A001494 -id:A001494 - cp's OEIS frontend

A322797 Powerful tau numbers.

1, 8, 9, 36, 72, 108, 128, 225, 288, 441, 625, 864, 972, 1089, 1152, 1521, 1800, 1944, 2000, 2025, 2601, 2700, 3249, 3456, 3528, 3600, 4500, 4761, 5000, 5292, 5625, 6561, 6912, 7569, 7776, 8100, 8649, 8712, 10000, 10800, 12168, 12321, 12348, 13068, 15129, 16000, 16200, 16641, 18000

Offset: 1

Torlach Rush, Jan 10 2019

nonn

If the largest exponent among the prime factors of a(n) does not exceed 2 then A046692(a(n)) = sqrt(a(n)), otherwise A046692(a(n)) = 0.

			1 is a term because A033950(1) = A001694(1) = 1.
8 is a term because A033950(8) divides A001694(3).
9 is a term because A033950(9) divides A001694(4).
36 is a term because A033950(36) divides A001694(9).

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

Intersection of A001694 (powerful numbers) and A033950 (tau numbers).

Cf. A001494, A033950, A046692.

powtauQ[1] = True; powtauQ[n_] := Min[e = (Last /@ FactorInteger[n])] > 1 && Divisible[n, Times @@ (e + 1)]; Select[Range[18000], powtauQ] (* Amiram Eldar, Dec 30 2019 *)

isok(n) = ispowerful(n) && ((n % numdiv(n)) == 0); \\ Michel Marcus, Jan 16 2019

Corrected and extended by Michel Marcus, Jan 16 2019

A330251 Numbers k such that phi(k) = phi(k+3), where phi (A000010) is Euler's totient function.

Original entry on oeis.org

3, 5, 8720288051472, 9134280520365, 41544070492925, 42466684755492, 51363581614342, 68616494581632, 113312918293575, 210911076210835, 215517565688425, 294988451482725, 383617980270525, 432759876053505, 442863123838135, 532068058516992, 892813363927485, 923102743748185, 929531173876305

Offset: 1

Michel Marcus and Giovanni Resta, Feb 29 2020

nonn

10^15 < a(20) <= 1089641067389872.
Also terms: 1248817919303952, 1332436545865422, 1394926716616125, 1868522795664525, 1950445682260072.
a(4) and a(9) appear in Kevin Ford's paper.

Kevin Ford, Solutions of phi(n)=phi(n+k) and sigma(n)=sigma(n+k), arXiv:2002.12155 [math.NT], 2020.
S. W. Graham, J. J. Holt, and C. Pomerance, On the solutions to phi(n) = phi(n+k), Number Theory in Progress, K. Gyory, H. Iwaniec, and J. Urbanowicz, eds., vol. 2, de Gruyter, Berlin and New York, 1999, pp. 867-882.
Mathematics StackExchange, Conjecture on the gap between integers having the same number of co-primes, Sep 25 2019.

Cf. A000010, A007015.
Cf. A001274, A001494, A179186, A179187, A179188, A179189, A179202, A330429.
Cf. A276503, A276504, A217139.

Select[Range[100000], EulerPhi[#] == EulerPhi[# + 3] &] (* Alonso del Arte, Mar 01 2020 *)

isok(k) = eulerphi(k) == eulerphi(k+3); \\ Michel Marcus, Feb 29 2020

A330429 Numbers k such that phi(k) = phi(k+9), where phi (A000010) is Euler's totient function.

Original entry on oeis.org

9, 15, 1005079920836, 13695542245376, 26160864154416, 27402841561095, 27599063056565, 110263115897935, 124632211478775, 127400054266476, 154090744843026, 205849483744896, 231019991767556, 339938754880725, 459718637643265, 632733228632505, 646552697065275, 683008674773416, 884965354448175

Offset: 1

Giovanni Resta, Mar 01 2020

nonn

a(20) > 10^15.

Kevin Ford, Solutions of phi(n)=phi(n+k) and sigma(n)=sigma(n+k), arXiv:2002.12155 [math.NT], 2020.
S. W. Graham, J. J. Holt, and C. Pomerance, On the solutions to phi(n) = phi(n+k), Number Theory in Progress, K. Gyory, H. Iwaniec, and J. Urbanowicz, eds., vol. 2, de Gruyter, Berlin and New York, 1999, pp. 867-882.

Cf. A000010, A007015.
Cf. A001274, A001494, A330251, A179186, A179187, A179188, A179189, A179202.
Cf. A276503, A276504, A217139.

