A247975 Least positive integer m such that m + n divides prime(m)^2 + prime(n)^2.

1, 8, 15479, 30, 29, 68, 51, 2, 73, 15, 39, 13, 12, 36, 10, 25, 33, 8, 15, 38, 40, 108, 42, 1, 16, 39, 31, 57, 5, 4, 27, 2, 17, 51, 30, 14, 36, 20, 11, 21, 32, 23, 39, 689, 29, 4, 27, 1873, 184, 7248, 7, 153, 132, 76, 75, 18, 28, 99, 2, 86, 13, 111, 202, 66, 1212

Offset

1,2

Comments

Conjecture: a(n) exists for any n > 0. - Zhi-Wei Sun, Sep 28 2014

If a(i) = j, then a(j) <= i. - Derek Orr, Sep 28 2014

Computational extension to n = 1..120000 (search bound a(n) <= 2*10^11; i.e., p_{a(n)} <= 2.25*10^12): 119995 values resolved, 5 remain unresolved (n = 37249, 66257, 76868, 98379, 117862), all with estimated ratio a(n)/(n*log(n)) > 145000. The global maximum ratio among resolved cases is at n = 23995 (ratio approximately 196577), with a(23995) = 47572503750. The largest resolved value is a(82300) = 81961045941. - Carlo Corti, Mar 21 2026

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..5000 from Zhi-Wei Sun)

Carlo Corti, Extended dataset: a(n) for n = 1..120000 (119995 resolved values, 5 gaps)

Carlo Corti, Logarithmic scatterplot of A247975 (n=1..120000)

Carlo Corti, Code and data repository for the computational study of A247975, GitHub, 2026.

Carlo Corti, Computational study of A247975: code and data archive, Zenodo, DOI:10.5281/zenodo.18920371, 2026.

Zhi-Wei Sun, Some New Problems in Additive Combinatorics, Nanjing Univ. J. Math. Biquarterly 36 (2019), 134-155; arXiv:1309.1679 [math.NT], 2013-2019.

Example

a(2) = 8 since 8 + 2 = 10 divides prime(8)^2 + prime(2)^2 = 19^2 + 3^2 = 370.
a(3) = 15479 since 15479 + 3 = 15482 divides prime(15479)^2 + prime(3)^2 = 169789^2 + 5^2 = 28828304546 = 15482*1862053.
a(4703) = 760027770 since 760027770 + 4703 = 760032473 divides prime(760027770)^2 + prime(4703)^2 = 17111249191^2 + 45329^2 = 292794848878552872722 = 760032473*385239919714.
a(11924) = 6119832581 since 6119832581 + 11924 = 6119844505 divides prime(6119832581)^2 + prime(11924)^2 = 151142662267^2 + 127289^2 = 22844104357172628068810 = 6119844505*3732791631962.

Mathematica

Do[m = 1; Label[aa]; If[Mod[Prime[m]^2 + Prime[n]^2, m + n] == 0, Print[n, " ", m]; Goto[bb]]; m = m + 1; Goto[aa]; Label[bb]; Continue, {n, 1, 60}]

Prog

(PARI) a(n) = my(m=1); while((prime(m)^2+prime(n)^2)%(m+n), m++); m
vector(75, n, a(n)) \\ Derek Orr, Sep 28 2014
(PARI) a(n) = my(m=1, q=prime(m), pn2=prime(n)^2); while((q^2+pn2)%(m+n), m++; q=nextprime(q+1)); m; \\ Michel Marcus, Mar 03 2026

Crossrefs

Cf. A000040, A247824.

Sequence in context: A246959 A079186 A391094 * A198404 A079597 A160743

Adjacent sequences: A247972 A247973 A247974 * A247976 A247977 A247978

Keyword

nonn

Author

Zhi-Wei Sun, Sep 28 2014

Status

approved