A246281 -id:A246281 - cp's OEIS frontend

A336836 Number of iterations of x -> A003961(x) needed before A003961(x) < 2x, when starting from x=n, or -1 if such a number is never reached.

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

Offset: 1

Antti Karttunen, Aug 07 2020

nonn

Starting from n, the number of prime shifts needed before a term of A246281 is reached.
It holds that a(n) >= A336835(n) for all n, because sigma(n) <= A003961(n) for all n (see A286385 for a proof).
Note that in contrast to abundancy used in A336835, the condition [A003961(x) > 2x] (= A252742) is not monotonic when iterating with A003961. For example, we have A003961(9) = 25 > 2*9, A003961(25) = 49 < 2*25, and then again A003961(49) = 121 > 2*49.
Question: Is the escape clause necessary in the definition?

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A246281 (positions of zeros, numbers k for which A003961(k) < 2*k).
Cf. A005940, A252742, A252743, A286385.
Cf. also A246271, A252459, A336835 for similar iterations.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336836(n) = for(i=0,oo,my(n2 = n+n); n = A003961(n); if(n < n2, return(i)));

A387710 Numbers k for which A003959(k) < 2*k, where A003959 is multiplicative with a(p^e) = (p+1)^e.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 75, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98, 99, 101, 103, 105, 106, 107, 109, 110, 111, 113, 115, 117, 118, 119, 121, 122, 123, 125

Offset: 1

Antti Karttunen, Sep 06 2025

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A003959, A387711.
Subsequence of A005100.
Subsequences: A000040, A001358\{4, 6}, A246281.
Positions of 0's in A387715.

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
is_A387710(n) = (A003959(n)<(2*n));

A387720 Numbers k for which A107758(k) < 2*k, where A107758 is sigma+, multiplicative function with a(p^e) = 1+sigma(p^e).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157, 159

Offset: 1

Antti Karttunen, Sep 06 2025

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A107758.
Complement of (A052396 U A387721).
Cf. also A005100, A246281, A387710.

A107758(n) =  { my(f = factor(n)); prod(k=1, #f~, 1+sigma(f[k, 1]^f[k, 2])); };
is_A387720(n) = (A107758(n)<(2*n));

cp's OEIS Frontend

A336836 Number of iterations of x -> A003961(x) needed before A003961(x) < 2x, when starting from x=n, or -1 if such a number is never reached.

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A387710 Numbers k for which A003959(k) < 2*k, where A003959 is multiplicative with a(p^e) = (p+1)^e.

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A387720 Numbers k for which A107758(k) < 2*k, where A107758 is sigma+, multiplicative function with a(p^e) = 1+sigma(p^e).

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