A006145 -id:A006145 - cp's OEIS frontend

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

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

A063969 Numbers k such that sopf(k) = sopf(k+3), where sopf(k) = A008472(k).

Original entry on oeis.org

7, 60, 147, 407, 470, 1053, 1175, 3431, 3822, 5126, 5960, 6280, 6321, 6897, 7200, 8687, 9243, 10760, 12614, 15093, 16153, 18080, 18818, 19668, 20433, 20976, 24648, 26826, 30804, 44016, 45878, 46221, 47423, 55965, 58506, 58682, 59645, 63897

Offset: 1

Jason Earls, Sep 05 2001

nonn

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Cf. A008472, A006145.

Select[Range[65000],Total[Transpose[FactorInteger[#]][[1]]] == Total[ Transpose[FactorInteger[#+3]][[1]]]&] (* Harvey P. Dale, Jan 19 2013 *)

sopf(n,s,fac,i)=fac=factor(n); for(i=1,matsize(fac)[1],s=s+fac[i,1]); return(s);
j=[]; for(n=1,100000, if(sopf(n)==sopf(n+3),j=concat(j,n))); j

sopf(n)= { local(f,s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(s) }
{ n=0; for (m=1, 10^9, if(sopf(m)==sopf(m + 3), write("b063969.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 04 2009

A045760 Smallest Maris-McGwire k-tuple (k>1) for each k: f(n) = f(n+1) = ... = f(n+k-1), where f is defined in comments.

Original entry on oeis.org

7, 212, 8126, 241995, 3539990, 1330820, 12222533493, 3249880870

Offset: 2

Mike Keith

A045759 generalizes to k consecutive integers that all have the same value of f(n), where f(n) = sum of digits of n plus sum of digits in prime factors of n. The sequence shows the integer which starts the smallest set of k (and no more than k) consecutive integers having this property.

H. Havermann, Maris-McGwire-Sosa 7-tuples, 8-tuples and 9-tuples

Cf. A045759, A006145.

Two more terms from Hans Havermann, Dec 13 2000

Offset changed to 2 by Hans Havermann, Jun 22 2014

A227654 Ruth-Aaron triples (1): sum of primes dividing n = sum of primes dividing (n+1) = sum of primes dividing (n+2).

Original entry on oeis.org

89460294, 151165960539, 3089285427491, 6999761340223, 7539504384825

Offset: 1

Giovanni Resta, Jul 19 2013

a(6) > 10^13. Taking into account primes multiplicity (A039752), the only triples up to 10^13 are those starting at 417162 and 6913943284.

			7539504384825 = 3^2 * 5^2 * 7^2 * 43 * 251 * 63361,
7539504384826 = 2 * 19 * 367 * 10181 * 53101,
7539504384827 = 17 * 457 * 26309 * 36887 and
3 + 5 + 7 + 43 + 251 + 63361 = 2 + 19 + 367 + 10181 + 53101 = 17 + 457 + 26309 + 36887.

Shyam Sunder Gupta, Smith Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 4, 127-157.
Carlos Rivera, Puzzle 358. Ruth-Aaron Pairs Revisited, The Prime Puzzles and Problems Connection.
Eric Weisstein's World of Mathematics, Ruth-Aaron Pair.
Wikipedia, Ruth-Aaron pair.

Cf. A006145, A039752.

A239290 Number of pairs of terms of the sequence A228126 less than 10^n.

Original entry on oeis.org

4, 7, 8, 19, 55, 149, 497, 1799, 6696, 26109, 106953

Offset: 1

Abhiram R Devesh, Jun 14 2014

Giovanni Resta, eRAPs: the 446139 terms < 10^12
Carlos Rivera, Extension to Ruth Aaron pairs

Cf. A001414, A006145 Ruth-Aaron numbers (1): sum of prime divisors of n = sum of prime divisors of n+1, A228126 Sum of prime divisors of n (with repetition) is one less than the sum of prime divisors (with repetition) of n+1.

A243902 Number of pairs of terms of the sequence A237929 less than 10^n.

Original entry on oeis.org

2, 3, 4, 9, 18, 45, 146, 469, 1655, 6095, 23775

Offset: 1

Abhiram R Devesh, Jun 14 2014

Giovanni Resta, eRAPs, the 446139 terms < 10^12.
Carlos Rivera, Extension to Ruth Aaron pairs

Cf. A001414, A006145 Ruth-Aaron numbers (1): sum of prime divisors of n = sum of prime divisors of n+1, A228126 Sum of prime divisors of n (with repetition) is one less than the sum of prime divisors (with repetition) of n+1, A237929 Numbers n such that (i) the sum of prime divisors of n (with repetition) is one less than the sum of prime divisors (with repetition) of n+1, and (ii) n and n+1 have the same number of prime divisors (with repetition).

A354603 Numbers k such that sum of distinct primes dividing k is equal to the sum of proper divisors of k+1.

Original entry on oeis.org

3, 7, 14, 31, 127, 206, 2974, 8191, 19358, 20490, 131071, 147454, 286122, 289650, 292332, 441276, 524287, 909498, 1207358, 1657968, 1782540, 2490042, 3368860, 9274806, 11367402, 14107852, 16776156, 18589386, 22910988, 24450316, 26867718, 28959606, 32674506, 33163372

Offset: 1

Metin Sariyar, Jul 08 2022

nonn

Numbers k such that A008472(k) = A001065(k+1). All Mersenne primes are terms.

			Example:  14 is a term because A008472(14) = 2+7 = A001065(15) = 1+3+5.

Cf. A000668, A001065, A008472, A006145.

Select[Range[19358],Sum[f, {f, Select[Divisors[#], PrimeQ]}]==DivisorSigma[1,#+1]-(#+1)&]

More terms from Amiram Eldar, Jul 08 2022

A362152 Numbers k such that k and k^2+1 have equal sums of distinct prime divisors.

Original entry on oeis.org

7, 1384230, 1437236, 1770802, 2090663, 4406787, 8493543, 8691863, 11576449, 16147463, 18216983, 22128632, 25156787, 32929141, 43106430, 43768187, 47500230, 50085263, 50497485, 59461592, 66419007, 66507421, 71182692, 95268412, 99848687, 164163693

Offset: 1

Max Alekseyev, Apr 18 2023

nonn

Numbers k such that A008472(k) = A008472(k^2+1).

Cf. A008472, A006145.

is_A362152(n) = vecsum(factor(n)[,1])==vecsum(factor(n^2+1)[,1]);

cp's OEIS Frontend

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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A063969 Numbers k such that sopf(k) = sopf(k+3), where sopf(k) = A008472(k).

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A045760 Smallest Maris-McGwire k-tuple (k>1) for each k: f(n) = f(n+1) = ... = f(n+k-1), where f is defined in comments.

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A227654 Ruth-Aaron triples (1): sum of primes dividing n = sum of primes dividing (n+1) = sum of primes dividing (n+2).

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A239290 Number of pairs of terms of the sequence A228126 less than 10^n.

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A243902 Number of pairs of terms of the sequence A237929 less than 10^n.

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A354603 Numbers k such that sum of distinct primes dividing k is equal to the sum of proper divisors of k+1.

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