A330702 Numbers k such that psi(k) = psi(k + 2) and phi(k) = phi(k + 2), where psi(k) is the Dedekind psi function (A001615) and phi(k) is the Euler totient function (A000010).

Original entry on oeis.org

70, 308, 572, 2132, 4292, 6764, 12212, 32804, 72836, 79292, 169724, 198596, 207692, 289052, 362972, 392426, 545876, 547724, 611612, 651932, 678812, 687812, 809252, 842012, 868436, 930932, 1030772, 1032956, 1122932, 1336052, 1627772, 1705892, 1722932, 2173772

Offset: 1

Amiram Eldar, Dec 26 2019

nonn

Sandor asked whether this sequence is infinite.

Apparently the only common solution to psi(n) = psi(n+1) and phi(n) = phi(n+1) is 15.

			70 is a term since psi(70) = psi (72) = 144 and phi(70) = phi(72) = 24.

Amiram Eldar, Table of n, a(n) for n = 1..600
Jozsef Sandor, On the composition of some arithmetic functions, II, Journal of Inequalities in Pure and Applied Mathematics, Vol. 6, No. 3 (2005), Article 73.

Intersection of A001494 and A330703.

Cf. A000010, A001615.

psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); Select[Range[10^5], psi[#] == psi[# + 2] && EulerPhi[#] == EulerPhi[#+2] &]

A330902 Odd numbers k such that s(k) = s(k+2), where s(k) is Schemmel's totient function of order 2 (A058026).

Original entry on oeis.org

1, 9359, 23933, 97405, 131493, 304589, 529205, 6005613, 6024473, 6057257, 7636517, 9566549, 11481581, 25143017, 25439117, 28542745, 40473869, 57712193, 58761197, 69502169, 77085497, 78481397, 81127109, 95223857, 99815303, 104092517, 112282481, 119954477, 130052613

Offset: 1

Amiram Eldar, May 01 2020

nonn

Since s(k) = 0 for all even numbers k, they are trivial solutions of the equation s(k) = s(k+2) and therefore they were excluded from this sequence.

Analogous to A001494 since Schemmel's totient functions are a generalization of the Euler totient function (A000010).

			1 is a term since s(1) = s(3) = 1.
9359 is a term since s(9359) = s(9361) = 6615.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 3, p. 276.

Amiram Eldar, Table of n, a(n) for n = 1..92
Victor Schemmel, Ueber relative Primzahlen, Journal für die reine und angewandte Mathematik, Vol. 70 (1869), pp. 191-192.

Cf. A001494, A058026.

f[p_, e_] := (p-2) * p^(e-1); s[1]=1; s[n_] := Times @@ (f @@@ FactorInteger[n]); seq={}; s1 = 1; Do[s2 = s[n]; If[s1 == s2, AppendTo[seq, n-2]]; s1 = s2, {n, 3, 10^6, 2}]; seq

A333742 Numbers k such that lambda(k) = lambda(k+2), where lambda is the Carmichael lambda function (A002322).

Original entry on oeis.org

4, 6, 7, 180, 208, 427, 1183, 1330, 1404, 1480, 1584, 1651, 1672, 2013, 2695, 2715, 3256, 3439, 5250, 5668, 5698, 5950, 5955, 7600, 12243, 13392, 13715, 14768, 22263, 22878, 23347, 24804, 26100, 30500, 32940, 43648, 45870, 46205, 52548, 54481, 59148, 59780, 62719

Offset: 1

Amiram Eldar, Apr 03 2020

nonn

			4 is a term since lambda(4) = lambda(6) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..5000

Cf. A001494, A002322.

Select[Range[10^4], CarmichaelLambda[#] == CarmichaelLambda[# + 2] &]
Position[Partition[CarmichaelLambda[Range[63000]],3,1],?(#[[3]]==#[[1]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, Aug 16 2024 *)

cp's OEIS Frontend

A322797 Powerful tau numbers.

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A330251 Numbers k such that phi(k) = phi(k+3), where phi (A000010) is Euler's totient function.

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A330429 Numbers k such that phi(k) = phi(k+9), where phi (A000010) is Euler's totient function.

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A330702 Numbers k such that psi(k) = psi(k + 2) and phi(k) = phi(k + 2), where psi(k) is the Dedekind psi function (A001615) and phi(k) is the Euler totient function (A000010).

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A330902 Odd numbers k such that s(k) = s(k+2), where s(k) is Schemmel's totient function of order 2 (A058026).

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A333742 Numbers k such that lambda(k) = lambda(k+2), where lambda is the Carmichael lambda function (A002322).

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