Walter Nissen - cp's OEIS frontend

A222588 Composites of the form 2^n-1 or 2^n+1 that are non-multiples of 3.

65, 511, 1025, 2047, 4097, 16385, 32767, 262145, 1048577, 2097151, 4194305, 8388607, 16777217, 33554431, 67108865, 134217727, 268435457, 536870911, 1073741825, 4294967297, 8589934591, 17179869185, 34359738367, 68719476737, 137438953471, 274877906945

Offset: 0

Walter Nissen, Feb 25 2013

Half the numbers of the proper form are divisible by 3 and thus excluded.
For 2^n-1, n must be odd to be in this sequence.
For 2^n+1, n must be even to be in this sequence.

			31 = 2^5-1 is prime and thus not a member of the sequence.
65 = 2^6+1 has 2 proper divisors, 5 and 13, thus is a(0) in the sequence.

Oystein Ore, Number Theory and Its History, McGraw-Hill, 1948, reprinted 1988, section 4-7, pp 69-75.

Wikipedia, Mersenne number
Wikipedia, Fermat number
George Woltman, GIMPS

Subsequence of both A014551 and A166977.

t = 2^Range[50]; u = Union[t - 1, t + 1]; Select[u, # > 1 && Mod[#, 3] != 0 && ! PrimeQ[#] &] (* T. D. Noe, Feb 26 2013 *)

A192397 Record holders for greatest prime factor of n^n + (n+1)^(n+1).

Original entry on oeis.org

2, 5, 31, 283, 743, 1600069, 60042893, 7438489991, 61215157711, 34041259347101651, 6564253087266573169, 22022174223585405703, 69454092876521107983605569601, 2360926164108571968813424783598971267, 462605180698333957063188362720170172617217, 14645575916792712592989131451003587034531413111, 214236369415820799335832514547376967536187180963

Offset: 1

Walter Nissen, Jun 29 2011

nonn

			60042893 = A056790(9) is in the sequence because all earlier members of A056790 are smaller than 60042893.

Walter Nissen, More terms with additional commentary

Cf. A056790, A056187.

fmax=0;for(k=0,35,my(x=factor(k^k+(k+1)^(k+1)),f=x[#x[,1],1]);if(f>fmax,print1(f,", ");fmax=f)) \\ Hugo Pfoertner, Aug 18 2019

A056790(m) < A056790(n), for all m < n

2 added by Arkadiusz Wesolowski, Jun 30 2011

A160405 Primes that are the concatenation of a 5-digit prime, a 7-digit prime, and a 5-digit prime.

Original entry on oeis.org

10007100000310037, 10007100000310163, 10007100000310247, 10007100000310271, 10007100000310289, 10007100000310321, 10007100000310433, 10007100000310463, 10007100000310477

Offset: 1

Walter Nissen, May 13 2009

Haiku-haiku-haiku primes. I would like to call these "Haiku primes" but it seems that name has been used by Geoffrey Caveney for a different concept. Another possible name would be haiku-formed primes, but maybe that should be reserved for primes which are formed from any number of primes of width 5 or 7. Note that if you associate the hyphens with the central word, Haiku-haiku-haiku is itself of the 5-7-5 form (in characters).

			10007, 1000003, 10037, and 10007100000310037 are all prime, so 10007100000310037 is in the sequence.

Wikipedia, Haiku

Cf. A000040 (primes), A006879 (number of primes with n digits).

[ a: p in PrimesInInterval(10000,10007), q in PrimesInInterval(1000000,1000003), r in PrimesInInterval(10000,12000) | IsPrime(a) where a is Seqint(Intseq(r) cat Intseq(q) cat Intseq(p)) ]; // Klaus Brockhaus, May 20 2009

Edited by Klaus Brockhaus, May 20 2009

A159611 Indices of the Fermat primes in the sequence of primes.

