Robert Bilinski - cp's OEIS frontend

A339470 Decimal expansion of log(phi)^2, where phi is the golden ratio (A002390^2).

2, 3, 1, 5, 6, 4, 8, 2, 0, 5, 7, 7, 1, 9, 4, 3, 9, 2, 4, 9, 6, 9, 2, 9, 0, 7, 1, 2, 3, 1, 5, 3, 2, 7, 6, 0, 0, 1, 6, 4, 0, 6, 3, 5, 0, 0, 4, 9, 2, 9, 8, 8, 7, 0, 8, 1, 5, 3, 0, 1, 2, 2, 8, 6, 8, 9, 7, 9, 5, 3, 4, 5, 5, 6, 6, 9, 6, 1, 8, 1, 2, 9, 8, 5, 0, 5, 4

Offset: 0

Robert Bilinski, Dec 06 2020

			0.2315648205771943924969290712315327600164063500492988708153012286...

Travis Sherman, Summation of Glaisher- and Apéry-like Series, University of Arizona, 2000.
Index entries for transcendental numbers

Cf. A002390, A001622, A086467.

RealDigits[Log[GoldenRatio]^2, 10, 100][[1]] (* Amiram Eldar, Dec 06 2020 *)

asinh(1/2)^2 \\ Michel Marcus, Dec 06 2020

Equals arcsinh(1/2)^2 = A002390^2.
Equals (1/2)*Sum_{k>=1} ((k!)^2*(-1)^(k+1))/((2*k)!*k^2) = A086467/2.
Equals (1/3)*(zeta(2) - Sum_{k>=1} ((k!)^2*(-1)^k)/((2*k)!*(2*k+1)^2)).
Equals (1/2)*Sum_{k>=1} (-1)^(k+1)/A002736(k).

A337789 Numbers k such that trajectory of k under repeated calculation of fecundity (x -> A070562(x)) eventually reaches 0.

Original entry on oeis.org

0, 1, 5, 10, 15, 18, 20, 21, 22, 24, 27, 30, 35, 40, 42, 44, 46, 48, 50, 51, 55, 59, 60, 63, 64, 66, 67, 69, 70, 74, 75, 77, 80, 83, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 115, 118, 120, 121, 122, 124, 127

Offset: 1

Robert Bilinski, Sep 21 2020

			5 is a term in the sequence because the fecundity of 5 is 1, the fecundity of 1 is 10 and the fecundity of 10 is 0.
7 is not a term in the sequence because the fecundity of 7 is 7 and therefore the fecundity will never become 0.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A070562, A070061, A070257.

fec:= proc(n) local k, x,t;
  x:= n;
  for k from 0 do
    t:= convert(convert(x,base,10),`*`);
    if t = 0 then return k fi;
    x:= x+t
  od
end proc:
filter:= proc(n) local v; option remember;
    v:= fec(n);
    if v = 0 then true
    elif v = n then false
    else procname(v)
    fi
end proc:
select(filter, [$0..1000]); # Robert Israel, Apr 12 2021

fec[n_] := Length @ FixedPointList[# + Times @@ IntegerDigits[#] &, n] - 2; Select[Range[0, 100], FixedPoint[fec, #] == 0 &] (* Amiram Eldar, Sep 22 2020 *)

from math import prod
from functools import lru_cache
def pd(n): return prod(map(int, str(n)))
def A070562(n):
  s = 0
  while pd(n) != 0: n, s = n + pd(n), s + 1
  return s
@lru_cache(maxsize=None)
def ok(n):
  fn = A070562(n)
  if fn == 0: return True
  if fn == n: return False
  return ok(fn)
print(list(filter(ok, range(128)))) # Michael S. Branicky, Apr 12 2021

More terms from Amiram Eldar, Sep 22 2020

Offset changed by Robert Israel, Apr 12 2021

A336605 a(n) is the number of partitions of the n-th tetrahedral number (A000292).

Original entry on oeis.org

1, 1, 5, 42, 627, 14883, 526823, 26543660, 1844349560, 172389800255, 21248279009367, 3397584011986773, 695143713458946040, 179855916453958267598, 58248417552751868050007, 23402165235974892374954302, 11571309261543787320061392679

Offset: 0

Robert Bilinski, Sep 13 2020

nonn

Cf. A000041, A000292, A066655.

a(n) = numbpart(n*(n+1)*(n+2)/6); \\ Michel Marcus, Sep 14 2020

a(n) = p(n*(n+1)*(n+2)/6).

a(n) = A000041(A000292(n)).

A334631 a(n) = number of letters in the sum of all previous terms, with a(1) = 1 (in US English, excluding spaces and hyphens).

Original entry on oeis.org

1, 3, 4, 5, 8, 9, 6, 9, 9, 9, 10, 12, 10, 10, 14, 18, 21, 20, 22, 10, 13, 21, 19, 20, 21, 16, 18, 23, 20, 21, 14, 18, 21, 20, 22, 22, 19, 22, 16, 21, 22, 18, 21, 20, 22, 12, 18, 18, 22, 19, 22, 18, 22, 20, 22, 23, 18, 21, 20, 22, 22

Offset: 1

Robert Bilinski, Sep 14 2020

			a(4) = 5, as a(1) + a(2) + a(3) = 8 and 8 has 5 letters in English.

Cf. A000655, A005589.

A328738 Numbers k such that at least one of sopfr(k-1) and sopfr(k+1) is greater than or equal to 4*sopfr(k).

