A228126 -id:A228126 - cp's OEIS frontend

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

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.

A020700 Numbers k such that k + sum of its prime factors = (k+1) + sum of its prime factors.

Original entry on oeis.org

7, 14, 63, 80, 224, 285, 351, 363, 475, 860, 902, 1088, 1479, 2013, 2023, 3478, 3689, 3925, 5984, 6715, 8493, 9456, 13224, 15520, 17227, 18569, 19502, 20490, 21804, 24435, 24476, 27335, 31899, 32390, 35815, 37406, 37582, 41876, 49468, 50609, 54137, 57239

Offset: 1

Enoch Haga

If p, (3/2)*(p+1), (3/2)*(p^2+p)+1 and (3/2)*(p^2+1)+2*p are all prime, then (3/2)*p*(3*p^2+4*p+3) is a term. The Generalized Bunyakovsky Conjecture implies that there are infinitely many of these. - Robert Israel, Apr 15 2022

			A075254(7) = 7+7 = 14 and A075254(8) = 8+2+2+2 = 14, so 7 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..2500

Cf. A020905, A228126.

SequencePosition[Table[n+Total[Times@@@FactorInteger[n]],{n,58000}],{x_,x_}][[;;,1]] (* Harvey P. Dale, Feb 26 2023 *)

A075254(n) = my(f = factor(n)); n + sum(i=1, #f~, f[i,1]*f[i,2]);
isok(n) = A075254(n) == A075254(n+1); \\ Michel Marcus, Jun 05 2014

from sympy import factorint
from itertools import count, islice
def sopf(n): return sum(p*e for p, e in factorint(n).items())
def agen(): # generator of terms
    sopfkplus1 = 2
    for k in count(2):
        sopfk, sopfkplus1 = sopfkplus1, sopf(k+1)
        if k + sopfk == k + 1 + sopfkplus1: yield k
print(list(islice(agen(), 42))) # Michael S. Branicky, Apr 15 2022

More terms from Michel Marcus, Jun 05 2014

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).

Original entry on oeis.org

2, 9, 98, 170, 1274, 4233, 4345, 7105, 7625, 14905, 21385, 30457, 34945, 66585, 69874, 77314, 82946, 98841, 175354, 177122, 233090, 236282, 238017, 263145, 265225, 295274, 298082, 322234, 335793, 336106

Offset: 1

Abhiram R Devesh, Feb 16 2014

nonn

The first term a(1)=2 is the only prime number in this sequence.

			For n=98: prime factors = 2,7,7; sum of prime factors = 16; number of prime divisors = 3
For n+1=99: prime factors = 3,3,11; sum of prime factors = 17; number of prime divisors=3.

Abhiram R Devesh, Table of n, a(n) for n = 1..96049

Cf. A001414, A006145 Ruth-Aaron numbers (1): sum of prime divisors of n = sum of prime divisors of n+1.
Cf. A228126 Sum of prime divisors of n (with repetition) is one less than the sum of prime divisors (with repetition) of n+1.
Cf. A045920 Numbers n such that factorizations of n and n+1 have same number of primes (including multiplicities).

Select[Partition[Table[{n,PrimeOmega[n],Total[Times@@@FactorInteger[n]]},{n,34*10^4}],2,1],#[[1,2]]==#[[2,2]]&&#[[1,3]]+1==#[[2,3]]&][[;;,1,1]] (* Harvey P. Dale, May 03 2024 *)

from sympy import primeomega
def is_A237929(n): return A001414(n) == A001414(n+1)-1 and primeomega(n) == primeomega(n+1) # David Radcliffe, Aug 08 2025

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

A352581 Numbers k such that A001414(k+1) = A001414(k)+1 and A001414(k)^2+3*A001414(k)+1 is prime.

Original entry on oeis.org

2, 3, 4, 20, 24, 1104, 1274, 2079, 4345, 13775, 14905, 20220, 23408, 25592, 35167, 49230, 61456, 66585, 68479, 75648, 76640, 121539, 172255, 194403, 200384, 229581, 233090, 236282, 238017, 247475, 263145, 283590, 287615, 295274, 295640, 326451, 386169, 422065, 429385, 429802, 475968, 585310

Offset: 1

J. M. Bergot and Robert Israel, Mar 21 2022

nonn

Numbers k such that A001414(k+1) = A001414(k)+1 and (A001414(k)+1)*(A001414(k+1)+1)-1 is prime.

			a(4) = 20 is a term because A001414(20) = 9, A001414(21) = 10 = 9+1, and 10*11-1 = 109 is prime.

Robert Israel, Table of n, a(n) for n = 1..1000

Intersection of A228126 and A352580. Cf. A001414.

spf:= proc(n) local t; option remember; add(t[1]*t[2], t=ifactors(n)[2]) end proc:
select(t -> (spf(t+1) = spf(t)+1) and isprime(spf(t)^2 + 3*spf(t)+1), [$1..10^6]);

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).

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

A020700 Numbers k such that k + sum of its prime factors = (k+1) + sum of its prime factors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A352581 Numbers k such that A001414(k+1) = A001414(k)+1 and A001414(k)^2+3*A001414(k)+1 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

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

Views

Author

Keywords

Links

Crossrefs