A172333 Numbers m such that m and m+22 have the same sum of divisors.

57, 85, 213, 224, 354, 476, 568, 594, 812, 1218, 1235, 1316, 1484, 2103, 2470, 2492, 2643, 2840, 2996, 3836, 3978, 4026, 4544, 4810, 4844, 5012, 6125, 6356, 6524, 7364, 7532, 7648, 8876, 9272, 9328, 10098, 11107, 11797, 12572, 12594, 13412, 13640

Offset: 1

Author

Michel Lagneau, Feb 01 2010

Keywords

• nonn

Comments

If 3*k-1 and 14*k-1 are both prime with k>1, then n = 28*(3*k-1) belongs to this sequence. The number of such integers n <= x would be asymptotically cx/(log x)^2 for some constant c > 0 from the Hardy-Littlewood conjecture D in Partitio Numerorum. - Tomohiro Yamada, Oct 03 2018

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 62, p. 22, Ellipses, Paris 2008.
• W. Sierpinski, A Selection of Problems in the Theory of Numbers. Macmillan, NY, 1964, p. 110.
• Tomohiro Yamada, On equations sigma(n) = sigma(n+k) and phi(n) = phi(n+k), J. Comb. Number Theory 9 (2017), 15-21.

Links

• Tomohiro Yamada, Table of n, a(n) for n = 1..46702 (All terms < 2^28, first 2000 terms from Muniru A Asiru)
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
• G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math. 44 (1923), 1-70.
• Tomohiro Yamada, On equations sigma(n) = sigma(n+k) and phi(n) = phi(n+k)<, arXiv:1001.2511 [math.NT], 2010.

Crossrefs

Cf. A000203, A015861, A002961, A015865, A015867, A015858, A015859, A015860.

Programs

GAP

Filtered([1..13700],k->Sigma(k)=Sigma(k+22)); # Muniru A Asiru, Oct 20 2018

Maple

with(numtheory):for n from 1 to 20000 do;if sigma(n) = sigma(n+22) then print(n); else fi ; od;

PARI

isok(k) = sigma(k)==sigma(k+22); \\ Altug Alkan, Oct 03 2018