A031214 -id:A031214 - cp's OEIS frontend

A100544 Outdated duplicate of A031214.

0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2, 1, 0, 1, 2, 0, 1, 0, 2, 2, 2, 1, 2, 1, 1, 2, 1, 0, 1, 1, 1, 0, 1, 2, 8, 14, 4, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 0, 4, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 3, 1, 0, 1, 1

Offset: 1

dead

A051070 a(n) is the n-th term in sequence A_n, respecting the offset, or a(n) = -1 if A_n has fewer than n terms.

Original entry on oeis.org

1, 2, 1, 0, 2, 3, 0, 7, 8, 4, 63, 1, 316, 78, 16, 2048, 7652, 26627, 8, 24000, 232919, 1145406, 3498690007594650042368, 2058537, 58, 26, 27, 59, 9272780, 3, 69273668, 4870847, 2387010102192469724605148123694256128, 1, 1, -53, 43, 0, -4696, 173, 44583, 111111111111111111111111111111111111111111, 30402457, 668803781, 1134903170, 382443020332

Offset: 1

Robert G. Wilson v, Aug 23 2000

a(58) = A000058(58) = 192523...920807 (58669977298272603 digits) is too large to include in the b-file. - Pontus von Brömssen, May 19 2022
Comment from N. J. A. Sloane, Dec 26 2022 (Start)
Note that a(n) = -1 can arise in two ways: either A_n has fewer than n terms, or A_n has at least n terms, but its n-th term is -1.
Here is a summary of the terms with n <= 80.
a(n) = -1 occurs just twice, for n = 53 and 54, in both cases because the relevant New York subway lines do not have enough stops.
a(1) though a(65) are known, although a(58) = = 192523...920807 has 58669977298272603 digits.
a(66) is the first unknown value.
Also unknown for n <= 80 are a(67), a(72), a(74), a(75), a(76), and a(77) (counts of numbers <= 2^n represented by various quadratic forms; some of these do not even have b-files), and a(80), which like a(66) is a graph-theory question. (End)

			a(19) = 8 because A000019(19) = 8.
a(20) = 24000 because A000020(20) = 24000.

Seth A. Troisi, Table of n, a(n) for n = 1..57 (terms 1..48 from Pontus von Brömssen).
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 21.
Index entries for sequences whose definition involves A_n (or An).

See A091967, A107357, A102288 for other versions. See also A031214, A031135.

Cf. A000019, A000020, A000058.

for m from 1 do
  url:= sprintf("https://oeis.org/A%06d/b%06d.txt",m,m);
  S:= URL:-Get(url);
  L:= StringTools[Split](S,"\n");
  for t in L do
    g:= sscanf(t, "%d %d");
    if nops(g) = 2 and g[1] = m then
      a[m]:= g[2];
      break
    fi;
  od;
  if not assigned(a[m]) then break fi;
od:
seq(a[i],i=1..m-1); # Robert Israel, May 31 2015

Rechecked and 4 more terms added by N. J. A. Sloane, May 25 2005
a(36) and a(42) corrected and a(43) to a(46) added by Robert Israel, May 31 2015
Definition revised by N. J. A. Sloane, Nov 27 2016

A100765 Numbers for which the values of the Moebius function (A008683) and the Mertens function (A002321) are both -1.

Original entry on oeis.org

3, 41, 59, 66, 102, 151, 165, 167, 233, 239, 255, 354, 357, 359, 367, 402, 406, 409, 421, 426, 429, 609, 638, 782, 786, 797, 826, 854, 885, 887, 890, 894, 897, 907, 911, 1015, 1019, 1221, 1259, 1281, 1283, 1298, 1301, 1303, 1307, 1319, 1327, 1493, 1526, 1533

Offset: 1

Alonso del Arte, Jan 03 2005

nonn

This sequence is a subsequence of A100306, Numbers for which the values of the Moebius function and the Mertens function agree.

			102 is in the sequence because it is a sphenic number (exactly 3 distinct prime factors, A007304) number, so the Mobius function yields -1 and the sum of that value and the previous Mobius values (the Mertens function) is also -1.

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

Cf. A100306, A100766, A100767.

(* If not already defined *) If[Names["Mertens"] == {}, Mertens[x_] := Plus @@ MoebiusMu[Range[1, x]]]; Select[Range[2500], MoebiusMu[ # ] == -1 && Mertens[ # ] == -1 &]

Offset corrected by Donovan Johnson, Jun 19 2012

A100306 Numbers for which the values of the Moebius function (A008683) and the Mertens function (A002321) agree.

Original entry on oeis.org

1, 3, 40, 41, 59, 66, 94, 102, 146, 150, 151, 160, 161, 164, 165, 167, 215, 232, 233, 236, 237, 239, 255, 330, 332, 333, 334, 354, 356, 357, 359, 363, 364, 365, 367, 394, 402, 404, 405, 406, 408, 409, 414, 415, 420, 421, 423, 424, 425, 426, 428, 429, 538, 542

Offset: 1

N. J. A. Sloane, Dec 30 2004

nonn

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

Cf. A100765, A100766, A100767.

with(numtheory): p:=proc(n) if mobius(n)=sum(mobius(k),k=1..n) then n else fi end: seq(p(n),n=1..700); # Emeric Deutsch, Feb 14 2005

Select[Range[500], Plus @@ MoebiusMu[Range[#]] == MoebiusMu[#] &] (* Carl Najafi, Aug 17 2011 *)

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

More terms from Emeric Deutsch, Feb 14 2005

A100766 Numbers for which the values of the Moebius function (A008683) and the Mertens function (A002321) are both 0.

