A031214 -id:A031214 - cp's OEIS frontend

A100669 Zeros of the Mertens function that are also prime.

2, 101, 149, 163, 331, 353, 401, 419, 541, 607, 811, 823, 853, 877, 883, 919, 1013, 1279, 1289, 1291, 1297, 1523, 1531, 1543, 1861, 2017, 2099, 2113, 2309, 2689, 3607, 3613, 4603, 4637, 4723, 5107, 5113, 5197, 5261, 5849, 6011, 6067, 6089, 7187, 7853

Offset: 0

Robert Happelberg (roberthappelberg(AT)yahoo.com), Dec 05 2004

nonn

A. Delarte, Mathematician reaches 100k milestone for online integer archive, The South End, November 11, 2004, page 5
PrimeFan, Esoteric Integer Sequences
PrimeFan, Esoteric Integer Sequences [Cached copy]

Cf. A028442.

Select[Range[4000], Plus @@ MoebiusMu[Range[#]] == 0 && PrimeQ[#] &] (* Carl Najafi, Aug 17 2011 *)

s=0;for(n=1,1e4,s+=moebius(n); if(!s && isprime(n), print1(n", "))) \\ Charles R Greathouse IV, Jan 28 2014

A209187 Sum of divisors of n minus cototient of n.

Original entry on oeis.org

1, 2, 3, 5, 5, 8, 7, 11, 10, 12, 11, 20, 13, 16, 17, 23, 17, 27, 19, 30, 23, 24, 23, 44, 26, 28, 31, 40, 29, 50, 31, 47, 35, 36, 37, 67, 37, 40, 41, 66, 41, 66, 43, 60, 57, 48, 47, 92, 50, 63, 53, 70, 53, 84, 57, 88, 59, 60, 59, 124, 61, 64, 77, 95, 67, 98, 67, 90, 71, 98, 71

Offset: 1

N. J. A. Sloane, Mar 05 2012

Ivan Neretin, Table of n, a(n) for n = 1..10000
PrimeFan, Listing of Esoteric Integer Sequences.
PrimeFan, Esoteric Integer Sequences [Cached copy]

Cf. A000203, A051953, A065387.

Cf. A013661, A059956.

Table[(Plus @@ Divisors[n]) - (n - EulerPhi[n]), {n, 50}]

a(n) = sigma(n) + eulerphi(n) - n; \\ Altug Alkan, Sep 30 2017

[sigma(n,1)+euler_phi(n)-n for n in range(1, 72)] # Stefano Spezia, Jul 20 2025

a(n) = A065387(n) - n. - Altug Alkan, Sep 30 2017
a(n) = A000203(n) - A051953(n). - Omar E. Pol, Sep 30 2017
Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n*log(n)), where c = 6/Pi^2 + Pi^2/6 - 1 = 1.252861... . - Amiram Eldar, Dec 08 2023

A209188 Smallest prime factor of n^2 + n - 1.

Original entry on oeis.org

5, 11, 19, 29, 41, 5, 71, 89, 109, 131, 5, 181, 11, 239, 271, 5, 11, 379, 419, 461, 5, 19, 599, 11, 701, 5, 811, 11, 929, 991, 5, 19, 29, 1259, 11, 5, 1481, 1559, 11, 1721, 5, 31, 1979, 2069, 2161, 5, 2351, 31, 2549

Offset: 2

N. J. A. Sloane, Mar 05 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
PrimeFan, Listing of Esoteric Integer Sequences
PrimeFan, Esoteric Integer Sequences [Cached copy]

Cf. A071182.

[Min(PrimeFactors(n^2+n-1)):n in[2..70]];// Vincenzo Librandi, Sep 18 2016

SPF[n_] := (FactorInteger[n])[[1]][[1]]; Flatten[Table[SPF[n^2 + (n - 1)], {n, 2, 50}]]
Table[FactorInteger[n^2+n-1][[1,1]],{n,2,50}] (* Harvey P. Dale, Aug 20 2014 *)

a(n) = vecmin(factor(n^2+n-1)[,1]); \\ Michel Marcus, Sep 18 2016

A209191 Numbers n such that digital root of n equals smallest prime factor of n.