Original entry on oeis.org

2, 3, 7, 55, 6543

Offset: 1

Walter Nissen, Apr 16 2009

If it exists, a(6) >= primepi(2^(2^33)+1) which has more than 2*10^9 decimal digits. - Amiram Eldar, Sep 27 2024

			3, the 1st Fermat prime is the 2nd prime, so a(1) = 2.
17, the 3rd Fermat prime is the 7th prime, so a(3) = 7.

Index entries for sequences that are related to primes dividing Fermat numbers.

Cf. A000040 (primes), A000720, A019434 (Fermat primes).

Cf. A098006.

import Data.List (elemIndices)
a159611 n = a159611_list !! (n-1)
a159611_list = map (+ 2) $ elemIndices 0 a098006_list
-- Reinhard Zumkeller, Mar 26 2013

PrimePi/@{3,5,17,257,65537} (* Harvey P. Dale, Aug 07 2022 *)

for(i=0, 10, isprime(f=2^2^i+1) & print1(primepi(f), ", ")) \\ Michel Marcus, Apr 28 2016

a152155(n) = centerlift(Mod(3, 2^(2^n)+1)^(2^(2^n-1)))
print1(2, ", "); for(x=0, oo, if(a152155(x)==-1, print1(primepi(2^(2^x)+1), ", "))) \\ Felix Fröhlich, Apr 30 2021

A098006(a(n)) = 0. - Reinhard Zumkeller, Mar 26 2013

a(n) = A000720(A019434(n)). - Michel Marcus, Apr 29 2021

Name edited by Felix Fröhlich, Apr 30 2021

A159939 Odd solutions of phi(sigma(k)) = sigma(phi(k)).

Original entry on oeis.org

1, 9, 225, 729, 18225, 65025, 140625, 531441, 5267025, 11390625, 13286025, 18792225, 40640625, 87890625, 1522170225, 2197265625, 3291890625, 3839661225, 5430953025, 7119140625, 8303765625, 11745140625, 25400390625

Offset: 1

Walter Nissen, Apr 26 2009

nonn

sigma is the multiplicative sum-of-divisors function.
phi is Euler's totient.
Complete through 25558816403.
All given here are products of powers of consecutive Fermat primes based on generalized repunit primes; see links.
It is conjectured (see links) that all odd solutions are of this form, for which at least 10130 solutions are known.
a(24) > 10^11, if it exists. - Amiram Eldar, Nov 21 2024

			sigma(9) = 13, phi(9) = 6, sigma(6) = phi(13) = 12, so 9 is in the sequence.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B42, pp. 150-152.
Oystein Ore, Number Theory and Its History, 1948, reprinted 1988, Dover, ISBN-10: 0486656209, pp. 88 et seq., 109 et seq.

Walter Nissen, phi(sigma(n)) = sigma(phi(n)).

Cf. A000203, A000010, A033632, A019434.

isok(n) = (n % 2) && (eulerphi(sigma(n)) == sigma(eulerphi(n))) \\ Michel Marcus, Jul 23 2013

Edited by Charles R Greathouse IV, Oct 28 2009

a(1) = 1 inserted by Amiram Eldar, Nov 21 2024

A164875 Record holders for n^2 - phi(n)*sigma(n).

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 14, 18, 22, 26, 30, 38, 42, 50, 54, 58, 60, 62, 66, 78, 90, 102, 114, 126, 130, 138, 150, 170, 174, 186, 210, 246, 258, 282, 294, 318, 330, 354, 366, 390, 426, 438, 462, 498, 510, 534, 546, 570, 606, 618, 642, 654, 678, 690, 714, 750, 762, 786

Offset: 1

Walter Nissen, Aug 29 2009

nonn

These numbers exhibit the largest differences between n^2 and sigma(n)*phi(n).

All of the differences are in A069249, and are guaranteed to be positive by Th. 329 in Hardy & Wright. The record differences are in A164876.

			sigma(10) = 18; phi(10) = 4; 10^2 - sigma(10)*phi(10) = 28. This difference, 28, exceeds the difference for every smaller n, so 10 is in this sequence and 28 is in A164876.

