A034939 -id:A034939 - cp's OEIS frontend

A218715 a(n) is smallest number such that a(n)^2 + 1 is divisible by 53^n.

0, 23, 500, 27590, 623098, 23048345, 5041394261, 416081467190, 11331029931180, 50928660480181, 6548598523124085, 2441875986594058601, 76594163421571591377, 7783548304686046882879, 252583670951378815076851, 4392422457122810120236558, 1165802007767335105471573954

Offset: 0

Michel Lagneau, Nov 04 2012

nonn

			a(3) = 27590 because 27590^2+1 =  53 ^ 3 * 5113.

Cf. A002522, A049532, A034939, A218709, A218710, A218712, A218713, A218714.

b=23;n53=53;jo=Join[{0,b},Table[n53=53*n53;b=PowerMod[b,53,n53];b=Min[b,n53-b],{99}]]

A218563 Numbers n such that n^2 + 1 is divisible by a 4th power.

Original entry on oeis.org

182, 239, 443, 807, 1068, 1432, 1693, 2057, 2318, 2682, 2943, 3307, 3568, 3932, 4193, 4557, 4818, 5182, 5443, 5807, 6068, 6432, 6693, 7057, 7318, 7682, 7943, 8307, 8568, 8932, 9193, 9557, 9818, 10182, 10443, 10807, 11068, 11432, 11693, 12057, 12318, 12682

Offset: 1

Michel Lagneau, Nov 02 2012

nonn

Includes all n == 182 or 443 (mod 625). In particular, the sequence has positive asymptotic density. # Robert Israel, Oct 06 2016

			239 is in the sequence because 239^2+1 = 57122 = 2*13^4;
27493 is in the sequence because 27493^2+1 = 755865050 = 2*5^2*17^4*181.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002522, A049532, A034939, A218562.

N:= 100000: # to get all terms <= N
res:= {}:
p:= 2;
while p^4 <= N^2+1 do
  for v in map(t -> subs(t,n), [msolve(n^2+1, p^4)]) do
    res:= res union {seq(k*p^4+v, k = 0 .. (N-v)/p^4)}
  od;
  p:= nextprime(p);
od:
sort(convert(res,list)); # Robert Israel, Oct 06 2016

Select[Range[2,13000],Max[Transpose[FactorInteger[#^2+1]][[2]]]>3&]

A218716 a(n) is smallest number such that a(n)^2 + 1 is divisible by 61^n.

Original entry on oeis.org

0, 11, 682, 51412, 6304056, 144762466, 9435321777, 988322434636, 71294762793847, 3138611770750343, 283798117998769727, 15409745938584647495, 320007169218635518122, 45443939732277600209579, 207359227164430355867160, 59053635973003478214807486

Offset: 0

Michel Lagneau, Nov 04 2012

nonn

			a(3) = 51412 because 51412^2+1 = 5 * 17 * 61 ^ 3 * 137.

Cf. A002522, A049532, A034939, A218709, A218710, A218712, A218713, A218714, A218715.

b=11;n61=61;jo=Join[{0,b},Table[n61=61*n61;b=PowerMod[b,61,n61];b=Min[b,n61-b],{99}]]

A218564 Numbers n such that n^2 + 1 is divisible by a 5th power.

Original entry on oeis.org

1068, 2057, 4193, 5182, 7318, 8307, 10443, 11432, 13568, 14557, 16693, 17682, 19818, 20807, 22943, 23932, 26068, 27057, 29193, 30182, 32318, 33307, 35443, 36432, 38568, 39557, 41693, 42682, 44818, 45807, 47943, 48932, 51068, 52057, 54193, 55182, 57318, 58307

Offset: 1

Michel Lagneau, Nov 02 2012

nonn

For each prime p == 1 (mod 4), there are two values of x (mod p^5) that solve x^2 + 1 == 0 (mod p^5), and then x + k*p^5 is in the sequence for every k. Thus the asymptotic density of this sequence should be 1 - Product_p (1 - 2/p^5), where the product is over all primes p == 1 (mod 4). - Robert Israel, Sep 04 2018

			1068 is in the sequence because 1068^2+1 = 1140625 = 5^6*73;
143044 is in the sequence because 143044^2+1 = 20461585937 = 13^5*55109;
390112 is in the sequence because 390112^2+1 = 152187372545 = 5*13*17^6*97.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002522, A049532, A034939, A218562, A218563.

N:= 10^5: # to get all terms <= N
P:= select(isprime,[seq(i,i=5..floor((N^2+1)^(1/5)),4)]):
g:= proc(x,r,N) local t; t:= rhs(op(x)); seq(t+r*k,k=0..(N-t)/r) end proc:
R:= `union`(seq(map(g, {msolve(n^2+1,p^5)},p^5,N),p=P)):
sort(convert(R,list)); # Robert Israel, Sep 04 2018

Select[Range[2,20000],Max[Transpose[FactorInteger[#^2+1]][[2]]]>4&]

isok(n) = vecmax(factor(n^2+1)[,2]) >= 5; \\ Michel Marcus, Sep 04 2018

A218717 a(n) is smallest number such that a(n)^2 + 1 is divisible by 73^n.

Original entry on oeis.org

0, 27, 776, 153765, 6459524, 404034898, 41865466758, 3219884218827, 239822883201307, 9110883894036198, 991706090146518323, 142813358470363920740, 8641533837443707913816, 586811715371303018585730, 2756887299416274753296336, 729513196939063257288876118

Offset: 0

Michel Lagneau, Nov 04 2012

nonn

			a(3) = 153765 because 153765^2+1 = 2 * 73 ^ 3 * 30389.

