A039752 -id:A039752 - cp's OEIS frontend

A129320 a(n) = A129319(n)/A129318(n).

1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 5, 15, 2, 5, 1, 3, 11, 2, 11, 1, 5, 5, 1, 4, 4, 1, 5, 2, 16, 10, 1, 6, 13, 4, 3, 9, 1, 1, 6, 2, 26, 2, 3, 23, 4, 1, 1, 6, 2, 3, 1, 11, 2, 1, 8, 14, 28, 8, 1, 7, 1, 23, 8, 20, 7, 2

Offset: 1

Walter Kehowski, Apr 09 2007

			a(6)=2 since A129319(n)/A129318(n)=30/15=2.

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

Cf. A001414, A006145, A039752, A129316, A129317, A129318, A129319.

A330999 Infinitary Ruth-Aaron numbers: numbers k such that A181894(k) = A181894(k+1).

Original entry on oeis.org

5, 77, 714, 948, 2431, 2491, 2996, 3450, 4293, 5405, 5560, 5885, 5959, 11124, 13869, 14587, 16932, 17080, 17346, 18468, 19551, 26642, 31931, 33019, 37925, 42250, 47544, 48635, 49240, 52554, 53192, 60048, 79248, 80837, 89979, 95709, 98119, 98644, 99163, 108458

Offset: 1

Amiram Eldar, Jan 05 2020

nonn

A variation of Ruth-Aaron numbers with "Fermi-Dirac primes" (or infinitary components) instead of prime divisors.

			5 is a term since A181894(5) = A181894(6) = 5.

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

Cf. A006145, A039752, A050376, A181894, A331000.

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); s[1] = 0; s[n] := Plus @@ (Flatten @ (f @@@ FactorInteger[n])); seq ={}; s1 = 0; Do[s2 = s[n]; If[s1 == s2, AppendTo[seq, n-1]]; s1 = s2, {n, 2, 10^5}]; seq

A331000 Unitary Ruth-Aaron numbers: numbers k such that A008475(k) = A008475(k+1).

Original entry on oeis.org

5, 77, 714, 948, 2491, 2996, 3450, 4293, 5405, 6669, 9125, 10807, 13869, 14587, 16932, 17346, 19511, 19967, 23323, 26642, 27104, 31931, 33019, 37925, 41124, 43616, 48635, 52554, 55499, 58077, 58695, 79248, 80837, 86088, 89979, 95709, 98644, 99163, 108458, 117467

Offset: 1

Amiram Eldar, Jan 05 2020

nonn

A variation of Ruth-Aaron numbers with unitary prime-power divisors instead of prime divisors.

			5 is a term since A008475(5) = A008475(6) = 5.

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

Cf. A006145, A039752, A008475, A330999.

s[1] = 0; s[n_] := Plus @@ (Power @@@ FactorInteger[n]); seq = {}; s1 = 0; Do[s2 = s[n]; If[s1 == s2, AppendTo[seq, n - 1]]; s1 = s2, {n, 2, 10^5}]; seq

A178214 Numbers in A039753 with neither of their Ruth-Aaron pairs squarefree.

Original entry on oeis.org

7129199, 9867275, 18918704, 43009524, 43882488, 45828324, 126511280, 132082191, 150786063, 252625743, 285816464, 303792200, 313887275, 330130475, 336945392, 337795524, 361652075, 380035664, 480297824, 579423924, 647037215, 650012724, 756098624, 986677500, 1000308015, 1001438136, 1139325668, 1140314075, 1205629524, 1315277147

Offset: 1

Hans Havermann, Dec 19 2010

nonn

			7129199 (7*11^2*19*443, with 7129200 = 2^4*3*5^2*13*457) is a member of both A006145 (7+11+19+443 = 2+3+5+13+457) and A039752 (7+11+11+19+443 = 2+2+2+2+3+5+5+13+457) but neither 7129199 nor 7129200 is squarefree, so 7129199 is a member of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..385

Cf. A039753, A006145, A039752, A013929.