Original entry on oeis.org

40, 150, 160, 164, 232, 236, 332, 333, 356, 363, 364, 404, 405, 408, 414, 420, 423, 424, 425, 428, 608, 636, 637, 796, 812, 824, 825, 850, 884, 896, 904, 916, 920, 1014, 1220, 1256, 1280, 1292, 1300, 1336, 1492, 1519, 1520, 1521, 1524, 1525, 1528, 1532, 1544

Offset: 1

Alonso del Arte, Jan 03 2005

nonn

This sequence is a subset of A100306, numbers for which the values of the Moebius function and the Mertens function agree and, in a different way, a subset of A028442, zeros of the Mertens function. There are no prime numbers in this sequence.

Numbers k such that k-1 and k are consecutive zeros of the Mertens function. - Amiram Eldar, Jun 13 2020

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

Cf. A100306, A100765, A100767.

(* If not already defined *) If[Names["Mertens"] == {}, Mertens[x_] := Plus @@ MoebiusMu[Range[1, x]]]; Select[Range[2500], MoebiusMu[ # ] == 0 && Mertens[ # ] == 0 &]

Offset corrected by Donovan Johnson, Jun 19 2012

A100767 Numbers for which the values of the Moebius function (A008683) and the Mertens function (A002321) are both 1.

Original entry on oeis.org

1, 94, 146, 161, 215, 237, 330, 334, 365, 394, 415, 538, 542, 794, 799, 813, 815, 851, 870, 878, 899, 905, 914, 917, 921, 1003, 1006, 1011, 1257, 1262, 1267, 1271, 1286, 1290, 1293, 1330, 1337, 1339, 1343, 1522, 1529, 1538, 1858, 1865, 1939, 2018, 2098

Offset: 1

Alonso del Arte, Jan 03 2005

nonn

This sequence is a subsequence of A100306, Numbers for which the values of the Moebius function and the Mertens function agree.

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

Cf. A100306, A100765, A100766.

(* If not already defined *) If[Names["Mertens"] == {}, Mertens[x_] := Plus @@ MoebiusMu[Range[1, x]]]; Select[Range[2500], MoebiusMu[ # ] == 1 && Mertens[ # ] == 1 &]

Offset corrected by Donovan Johnson, Jun 19 2012

A209190 Least prime factor of reversal of digits of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 3, 1, 11, 3, 31, 41, 3, 61, 71, 3, 7, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 13, 23, 3, 43, 53, 3, 73, 83, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 3, 5, 5, 3, 5, 5, 3, 5, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 7, 17, 3, 37, 47, 3, 67, 7, 3, 97, 2, 2, 2

Offset: 1

N. J. A. Sloane, Mar 05 2012

This is to lpf (A020639) as A210437 is to gpf (A006530). - Jonathan Vos Post, Mar 22 2012

Alois P. Heinz, Table of n, a(n) for n = 1..12000
PrimeFan, Listing of Esoteric Integer Sequences
PrimeFan, Esoteric Integer Sequences [Cached copy]

Cf. A004086, A020639, A210437.

read("transforms") ;
A209190 := proc(n)
        A020639(digrev(n)) ;
end proc:
seq(A209190(n),n=1..100) ; # R. J. Mathar, Mar 22 2012

a[n_] := FactorInteger[IntegerReverse[n]][[1, 1]];
Array[a, 100] (* Jean-François Alcover, Jan 14 2021 *)

lpf(n) = if (n==1, 1, vecmin(factor(n)[,1]));
a(n) = lpf(fromdigits(Vecrev(digits(n)))); \\ Michel Marcus, Feb 11 2020

a(n) = A020639(A004086(n)) = lpf(R(n)). - Jonathan Vos Post, Mar 22 2012

A039928 Sum of first n terms of A_n (using absolute values of terms).

Original entry on oeis.org

0, 3, 3, 0, 10, 12, 1, 24, 25, 32, 116, 12, 412, 109, 126, 2389, 12497, 28772, 126, 72795, 247786, 770213, 159378001963452599318, 2169128, 442, 311, 378, 789, 10015050, 75, 74253544, 7881195, 2461717833658872781238383813854943728, 51, 17, 824, 855, 2, 29981, 3087, 215308, 123456790123456790123456790123456790123452, 132813776, 1086162642, 1836311902, 400276874544

Offset: 1

Russ Cox

Since the sequences in the OEIS occasionally change their initial terms (for editorial reasons), this is an especially ill-defined sequence! - N. J. A. Sloane, Jan 01 2005

The next term, a(47), is currently unknown. - Jianing Song, Oct 07 2018

			A000001 (Number of groups of order n) begins 0,... -> a(1) = 0
A000002 (Kolakoski sequence) begins 1, 2,... -> a(2) = 3
A000003 begins 1, 1, 1,... -> a(3) = 3
A000004 (The zero sequence) begins 0, 0, 0, 0,... -> a(4) = 0
A000005 (The number of divisors) begins 1, 2, 2, 3, 2, ... -> a(5) = 10
...
A000010 (Euler totient function) begins 1, 1, 2, 2, 4, 2, 6, 4, 6, ... so a(10) = 1 + 1 + 2 + 2 + 4 + 2 + 6 + 4 + 6 + 4 = 32.