Original entry on oeis.org

2, 3, 5, 7, 20, 21, 38, 39, 56, 57, 74, 75, 92, 93, 95, 110, 111, 128, 129, 133, 146, 147, 164, 165, 182, 183, 185, 200, 201, 218, 219, 236, 237, 254, 255, 259, 272, 273, 275, 290, 291, 308, 309, 326, 327, 344, 345, 362, 363, 365, 380, 381, 398

Offset: 1

N. J. A. Sloane, Mar 05 2012

Harvey P. Dale, Table of n, a(n) for n = 1..1000
PrimeFan, Listing of Esoteric Integer Sequences
PrimeFan, Esoteric Integer Sequences [Cached copy]

drspfQ[n_]:=NestWhile[Total[IntegerDigits[#]]&,n,#>9&] == FactorInteger[ n][[1,1]]; Select[Range[2,400],drspfQ] (* Harvey P. Dale, Nov 11 2017 *)

Prior Mathematica program replaced by Harvey P. Dale, Nov 11 2017

A209192 Numbers n such that digital root of n equals largest prime factor of n.

Original entry on oeis.org

2, 3, 5, 7, 12, 48, 50, 70, 128, 192, 196, 320, 448, 500, 700, 768, 1960, 2401, 3072, 3200, 4480, 5000, 5488, 7000, 8192, 8575, 12288, 12544, 19600, 20480, 24010, 28672, 30625, 32000, 44800, 49152, 50000, 54880, 67228, 70000, 78125, 85750, 109375, 125440

Offset: 1

N. J. A. Sloane, Mar 05 2012

Donovan Johnson, Table of n, a(n) for n = 1..1000
PrimeFan, Listing of Esoteric Integer Sequences
PrimeFan, Esoteric Integer Sequences [Cached copy]

A010888 := proc(n)
    d := (add(d,d=convert(n,base,10))) mod 9 ;
    if d = 0 then
        return 9;
    else
        return d;
    end if;
end proc:
isA209192 := proc(n)
    A010888(n) = A006530(n) ;
end proc:
for n from 2 to 40000 do    if isA209192(n) then   printf("%d,",n) ;    end if; end do: # R. J. Mathar, Mar 22 2012

drlpfQ[n_]:=NestWhile[Total[IntegerDigits[#]]&,n,#>9&]==FactorInteger[n][[-1,1]]; Select[Range[2,126000],drlpfQ] (* Harvey P. Dale, Sep 26 2024 *)

A111157 Numbers k such that sequence A_k in this database does not contain a prime.

Original entry on oeis.org

4, 7, 12, 35, 56, 66, 82, 118, 132, 141, 143, 144, 145, 152, 156, 173, 252

Offset: 1

N. J. A. Sloane, following a suggestion of Alonso del Arte, Oct 21 2005

That is, the complete sequence A_k, not just the terms that are shown in the entry, does not contain a prime or the negative of a prime. The only exception to this is for sequences with keyword "dead", when only the visible terms are to be considered.
No other terms below 287. Is 287 a member? This requires proof.
I've checked A000287 up to a(25000) and can report that I found no prime. - Robert G. Wilson v, Jun 23 2014
A search of every term of A000287 of form a(2^k-2) has yielded no primes up to a(2^24-2). It shouldn't have any odd terms outside this subsequence, and assuming that its terms' factors are otherwise random, it has a 2.15*10^-8 probability of containing any prime at all. After A000287, the next difficult sequence is A000373, which should contain infinitely many primes, but has none within its first 10000 terms. - Matthew House, Nov 02 2024

PrimeFan, Esoteric Integer Sequences (Gives a different version.)
PrimeFan, Esoteric Integer Sequences [Cached copy]
Index entries for sequences whose definition involves A_n (or An).