A327250 Numbers k such that s(k) = s(k+1), where s(k) is A059975.

Original entry on oeis.org

3, 80, 175, 272, 492, 860, 943, 6556, 6867, 7104, 7215, 14672, 17459, 21804, 22672, 24435, 24476, 26128, 30899, 34595, 39215, 41327, 45548, 49468, 56563, 57075, 63440, 63744, 67123, 72556, 78524, 87615, 90243, 104111, 109939, 113283, 113296, 115344, 121539, 131651

Offset: 1

Amiram Eldar, Sep 15 2019

nonn

Madeleine Farris named these numbers "Euler-totient Ruth-Aaron numbers" (in analogy to the Ruth-Aaron numbers, A039752). She proved that the number of terms <= x is O(x*(log(log(x))^4)/(log(x))^2) and that the sum of their reciprocals is bounded.

			3 is in the sequence since A059975(3) = A059975(4) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Madeleine Farris, Ruth-Aaron Numbers: An Exploration in Analytic Number Theory (thesis), Wellesley College, 2019.

Cf. A059975, A006145, A039752.

f[p_,e_] := e * (p-1); a[n_] := Plus @@ (f @@@ FactorInteger[n]); aQ[n_] := a[n] == a[n+1]; Select[Range[10^5], aQ]

s(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i, 2] * (f[i, 1] - 1));}
lista(kmax) = {my(s1 = s(1), s2); for(k=2, kmax, s2 = s(k); if(s1 == s2, print1(k-1, ", ")); s1 = s2);} \\ Amiram Eldar, Apr 06 2023

A333801 Numbers k such that A008475(k)+1 = A008475(k+1).

Original entry on oeis.org

2, 3, 4, 7, 8, 16, 20, 31, 35, 127, 143, 208, 256, 650, 1479, 2464, 2623, 4233, 4345, 5183, 8099, 8191, 9424, 11024, 11919, 12099, 14905, 16159, 20220, 20800, 21716, 22194, 24335, 26123, 27335, 27390, 30457, 34945, 38180, 40425, 52206, 56563, 65536, 67123, 68264

Offset: 1

Amiram Eldar, Apr 05 2020

nonn

A variation of A064111 and A228126 with unitary prime-power divisors instead of prime divisors.

			4 is a term since A008475(4) + 1 = 4 + 1 = 5 = A008475(5).

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

Cf. A006145, A008475, A039752, A064111, A228126, A331000, A333802.

s[1] = 0; s[n_] := Plus @@ (Power @@@ FactorInteger[n]); seq = {}; s1 = 0; Do[s2 = s[n]; If[s1 + 1 == s2, AppendTo[seq, n - 1]]; s1 = s2, {n, 2, 10^5}]; seq

A333802 Numbers k such that A181894(k)+1 = A181894(k+1).

Original entry on oeis.org

2, 3, 4, 16, 20, 35, 143, 152, 208, 256, 650, 1624, 2232, 4233, 4345, 5368, 8099, 9424, 11024, 11919, 12099, 14905, 18424, 20220, 21716, 22194, 24335, 25592, 26123, 27390, 30457, 34945, 38180, 40425, 51992, 52206, 52947, 56563, 63712, 65536, 67123, 71154, 71284

Offset: 1

Amiram Eldar, Apr 05 2020

nonn

A variation of A064111 and A228126 with "Fermi-Dirac primes" (or infinitary components) instead of prime divisors.

			4 is a term since A181894(4) + 1 = 4 + 1 = 5 = A181894(5).

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

Cf. A006145, A039752, A050376, A064111, A181894, A228126, A330999, A333801.

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], 1])); s[1] = 0; s[n_] := Plus @@ (Flatten @ (f @@@ FactorInteger[n])); seq = {}; s1 = 0; Do[s2 = s[n]; If[s1 + 1 == s2, AppendTo[seq, n - 1]]; s1 = s2, {n, 2, 10^5}]; seq

A101805 Number of Ruth-Aaron numbers (with multiplicity) less than 10^n.