Index entries for sequences whose definition involves A_n (or An).

Cf. A031135, A031214, A100543 (uses signed values).

Corrected and extended by C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 27 2004

a(1) changed from 1 to 0 and extended by Jianing Song, Oct 06 2018

A105440 Highly cototient numbers that are prime, or intersection of A000040 and A100827.

Original entry on oeis.org

2, 23, 47, 59, 83, 89, 113, 167, 269, 389, 419, 509, 659, 839, 1049, 1259, 1889, 2099, 2309, 2729, 3359, 3989, 4289, 4409, 5879, 6089, 6719, 9029, 9239, 10289, 10709, 11549, 13649, 13859, 15329, 15959, 20789, 21839, 23099, 25409, 27299, 30029, 34649, 43889, 51869

Offset: 1

Paul Muljadi, Sep 22 2005

nonn

2 is definitely a "highly cototient prime". Oddly enough, though, PrimeFan does not list it as safe prime that is also a highly cototient number in RNTFS00013 even though he considers 1 to be a prime. - Robert Happelberg (roberthappelberg(AT)yahoo.com), Oct 04 2005

115 of the first 229 terms (just over 50%) of A100827 are prime. - Jud McCranie, May 18 2017

			269 (after 59 & 89) is the first prime in a sequence of primes which are congruent to 9 mod 10.

Amiram Eldar, Table of n, a(n) for n = 1..115
PrimeFan, Listing of Esoteric Integer Sequences, See RNTFS00013, RNTFS00014.
PrimeFan, Esoteric Integer Sequences [Cached copy]

Cf. A000040, A100827.

More terms from Robert Happelberg (roberthappelberg(AT)yahoo.com), Oct 04 2005
More terms from T. D. Noe, Mar 16 2010
More terms from Amiram Eldar, Dec 24 2020

A100543 Sum of first n terms of A_n (signed values).

Original entry on oeis.org

0, 3, 3, 0, 10, 12, 1, 24, 25, 32, 116, 12, 412, 109, 126, 2389, 12497, 28772, 126, 72795, 247786, 770213, 159378001963452599312, 2169128, -26, 311, 378, 789, 10015050, 75, 74253544, 7881195, 2461717833658872781238383813854943728, 51, 17, -250, 855, 2, -29979, 3087, 215308, 123456790123456790123456790123456790123452, 132813776, 1086162642, 1836311902, 400276874544

Offset: 1

Russ Cox

Since the sequences in the OEIS occasionally change their initial terms (for editorial reasons), this is an especially ill-defined sequence! - N. J. A. Sloane, Jan 01 2005

The next term, a(47), is currently unknown. - Jianing Song, Oct 07 2018

			A000001 (Number of groups of order n) begins 0,... -> a(1) = 0
A000002 (Kolakoski sequence) begins 1, 2,... -> a(2) = 3
A000003 begins 1, 1, 1,... -> a(3) = 3
A000004 (The zero sequence) begins 0, 0, 0, 0,... -> a(4) = 0
A000005 (The number of divisors) begins 1, 2, 2, 3, 2, ... -> a(5) = 10
...
A000010 (Euler totient function) begins 1, 1, 2, 2, 4, 2, 6, 4, 6, ... so a(10) = 1 + 1 + 2 + 2 + 4 + 2 + 6 + 4 + 6 + 4 = 32.

Index entries for sequences whose definition involves A_n (or An).

Cf. A031135, A031214, A039928.

Corrected and extended by C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 27 2004
a(1) changed from 1 to 0 by Jianing Song, Jul 02 2018
Extend by Jianing Song, Oct 07 2018

cp's OEIS Frontend

A100544 Outdated duplicate of A031214.

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A051070 a(n) is the n-th term in sequence A_n, respecting the offset, or a(n) = -1 if A_n has fewer than n terms.

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A100765 Numbers for which the values of the Moebius function (A008683) and the Mertens function (A002321) are both -1.

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A100306 Numbers for which the values of the Moebius function (A008683) and the Mertens function (A002321) agree.

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A100766 Numbers for which the values of the Moebius function (A008683) and the Mertens function (A002321) are both 0.

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A100767 Numbers for which the values of the Moebius function (A008683) and the Mertens function (A002321) are both 1.

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A209190 Least prime factor of reversal of digits of n.

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A039928 Sum of first n terms of A_n (using absolute values of terms).

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A105440 Highly cototient numbers that are prime, or intersection of A000040 and A100827.