Cf. A002522, A049532, A034939, A218709, A218710, A218712, A218713, A218714, A218715, A218716.

b=27;n73=73;jo=Join[{0,b},Table[n73=73*n73;b=PowerMod[b,73,n73];b=Min[b,n73-b],{99}]]

A034944 Successive approximations to 13-adic integer sqrt(-1).

Original entry on oeis.org

0, 5, 70, 239, 143044, 1999509, 6826318, 822557039, 85658552023, 1188526486815, 11941488851037, 291518510320809, 2108769149874327, 13920898306972194, 2675587335039691558, 63228498770709057089

Offset: 0

N. J. A. Sloane

K. Mahler, Introduction to p-Adic Numbers and Their Functions, Cambridge, 1973, p. 35.

Seiichi Manyama, Table of n, a(n) for n = 0..806

Cf. A034935, A034939, A218709, A286840.

seq(n)={my(v=vector(n), i=1, k=0); while(i<#v, k++; my(t=truncate(sqrt(-1 + O(13^k)))); if(t > v[i], i++; v[i]=t)); v} \\ Andrew Howroyd, Nov 10 2018

A218565 Numbers k such that k^2 + 1 is divisible by a 6th power.

Original entry on oeis.org

1068, 14557, 16693, 30182, 32318, 45807, 47943, 61432, 63568, 77057, 79193, 92682, 94818, 108307, 110443, 123932, 126068, 139557, 141693, 155182, 157318, 170807, 172943, 186432, 188568, 202057, 204193, 217682, 219818, 233307, 235443, 248932, 251068, 264557

Offset: 1

Michel Lagneau, Nov 02 2012

nonn

			1068 is in the sequence because 1068^2 + 1 = 5^6 * 73.
390112 is in the sequence because 390112^2 + 1 = 5 * 13 * 17 ^ 6 * 97.
1999509 is in the sequence because 1999509^2 + 1 = 2 * 13 ^ 6 * 29 * 14281.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002522, A049532, A034939, A218562, A218563, A218564.

Cf. A001014 (6th powers).

Select[Range[2,27000],Max[Transpose[FactorInteger[#^2+1]][[2]]]>5&]

A218718 a(n) is smallest number such that a(n)^2 + 1 is divisible by 89^n.

Original entry on oeis.org

0, 34, 3861, 344464, 20099637, 2153335831, 102666405913, 4867146503697, 923990886302412, 50251663587824641, 5655954122907587985, 909925832091926912414, 85120439454684773642745, 2631773999763198769695986, 41332517834853462204330752

Offset: 0

Michel Lagneau, Nov 04 2012

nonn

			a(3) = 344464 because 344464^2+1 = 37 * 89 ^ 3 * 4549.

Cf. A002522, A049532, A034939, A218709, A218710, A218712, A218713, A218714, A218715, A218716, A218717.

b=34;n89=89;jo=Join[{0,b},Table[n89=89*n89;b=PowerMod[b, 89,n89];b=Min[b,n89-b],{99}]]

A218574 Numbers k such that k^2 + 1 is divisible by a 7th power.

Original entry on oeis.org

32318, 45807, 110443, 123932, 188568, 202057, 266693, 280182, 344818, 358307, 422943, 436432, 501068, 514557, 579193, 592682, 657318, 670807, 735443, 748932, 813568, 827057, 891693, 905182, 969818, 983307

Offset: 1

Michel Lagneau, Nov 02 2012

nonn

			32318 is in the sequence because 32318^2 + 1 =  5 ^ 7 * 29 * 461.
6826318 is in the sequence because 6826318^2 + 1 = 5 ^ 3 * 13 ^ 8 * 457.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002522, A049532, A034939, A218562, A218563, A218564, A218565.

Cf. A001015 (7th powers).

Select[Range[2,1500000],Max[Transpose[FactorInteger[#^2+1]][[2]]]>6&]

A218719 a(n) is smallest number such that a(n)^2 + 1 is divisible by 97^n.

Original entry on oeis.org

0, 22, 4052, 107551, 22709274, 331407850, 197177418061, 26457926739667, 2369608176604944, 76004727767164666, 25163629663367816827, 1965881512952938486496, 191165497320828772935835, 21700278688179406782082106, 560121950820639295011033922

Offset: 0

Michel Lagneau, Nov 04 2012

nonn

			a(3) = 107551 because 107551^2+1 = 2 * 97 ^ 3 * 6337.

Cf. A002522, A049532, A034939, A218709, A218710, A218712, A218713, A218714, A218715, A218716, A218717, A218718.

b=22;n97=97;jo=Join[{0,b},Table[n97=97*n97;b=PowerMod[b, 97,n97];b=Min[b,n97-b],{99}]]

cp's OEIS Frontend

A218715 a(n) is smallest number such that a(n)^2 + 1 is divisible by 53^n.

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A218563 Numbers n such that n^2 + 1 is divisible by a 4th power.

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A218716 a(n) is smallest number such that a(n)^2 + 1 is divisible by 61^n.

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A218564 Numbers n such that n^2 + 1 is divisible by a 5th power.

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A218717 a(n) is smallest number such that a(n)^2 + 1 is divisible by 73^n.

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A034944 Successive approximations to 13-adic integer sqrt(-1).

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A218565 Numbers k such that k^2 + 1 is divisible by a 6th power.

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Mathematica

A218718 a(n) is smallest number such that a(n)^2 + 1 is divisible by 89^n.

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Mathematica

A218574 Numbers k such that k^2 + 1 is divisible by a 7th power.

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