - d(k), where d(k) is the number of divisors of k (A000005). This count includes n itself if it is odd.

Antti Karttunen, Table of n, a(n) for n = 0..10080

Cf. A000005, A000035, A049820, A155043, A262676, A262680.

a(0) = 0; for n >= 1, a(n) = A000035(n) + a(A049820(n)).
Other identities. For all n >= 0:
A155043(n) = A262676(n) + a(n).

A263279 a(n) = A263259(A259934(n)); one-based position of A259934(n) at row n of table A263265.

Original entry on oeis.org

1, 2, 3, 5, 4, 5, 6, 4, 4, 3, 6, 3, 2, 4, 5, 6, 5, 5, 5, 7, 4, 4, 5, 6, 9, 6, 7, 5, 6, 4, 4, 3, 5, 5, 4, 5, 3, 3, 4, 4, 5, 5, 5, 5, 5, 4, 4, 3, 5, 7, 4, 4, 5, 7, 8, 5, 7, 5, 7, 8, 8, 6, 8, 6, 7, 7, 7, 9, 10, 10, 11, 9, 10, 9, 11, 10, 9, 7, 8, 9, 7, 6, 9, 8, 10, 6, 5, 6, 5, 9, 8, 8, 8, 7, 6, 4, 3, 3, 3, 5, 4, 7, 6, 6, 7, 9, 5, 5, 10, 10, 11, 6, 7, 9, 9, 7, 9, 13, 7, 6, 8

Offset: 0

Antti Karttunen, Nov 24 2015

nonn

a(n) gives the number of integers k <= A259934(n) for which A155043(k) = n = A155043(A259934(n)).

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A155043, A259934, A263259, A263265, A263280.

Cf. also A261103.

(define (A263267 n) (A263259 (A259934 n)))

a(n) = A263259(A259934(n)).

A322996 Number of iterations of A049820(x) = x - A000005(x) needed to reach an odd number or zero, when starting from x = n.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 3, 0, 3, 0, 4, 0, 1, 0, 4, 0, 5, 0, 5, 0, 2, 0, 6, 0, 6, 0, 6, 0, 7, 0, 7, 0, 1, 0, 8, 0, 8, 0, 8, 0, 9, 0, 9, 0, 9, 0, 10, 0, 10, 0, 10, 0, 10, 0, 11, 0, 10, 0, 12, 0, 1, 0, 12, 0, 13, 0, 13, 0, 11, 0, 14, 0, 14, 0, 14, 0, 14, 0, 15, 0, 12, 0, 16, 0, 15, 0, 15, 0, 17, 0, 16, 0, 13, 0, 18, 0, 1, 0, 17, 0, 14, 0

Offset: 0

Antti Karttunen, Jan 05 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..110880

Cf. A000005, A049820, A155043, A259934, A322987, A322997 (bisection), A323073.

Cf. also A322983.

A322996[n_] := A322996[n] = If[n == 0 || OddQ[n], 0, 1 + A322996[n - DivisorSigma[0, n]]];
Array[A322996, 100, 0] (* Paolo Xausa, Jan 15 2025 *)

A322996(n) = if((!n)||(n%2),0,1+A322996(n-numdiv(n)));

A322996(n) = { for(j=0,oo,if((!n)||(n%2),return(j)); n -= numdiv(n)); };

a(0) = 0; for n >= 1, for n odd, a(n) = 0, and for n even, a(n) = 1 + a(n-A000005(n)).
a(n) <= A155043(n).
For n >= 83, a(2*n) = 1+A322987(2*n).

A323073 Number of iterations of A049820(x) = x - A000005(x) needed to reach either zero or such x that x and A049820(x) are coprime, when starting from x = n.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 2, 0, 1, 3, 3, 0, 3, 0, 4, 0, 0, 0, 4, 0, 5, 0, 5, 0, 1, 0, 6, 0, 6, 0, 6, 0, 7, 0, 7, 0, 1, 0, 8, 0, 8, 0, 8, 0, 9, 1, 9, 0, 9, 0, 10, 0, 10, 0, 10, 0, 10, 0, 11, 0, 10, 0, 12, 1, 0, 0, 12, 0, 13, 0, 13, 0, 11, 0, 14, 1, 14, 0, 14, 0, 14, 0, 15, 0, 12, 0, 16, 0, 15, 0, 15, 0, 17, 0, 16, 0, 13, 0

Offset: 0

Antti Karttunen, Jan 05 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..110880

Cf. A000005, A009191, A049820, A155043, A259934, A286540, A320009, A322974, A322987, A322996.

Cf. A046642 (positions of zeros after the initial a(0)=0).

