A073935 -id:A073935 - cp's OEIS frontend

A286499 Primes which divide a term of A073935.

2, 3, 5, 7, 17, 19, 43, 101, 163, 257, 487, 1459, 14407, 26407, 39367, 62501, 65537, 77659, 1020101, 1336337, 86093443, 242121643, 258280327, 3103616899, 4528177054183, 15258789062501, 411782264189299, 21108889701347407, 953735353027359375062501

Offset: 1

John Machacek, May 27 2017

nonn

A prime p is in this sequence if and only if p-1 = Product_{i} (p_i)^(a_i) with p_j - 1 = Product_{j

This sequence contains all Fermat primes (A019434).

			p = 43 is in the sequence because 43-1 = 42 = 2*3*7, 7-1 = 6 = 2*3, 3-1 = 2.

John Machacek, Egyptian Fractions and Prime Power Divisors, arXiv:1706.01008 [math.NT], 2017.

Cf. A073935.

upTo[mx_] := Block[{ric}, ric[n_, p_] := If[n < mx, Block[{m = n p}, If[PrimeQ[n + 1], Sow[n+1]; ric[n (n + 1), n+1]]; If[IntegerExponent[n, p] == 1, While[m < mx, ric[m, p]; m *= p]]]]; Sort[Reap[ric[1, 2]][[2, 1]]]]; upTo[10^20] (* Giovanni Resta, May 27 2017 *)

a(20)-a(29) from Giovanni Resta, May 27 2017

A073934 Sum of terms in n-th row of triangle in A073932.

Original entry on oeis.org

1, 3, 6, 7, 12, 12, 19, 15, 21, 22, 33, 24, 37, 33, 37, 31, 48, 39, 58, 42, 54, 55, 78, 48, 67, 63, 66, 61, 90, 67, 98, 63, 88, 82, 96, 75, 112, 96, 102, 82, 123, 96, 139, 99, 112, 124, 171, 96, 145, 117, 133, 115, 168, 120, 154, 117, 153, 148, 207, 127, 188, 160

Offset: 1

Amarnath Murthy, Aug 19 2002

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A032742, A060681, A073932, A073933, A073935.

a[1] := 1:for i from 2 to 500 do n := i:s := n:while(n>1) do if isprime(n) then r := n-1: else r := n-n/ifactors(n)[2][1][1]; fi; n := r:s := s+n:od:a[i] := s:od:seq(a[k],k=1..500);

Array[If[# == 1, 1, Total@ NestWhileList[If[PrimeQ@ #, # - 1, # - #/FactorInteger[#][[1, 1]] ] &, #, # > 1 &]] &, 62]

(define (A073934 n) (if (= 1 n) n (+ n (A073934 (A060681 n)))))
(define (A060681 n) (- n (A032742 n))) ;; See also code under A032742
;; Antti Karttunen, Aug 23 2017

a(1) = 1; for n > 1, a(n) = n + a(A060681(n)). - Antti Karttunen, Aug 23 2017

More terms from Sascha Kurz, Aug 23 2002

Offset corrected from 0 to 1 by Antti Karttunen, Aug 23 2017

A073933 Number of terms in n-th row of triangle in A073932.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 6, 5, 6, 6, 7, 6, 7, 7, 8, 6, 7, 7, 7, 7, 8, 7, 8, 6, 8, 7, 8, 7, 8, 8, 8, 7, 8, 8, 9, 8, 8, 9, 10, 7, 9, 8, 8, 8, 9, 8, 9, 8, 9, 9, 10, 8, 9, 9, 9, 7, 9, 9, 10, 8, 10, 9, 10, 8, 9, 9, 9, 9, 10, 9, 10, 8, 9, 9, 10, 9, 9, 10, 10, 9, 10, 9, 10, 10, 10, 11, 10, 8

Offset: 1

Amarnath Murthy, Aug 19 2002

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A032742, A060681, A073932, A073934, A073935.

One more than A064097.

a[1] := 1:for i from 2 to 500 do n := i:s := 1:while(n>1) do if isprime(n) then r := n-1: else r := n-n/ifactors(n)[2][1][1]; fi; n := r:s := s+1:od:a[i] := s:od:seq(a[k],k=1..500);

(define (A073933 n) (if (= 1 n) n (+ 1 (A073933 (A060681 n)))))
(define (A060681 n) (- n (A032742 n))) ;; See also code under A032742
;; Antti Karttunen, Aug 23 2017

From Antti Karttunen, Aug 23 2017: (Start)
a(1) = 1; for n > 1, a(n) = 1 + a(A060681(n)).
a(n) = 1 + A064097(n).
(End)

More terms from Sascha Kurz, Aug 23 2002

Offset corrected from 0 to 1 by Antti Karttunen, Aug 23 2017

A073932 Define f(n) = n - largest nontrivial divisor of n or f(n) = n-1 if n is a prime [that is, f(n) = A060681(n)]. Form a triangle in which the n-th row contains terms n, f(n), f(f(n)), ... until a 1 is reached; sequence gives triangle read by rows.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 2, 1, 5, 4, 2, 1, 6, 3, 2, 1, 7, 6, 3, 2, 1, 8, 4, 2, 1, 9, 6, 3, 2, 1, 10, 5, 4, 2, 1, 11, 10, 5, 4, 2, 1, 12, 6, 3, 2, 1, 13, 12, 6, 3, 2, 1, 14, 7, 6, 3, 2, 1, 15, 10, 5, 4, 2, 1, 16, 8, 4, 2, 1, 17, 16, 8, 4, 2, 1, 18, 9, 6, 3, 2, 1, 19, 18, 9, 6, 3, 2, 1, 20, 10, 5, 4, 2, 1