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

Cf. A069249, A164876, A000203, A000010.

f[n_] := n^2 - EulerPhi[n] * DivisorSigma[1, n]; s = {}; fm = -1; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 786}]; s (* Amiram Eldar, Aug 29 2019 *)

A164876 Record differences for n^2 - phi(n)*sigma(n).

Original entry on oeis.org

0, 1, 2, 12, 28, 32, 52, 90, 124, 172, 324, 364, 612, 640, 756, 844, 912, 964, 1476, 2052, 2484, 3492, 4356, 4644, 4804, 6372, 7620, 8164, 10116, 11556, 16452, 20196, 22212, 26532, 28980, 33732, 39780, 41796, 44676, 55332, 60516, 63972, 75204, 82692

Offset: 1

Walter Nissen, Aug 29 2009

nonn

These are the largest differences between n^2 and sigma(n)*phi(n).

All of the differences are in A069249.

			sigma(10) = 18; phi(10) = 4; 10^2 - sigma(10)*phi(10) = 28. This difference, 28, exceeds the difference for every smaller n, so 28 is in this sequence and 10 is in A164875.

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

Cf. A069249, A164875, A000203, A000010.

f[n_] := n^2 - EulerPhi[n] * DivisorSigma[1, n]; s = {}; fm = -1; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, fm]], {n, 1, 500}]; s (* Amiram Eldar, Aug 29 2019 *)

a(n) = A069249(A164875(n)). - Amiram Eldar, Aug 29 2019

a(1) = 0 added by Amiram Eldar, Aug 29 2019

A173430 Last of consecutive coprime iterations of sum-of-divisors function.

Original entry on oeis.org

1, 15, 15, 15, 6, 6, 15, 15, 14, 10, 12, 12, 14, 14, 15, 104, 18, 18, 20, 20, 104, 22, 24, 24, 104, 26, 40, 28, 30, 30, 104, 104, 33, 34, 48, 91, 38, 38, 56, 40, 42, 42, 44, 44, 45, 46, 48, 48, 80, 255, 51, 52, 54, 54, 72, 56, 80, 58, 60, 60, 62, 62, 104, 255, 84, 66, 68, 68

Offset: 1

Walter Nissen, Feb 18 2010

			Calculating sum-of-divisors ( ... sum-of-divisors ( sum-of-divisors ( 4 ) ) ... ) the iterates are 4, 7, 8, 15, 24, ... .
The initial, consecutive, pairwise, coprime iterates are 4, 7, 8, 15, so a(4) = 15 .
Here sigma ( 4 ) = 7, sigma ( sigma ( 4 ) ) = sigma ( 7 ) = 8, etc.

Oystein Ore, Number Theory and Its History, 1988, Dover Publications, ISBN 0486656209, pp. 88-96.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, Vol. 5, No. 2 (1996), pp. 91-100.
Leonard Eugene Dickson, History of the Theory of Numbers, Volume I, Divisibility and Primality, Carnegie Institution of Washington, 1919, Chapters II and X

Cf. A129246 and the references there, A019294, A019295, A000203, A051027, A019284, A019277.

a[1] = 1; a[n_] := Module[{k = n}, While[CoprimeQ[k, (s = DivisorSigma[1, k])], k = s]; k]; Array[a, 68] (* Amiram Eldar, Sep 02 2019 *)

A173431 Count of consecutive coprime iterations of sum-of-divisors function.

Original entry on oeis.org

1, 6, 5, 4, 2, 1, 3, 2, 3, 1, 2, 1, 2, 1, 1, 5, 2, 1, 2, 1, 4, 1, 2, 1, 5, 1, 2, 1, 2, 1, 4, 3, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 4, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 4, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 4, 3, 1, 5, 2, 1, 2, 1, 1

Offset: 1

Walter Nissen, Feb 18 2010

The last of these iterates is the value in A173430.

			Calculating sum-of-divisors ( ... sum-of-divisors ( sum-of-divisors ( 7 ) ) ... ) the iterates are 7, 8, 15, 24, ... .
The initial, consecutive, pairwise, coprime iterates are 7, 8, 15, and there are 3 of these, so a(7) = 3.
Here sigma ( 7 ) = 8, sigma ( sigma ( 7 ) ) = sigma ( 8 ) = 15, etc.

Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
Oystein Ore, Number Theory and Its History, 1988, Dover Publications, ISBN 0486656209, pp. 88-96.

Leonard Eugene Dickson, History of the Theory of Numbers, Volume I, Divisibility and Primality, Carnegie Institution of Washington, 1919, Chapters II and X

Cf. A173430, A129246 and the references there, A019294, A019295, A000203, A051027, A019284, A019277.

a(n)=my(t,s);if(n==1,1,while(1,s++;t=sigma(n);if(gcd(t,n)==1,n=t,return(s)))) \\ Charles R Greathouse IV, Feb 06 2012

A140688 List of numbers which are both amicable and friendly.

Original entry on oeis.org

12285, 67095, 71145, 87633, 142310

Offset: 1

Walter Nissen, Jul 11 2008

There are almost certainly many other amicables both within and without the displayed range which also are friendly, but they have not yet been identified.
The usual OEIS policy is to display sequences only as far as they are known to be complete. How far is this sequence known to be complete? - N. J. A. Sloane, Jul 12 2008. The answer appears to be that nothing is known for certain. It may well be that 12285 is not even the first term! - N. J. A. Sloane, Dec 04 2008
It is not known if the two smallest amicable numbers, 220 and 284, are friendly. In fact it is not even known if 10 is friendly. - Walter Nissen, Dec 02 2008
The following numbers are all known to be members of this sequence: 12285, 67095, 71145, 87633, 142310, 525915, 863835, 947835, 1125765, 1798875, 3606850, 5357625, 5684679, 5730615, 6088905, 9206925, 9478910, 9491625, 10634085, 12361622, 13671735, 14426230, 17041010, 17257695, 17754165, 20308995, 20955645, 22227075, 22508145, 23111055, 23389695, 25132545, 34765731, 35115795, 36939357, 43266285, 53011395, 66595130, 74769345, 80422335, 82824255, 82977345, 84591405. - Dean Hickerson, Dec 02 2008 (communicated by Walter Nissen). However, until we have more definite information about the correctness of the first five terms (there could be additional terms less than 142310), there is no point in adding these terms to the "DATA" line. - N. J. A. Sloane, Nov 23 2011

			(69615, 87633) are an amicable pair with sigma = 157248. { 14445, 87633 } are a friendly pair of abundancy = 192/107. Therefore 87633 is a member of the sequence.
The smallest friend of 3606850 is 7521154875.

Dean Hickerson, "Re: Friendly number", post to sci.math newsgroup, 2000, available through groups.google.com.
Ore, Oystein, Number Theory and Its History, McGraw-Hill, 1948, reprinted 1988, section 5-3, pp 96-100.

Claude W. Anderson and Dean Hickerson, Problem 6020: Friendly Integers, Amer. Math. Monthly 84 (1977) pp. 65-66.
Walter Nissen, Abundancy : Some Resources
Walter Nissen, Primitive Friendly Integers and Exclusive Multiples
J. O. M. Pedersen, Known Amicable Pairs [Broken link]
Eric Weisstein's World of Mathematics, Friendly Pair

Cf. A063990, A074902.

Edited by N. J. A. Sloane, Dec 04 2008 and Nov 23 2011

cp's OEIS Frontend

User: Walter Nissen

A222588 Composites of the form 2^n-1 or 2^n+1 that are non-multiples of 3.

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A192397 Record holders for greatest prime factor of n^n + (n+1)^(n+1).

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PARI

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A160405 Primes that are the concatenation of a 5-digit prime, a 7-digit prime, and a 5-digit prime.

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Magma

Extensions

A159611 Indices of the Fermat primes in the sequence of primes.

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Haskell

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PARI

PARI

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A159939 Odd solutions of phi(sigma(k)) = sigma(phi(k)).

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PARI

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A164875 Record holders for n^2 - phi(n)*sigma(n).

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A164876 Record differences for n^2 - phi(n)*sigma(n).

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