Original entry on oeis.org

2, 4, 7, 20, 57, 149, 523, 1835, 6810, 26222, 106670, 444085

Offset: 1

Eric W. Weisstein, Dec 16 2004

Eric Weisstein's World of Mathematics, Ruth-Aaron Pair

Cf. A039752.

Offset corrected and a(9)-a(12) from Donovan Johnson, Dec 14 2010

A175513 Numbers k for which 6k+1, 24k+5, 432k^2+72k-1, and 432k^2+90k-1 are all prime.

Original entry on oeis.org

1, 2, 13, 753, 767, 1336, 1771, 1773, 1911, 2487, 3527, 4192, 5061, 5343, 5973, 6341, 7062, 7777, 8932, 9153, 15301, 17976, 18713, 19543, 20318, 22253, 24068, 27461, 29416, 29502, 31383, 31593, 31616, 31693, 36026, 36087, 41456, 42966, 44711, 45453, 45493, 46766, 49067, 50602, 51212, 51393, 53193, 56762, 58267, 60332, 60918, 64126, 65727, 67872, 71266, 72011, 75861, 78728, 79652, 82978, 85246, 86207, 86988, 87793, 90873, 91753, 94173, 97297

Offset: 1

Hans Havermann, Dec 03 2010

nonn

10368k^3+3888k^2+336k-5 is a Ruth-Aaron number (2, A039752).

C. Nelson, D. E. Penney and C. Pomerance, "714 and 715", J. Recreational Math. 7 (No. 2) 1974, 87-89.

Terrel Trotter, Jr., 1974 article providing the basis for this sequence (for defined variables s, p, q, r, replace x with 3k)

Cf. A039752.

Select[Range[100000], PrimeQ[6 # + 1] && PrimeQ[24 # + 5] && PrimeQ[432*#^2 + 72*# - 1] && PrimeQ[432 #^2 + 90 # - 1] &]
Select[Range[100000],AllTrue[{6#+1,24#+5,432#^2+72#-1,432#^2+90#-1}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 24 2019 *)

A238605 Semiprimes n such that (n+1)/4 also is a semiprime.

Original entry on oeis.org

15, 35, 39, 55, 87, 155, 183, 203, 219, 247, 259, 295, 327, 339, 371, 471, 515, 535, 579, 583, 635, 707, 731, 803, 807, 835, 851, 871, 939, 995, 1011, 1047, 1059, 1067, 1111, 1147, 1191, 1195, 1203, 1207, 1211, 1219, 1255, 1315, 1339, 1355, 1363, 1383, 1507, 1527, 1563, 1591, 1643, 1651, 1687, 1707

Offset: 1

M. F. Hasler, Mar 01 2014

nonn

A subsequence of Ruth-Aaron numbers A039752. The terms are by definition of the form n = 4k+3.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001414.

Select[Range[2000],PrimeOmega[#]==PrimeOmega[(#+1)/4]==2&] (* Harvey P. Dale, Oct 17 2021 *)

forstep( n=3,999,4,bigomega(n)==2 & bigomega((n+1)/4)==2 && print1(n","))

cp's OEIS Frontend

A129320 a(n) = A129319(n)/A129318(n).

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A330999 Infinitary Ruth-Aaron numbers: numbers k such that A181894(k) = A181894(k+1).

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A331000 Unitary Ruth-Aaron numbers: numbers k such that A008475(k) = A008475(k+1).

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A178214 Numbers in A039753 with neither of their Ruth-Aaron pairs squarefree.

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A327250 Numbers k such that s(k) = s(k+1), where s(k) is A059975.

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A333801 Numbers k such that A008475(k)+1 = A008475(k+1).

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A333802 Numbers k such that A181894(k)+1 = A181894(k+1).

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A101805 Number of Ruth-Aaron numbers (with multiplicity) less than 10^n.

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A175513 Numbers k for which 6k+1, 24k+5, 432k^2+72k-1, and 432k^2+90k-1 are all prime.

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