Offset: 1

Amarnath Murthy, Aug 19 2002

			Triangle begins:
   1;
   2, 1;
   3, 2, 1;
   4, 2, 1;
   5, 4, 2, 1;
   6, 3, 2, 1;
   7, 6, 3, 2, 1;
   8, 4, 2, 1;
   9, 6, 3, 2, 1;
  10, 5, 4, 2, 1;

Michael De Vlieger, Table of n, a(n) for n = 1..12386 (rows 1 <= n <= 1000, flattened)
John Machacek, Egyptian Fractions and Prime Power Divisors, Journal of Integer Sequences, Vol. 21 (2018), Article 18.3.7.

Cf. A032742, A060681, A073933, A073934, A073935.

j := 1:a[1] := 1:for i from 2 to 50 do n := i:j := j+1:a[j] := n:while(n>1) do if isprime(n) then r := n-1: else r := n-n/ifactors(n)[2][1][1]; fi; n := r:j := j+1:a[j] := n: od:od:seq(a[k],k=1..j);

Array[If[# == 1, {1}, NestWhileList[If[PrimeQ@ #, # - 1, # - #/FactorInteger[#][[1, 1]] ] &, #, # > 1 &]] &, 20] // Flatten  (* Michael De Vlieger, Apr 15 2020 *)

More terms from Sascha Kurz, Aug 23 2002

Offset corrected from 0 to 1 by Antti Karttunen, Aug 23 2017

A283423 Prime power pseudoperfect numbers: numbers m > 1 such that 1/m + Sum 1/p^k = 1, where the sum is over the prime powers p^k | m.

Original entry on oeis.org

2, 4, 6, 8, 16, 18, 20, 32, 42, 54, 64, 100, 128, 162, 256, 272, 294, 342, 486, 500, 512, 1024, 1458, 1806, 2048, 2058, 2500, 4096, 4374, 4624, 6498, 8192, 10100, 12500, 13122, 14406, 16384, 23994, 26406, 32768, 34362, 39366, 47058

Offset: 1

John Machacek, May 27 2017

nonn

Since primary pseudoperfect numbers (A054377) must be squarefree, it follows that primary pseudoperfect numbers are contained in this sequence.
This sequence contains all powers of 2. With the exception of the powers of 2, every prime power pseudoperfect number is a pseudoperfect number (A005835).
Every number in A073935 is a prime power pseudoperfect number (note: this sequence and A073935 agree for many terms but eventually differ starting at 23994 the 38th term of this sequence).
The number 2^k(2^k+1) is the sequence whenever 2^k+1 is a Fermat prime (A019434).

			m = 18 is in the sequence because 1/18 + 1/2 + 1/3 + 1/9 = 1.
m = 12 is NOT in the sequence because 1/12 + 1/2 + 1/4 + 1/3 != 1.

Giovanni Resta, Table of n, a(n) for n = 1..131 (terms < 10^11, first 101 terms from Amiram Eldar)
John Machacek, Egyptian Fractions and Prime Power Divisors, arXiv:1706.01008 [math.NT], 2017.

Cf. A005835, A054377, A073935.

ok[n_] := Total[n/Flatten@ Table[e[[1]] ^ Range[e[[2]]], {e, FactorInteger[n]}]] + 1 == n; Select[ Range[10^5], ok] (* Giovanni Resta, May 27 2017 *)

A074093 Number of values of k such that n = k - largest divisor of k (

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 1, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 3, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 4, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 4, 1, 1, 1

Offset: 1

Amarnath Murthy, Aug 19 2002

nonn

			a(6) = 3 and the three values of k are 7,9 and 12.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A073932, A073933, A073934, A073935, A060681.

f1[n_] := Count[Range[n, 2 n], ?(Differences[Take[Divisors[#], -2]] == {n} &)]; Join[{1}, Table[f1[n], {n, 2, 105}]] (* _Jayanta Basu, Jul 30 2013 *)

a(n)=sum(k=2,2*n,if(k- component(divisors(k),numdiv(k)-1)-n,0,1))

a(2n+1)=1; sum(k=1, n, a(k)) seems to be asymptotic to C*n with C=1.6... - Benoit Cloitre, Aug 21 2002

More terms from Benoit Cloitre and Vladeta Jovovic, Aug 21 2002

Showing 1-6 of 6 results.

cp's OEIS Frontend

A286499 Primes which divide a term of A073935.

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A073934 Sum of terms in n-th row of triangle in A073932.

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A073933 Number of terms in n-th row of triangle in A073932.

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A073932 Define f(n) = n - largest nontrivial divisor of n or f(n) = n-1 if n is a prime [that is, f(n) = A060681(n)]. Form a triangle in which the n-th row contains terms n, f(n), f(f(n)), ... until a 1 is reached; sequence gives triangle read by rows.

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A283423 Prime power pseudoperfect numbers: numbers m > 1 such that 1/m + Sum 1/p^k = 1, where the sum is over the prime powers p^k | m.

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A074093 Number of values of k such that n = k - largest divisor of k (

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