A323073(n) = if(!n,0,my(nn=(n-numdiv(n))); if(1==gcd(n,nn),0,1+A323073(nn)));

A323073(n) = if(!n,0,for(j=0,oo,my(nn=(n-numdiv(n))); if((0==nn)||(1==gcd(n,nn)),return(j+(2==n)),n = nn)));

a(0) = 0; for n > 0, if A009191(n) == 1, a(n) = 0, otherwise a(n) = 1 + a(n-A000005(n)).

a(n) <= A155043(n).

A263270 Each n occurs A262507(n) times.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 18

Offset: 0

Antti Karttunen, Nov 24 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..12917

Cf. A155043, A262507, A263260, A263265, A263266.

Other identities. For all n >= 0:
a(n) = A155043(A263265(n)).
a(A263260(n)) = n+1. [The sequence is one more than the least monotonic left inverse of A263260.]

A261104 a(0)=0; for n >= 1, a(n) = 1 + a(n-A070319(n)), where A070319(n) is the maximum value for A000005 (number of divisors) in range 1 .. n.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 5, 5, 5, 6, 5, 6, 5, 6, 6, 6, 6, 7, 6, 7, 6, 7, 7, 7, 7, 7, 8, 7, 8, 7, 8, 8, 8, 8, 8, 8, 9, 8, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 9, 10, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 11, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 11, 12, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 12, 13, 12, 13, 13, 13, 13, 13, 13, 13, 13, 14, 13, 14, 13, 14, 14, 14, 14, 14, 14, 14, 14

Offset: 0

Antti Karttunen, Sep 24 2015

nonn

Number of steps needed to reach zero when starting from k = n and repeatedly applying the map that replaces k by k - A070319(k), where A070319(k) is the maximum value for A000005 (number of divisors) in range 1 .. k.

Antti Karttunen, Table of n, a(n) for n = 0..10080

Cf. A155043, A070319.

Cf. A262502 (positions of records).

a(0)=0; for n >= 1, a(n) = 1 + a(n-A070319(n)).
Other identities. For all n >= 0:
a(A262502(n)) = n.

A262520 a(n) = A262519(n) - A262518(n).

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 2, 1, 1, 3, 2, 3, 0, 3, 4, 5, 4, 5, 3, 5, 6, 7, 5, 6, 1, 6, 7, 7, 8, 8, 10, 7, 2, 10, 9, 10, 13, 9, 11, 12, 1, 1, 4, 1, 3, 3, 2, 3, 7, 2, 2, 5, 7, 4, 9, 5, 6, 5, 5, 5, 6, 5, 1, 3, 7, 2, 8, 1, 8, 3, 9, 3, 3, 2, 3, 5, 3, 4, 6, 4, 6, 7, 4, 6, 2, 6, 6, 1, 7, 7, 10, 8, 9, 8, 8, 9, 10, 8, 1, 10, 10, 10, 11, 9, 11, 12, 10, 12, 13, 12, 13, 13, -2, -1, 2, 13, 13, 14, 14, 15

Offset: 0

Antti Karttunen, Oct 02 2015

sign

a(n) = How many steps more are needed to reach zero when starting from k = 2*n + 1 than when starting from k = 2*n and repeatedly applying the map that replaces k by k - d(k)? [Here d(k) is the number of divisors of k (A000005)]. If it takes more steps when starting from 2n than from 2n+1, then a(n) is negative.

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A000005, A049820, A155043, A262518, A262519, A262521 (positions of negative values).

(define (A262520 n) (- (A262519 n) (A262518 n)))

a(n) = A262519(n) - A262518(n).

cp's OEIS Frontend

A261088 Number of steps needed to reach zero when starting from k = n^2 and repeatedly applying the map that replaces k with k - d(k), where d(k) is the number of divisors of k (A000005).

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A261085 Number of steps needed to reach zero when starting from the n-th prime [i.e., setting k to A000040(n)] and repeatedly applying the map that replaces k with k - d(k), where d(k) is the number of divisors of k (A000005).

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A261103 a(n) = A259934(n) - A261089(n).

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A262677 Number of odd numbers encountered when iterating A049820 starting from n: a(0) = 0 and for n >= 1, a(n) = A000035(n) + a(A049820(n)).

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A263279 a(n) = A263259(A259934(n)); one-based position of A259934(n) at row n of table A263265.

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A322996 Number of iterations of A049820(x) = x - A000005(x) needed to reach an odd number or zero, when starting from x = n.

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A323073 Number of iterations of A049820(x) = x - A000005(x) needed to reach either zero or such x that x and A049820(x) are coprime, when starting from x = n.

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A263270 Each n occurs A262507(n) times.

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A261104 a(0)=0; for n >= 1, a(n) = 1 + a(n-A070319(n)), where A070319(n) is the maximum value for A000005 (number of divisors) in range 1 .. n.