Cf. A111198.

a(16) = 173 inserted by Matthew House, Nov 02 2024

A209186 Admission order to the USA of the n-th state in alphabetical order.

Original entry on oeis.org

22, 49, 48, 25, 31, 38, 5, 1, 27, 4, 50, 43, 21, 19, 29, 34, 15, 18, 23, 7, 6, 26, 32, 20, 24, 41, 37, 36, 9, 3, 47, 11, 12, 39, 17, 46, 33, 2, 13, 8, 40, 16, 28, 45, 14, 10, 42, 35, 30, 44

Offset: 1

N. J. A. Sloane, Mar 05 2012

PrimeFan, Listing of Esoteric Integer Sequences
PrimeFan, Esoteric Integer Sequences [Cached copy]

A209189 Smallest prime factor of n^2 + n + 1.

Original entry on oeis.org

7, 13, 3, 31, 43, 3, 73, 7, 3, 7, 157, 3, 211, 241, 3, 307, 7, 3, 421, 463, 3, 7, 601, 3, 19, 757, 3, 13, 7, 3, 7, 1123, 3, 13, 31, 3, 1483, 7, 3, 1723, 13, 3, 7, 19, 3, 37, 13, 3, 2551

Offset: 2

N. J. A. Sloane, Mar 05 2012

nonn

PrimeFan, Listing of Esoteric Integer Sequences
PrimeFan, Esoteric Integer Sequences [Cached copy]

Flatten[Table[SPF[n^2 + (n + 1)], {n, 2, 50}]]

A209193 Safe primes that are also highly cototient numbers.

Original entry on oeis.org

23, 47, 59, 83, 167, 839, 2099, 5879, 6719, 23099, 27299, 180179, 414959, 420419, 628319, 660659, 840839, 1141139, 1681679, 2042039, 2277659, 4084079, 5525519, 5705699

Offset: 1

N. J. A. Sloane, Mar 05 2012

PrimeFan, Listing of Esoteric Integer Sequences
PrimeFan, Esoteric Integer Sequences [Cached copy]

A005385 intersect A100827.

a(19)-a(24) calculated from the b-file at A100827 by Amiram Eldar, Apr 09 2020

A209194 Sophie Germain primes that are also highly cototient numbers.

Original entry on oeis.org

2, 23, 83, 89, 113, 419, 509, 659, 1049, 1889, 3359, 4409, 9029, 10709, 11549, 13649, 20789, 23099, 57329, 84629, 90089, 113189, 115499, 207479, 210209, 270269, 314159, 324869, 330329, 420419, 570569, 658349, 660659, 810809, 840839, 1021019, 1138829

Offset: 1

N. J. A. Sloane, Mar 05 2012

nonn

PrimeFan, Listing of Esoteric Integer Sequences
PrimeFan, Esoteric Integer Sequences [Cached copy]

A005384 INTERSECT A100827.

cp's OEIS Frontend

A100669 Zeros of the Mertens function that are also prime.

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Mathematica

PARI

A209187 Sum of divisors of n minus cototient of n.

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SageMath

Formula

A209188 Smallest prime factor of n^2 + n - 1.

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Magma

Mathematica

PARI

A209191 Numbers n such that digital root of n equals smallest prime factor of n.

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Extensions

A209192 Numbers n such that digital root of n equals largest prime factor of n.

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Maple

Mathematica

A111157 Numbers k such that sequence A_k in this database does not contain a prime.

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A209186 Admission order to the USA of the n-th state in alphabetical order.

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A209189 Smallest prime factor of n^2 + n + 1.

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Mathematica

A209193 Safe primes that are also highly cototient numbers.

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Extensions

A209194 Sophie Germain primes that are also highly cototient numbers.

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Keywords

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