A066435 -id:A066435 - cp's OEIS frontend

A110176 Least k such that sigma(n) = sigma(k) + sigma(n-k) for 0

0, 0, 1, 0, 0, 0, 0, 2, 4, 2, 0, 0, 0, 0, 5, 0, 0, 0, 0, 2, 7, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 4, 11, 0, 0, 0, 0, 0, 13, 10, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 17, 0, 0, 0, 12, 14, 19, 0, 0, 0, 0, 17, 19, 0, 0, 0, 0, 0, 14, 14, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 38, 0, 22, 22, 0, 18, 0, 30, 31, 19, 0, 12

Offset: 1

T. D. Noe, Jul 15 2005

nonn

Sequence A110177 gives the number of solutions 0A110178.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A066435 (least k such that sigma(n)+sigma(k)=sigma(n+k)), A110177.

Cf. also A110173.

a[n_] := Select[Range[n-1], DivisorSigma[1, n]==DivisorSigma[1, n-# ]+DivisorSigma[1, # ]&]; Table[s=a[n]; If[Length[s]==0, 0, First[s]], {n, 150}]

A110176(n) = { my(x=sigma(n)); for(k=1,n-1,if(x == (sigma(k)+sigma(n-k)), return(k))); (0); }; \\ Antti Karttunen, Feb 20 2023

A091554 Primes p such that k = 2p is the smallest positive solution to the equation sigma(p+k) = sigma(p) + sigma(k).

Original entry on oeis.org

7, 11, 13, 17, 19, 23, 31, 53, 59, 73, 79, 89, 97, 103, 109, 113, 137, 139, 149, 157, 163, 181, 193, 211, 223, 227, 269, 281, 293, 313, 331, 337, 373, 389, 397, 409, 419, 421, 433, 463, 467, 487, 499, 509, 521, 523, 541, 547, 571, 599, 601, 617, 631, 641, 643

Offset: 1

T. D. Noe, Jan 20 2004

nonn

Note that for all primes p > 3, sigma(3p) = sigma(p) + sigma(2p).

Cf. A066435 (least k such that sigma(n+k)=sigma(n)+sigma(k)).

lst={}; Do[p=Prime[n]; k=1; While[DivisorSigma[1, p+k]!=DivisorSigma[1, p]+DivisorSigma[1, k], k++ ]; If[k==2p, AppendTo[lst, p]], {n, 3, 200}]; lst

A110108 Conjectured values of n such that sigma(n)+sigma(k)=sigma(n+k) has no solution.

Original entry on oeis.org

3, 9, 15, 21, 27, 39, 57, 63, 81, 93, 105, 117, 165, 171, 183, 189, 201, 219, 225, 243, 256, 279, 303, 309, 333, 351, 363, 375, 381, 387, 405, 417, 441, 453, 471, 489, 512, 543, 549, 567, 579, 603, 633, 651, 657, 669, 687, 693, 723, 729, 741, 777

Offset: 1

T. D. Noe, Jul 12 2005

nonn

All k<10^10 have been checked. Note that this sequence appears to contain all powers of 3. It also contains 256 and 512, which are the only even numbers listed here.

All k<10^11 have been checked. 65 n values remain < 1000. - Donovan Johnson, Aug 29 2012

R. K. Guy, Unsolved Problems in Number Theory, 3rd Ed., New York, Springer-Verlag, 2004, Section B15.

Cf. A066435 (least k such that sigma(n)+sigma(k)=sigma(n+k)).

Two terms removed (465 and 711) by Donovan Johnson, Aug 29 2012

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A110176 Least k such that sigma(n) = sigma(k) + sigma(n-k) for 0

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A091554 Primes p such that k = 2p is the smallest positive solution to the equation sigma(p+k) = sigma(p) + sigma(k).

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A110108 Conjectured values of n such that sigma(n)+sigma(k)=sigma(n+k) has no solution.

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