Original entry on oeis.org

48, 54, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 98, 100, 102, 104, 108, 110, 112, 114, 126, 128, 130, 132, 135, 136, 138, 140, 147, 150, 152, 156, 162, 165, 168, 174, 180, 182, 190, 192, 195, 196, 198, 200, 210, 222, 224, 225, 228, 230, 232, 234

Offset: 1

Robert Bilinski, Oct 26 2019

nonn

k-1 or k+1 or both have sums of prime factors at least 4 times the sum of prime factors of k.

Cf. A001414, A328736, A328737.

A328737 Numbers k such that at least one of sopfr(k-1) and sopfr(k+1) is greater than or equal to 3*sopfr(k).

Original entry on oeis.org

30, 32, 36, 40, 42, 48, 52, 54, 60, 66, 68, 70, 72, 75, 78, 80, 81, 84, 88, 90, 96, 98, 100, 102, 104, 105, 108, 110, 112, 114, 117, 126, 128, 130, 132, 135, 136, 138, 140, 143, 147, 148, 150, 152, 156, 160, 162, 164, 165, 168, 172, 174, 176, 180

Offset: 1

Robert Bilinski, Oct 26 2019

nonn

k-1 or k+1 or both have sums of prime factors at least 3 times the sum of prime factors of k.

Cf. A001414, A328736, A328738.

A328736 Numbers k such that at least one of sopfr(k-1) and sopfr(k+1) is greater than or equal to 2*sopfr(k).

Original entry on oeis.org

16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 45, 48, 52, 54, 60, 63, 66, 68, 70, 72, 75, 78, 80, 81, 84, 85, 88, 90, 95, 96, 98, 100, 102, 104, 105, 108, 110, 112, 114, 117, 119, 121, 125, 126, 128, 130, 132, 133, 135, 136, 138, 140, 143, 144, 145

Offset: 1

Robert Bilinski, Oct 26 2019

nonn

k-1 or k+1 or both have sums of prime factors at least twice the sum of prime factors of k.

Cf. A001414, A328737, A328738.

sop[n_] := If[n<2, 0, Total[Times @@@ FactorInteger@ n]]; Select[ Range[2, 150], (s = 2 sop[#]; sop[#+1] >= s || sop[#-1] >= s) &] (* Giovanni Resta, Oct 27 2019 *)

A328263 a(n) = number of letters in a(n-1) (in Polish), with a(1) = 1.

Original entry on oeis.org

1, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6, 5, 4, 6

Offset: 1

Robert Bilinski, Oct 09 2019

a(1) = 1; for n>1, a(n) = numbers of letters in Polish name for a(n-1).

Decimal expansion of 515/333. - Elmo R. Oliveira, May 05 2024

			Jeden, pięć, cztery, sześć, pięć, ...

Robert Bilinski, Table of n, a(n) for n = 1..99
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A008962 (number of letters).

Cf. A000655 (English), A061504 (French), A101432 (Spanish).

a(n) = a(n-3) for n > 4. - Elmo R. Oliveira, May 05 2024

A327749 Natural numbers whose sum of prime factors (with repetition) is palindromic in base 10.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 24, 27, 28, 40, 45, 48, 54, 57, 62, 85, 101, 102, 106, 116, 121, 123, 131, 151, 181, 182, 191, 194, 218, 259, 260, 278, 292, 298, 305, 308, 312, 313, 351, 353, 358, 366, 370, 373, 383, 388, 403, 413, 415, 428, 440, 444, 483, 495, 498

Offset: 1

Robert Bilinski, Sep 23 2019

Union of 1, A046352 and the palindromic primes (A002385). - Corrected by Robert Israel, Nov 20 2020

Karl G. Kröber, "Palindrome, Perioden und Chaoten: 66 Streifzüge durch die palindromischen Gefilde" (1997, Deutsch-Taschenbücher; Bd. 99) ISBN 3-8171-1522-9.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 71.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001414, A002113, A002385, A046352, A046355.

[1] cat [k: k in [2..500]| Intseq(a) eq Reverse(Intseq(a)) where a is &+[m[1]*m[2]: m in Factorization(k)]]; // Marius A. Burtea, Sep 27 2019

ispali:= proc(n) option remember; local L; L:= convert(n,base,10); evalb(L = ListTools:-Reverse(L)) end proc:
spf:= proc(n) add(t[1]*t[2],t=ifactors(n)[2]) end proc:
select(t -> ispali(spf(t)), [$0..1000]); # Robert Israel, Nov 20 2020

sopfr[1] = 0; sopfr[n_] := Plus @@ (Times @@@ FactorInteger[n]); aQ[n_] := PalindromeQ[sopfr[n]]; Select[Range[500], aQ] (* Amiram Eldar, Sep 23 2019 *)

sopfr(n) = (n=factor(n))[, 1]~*n[, 2]; \\ A001414
isok(n) = my(d=digits(sopfr(n))); d == Vecrev(d); \\ Michel Marcus, Sep 27 2019

A327687 Partial sums of Pisano periods (A001175).

Original entry on oeis.org

1, 4, 12, 18, 38, 62, 78, 90, 114, 174, 184, 208, 236, 284, 324, 348, 384, 408, 426, 486, 502, 532, 580, 604, 704, 788, 860, 908, 922, 1042, 1072, 1120, 1160, 1196, 1276, 1300, 1376, 1394, 1450, 1510, 1550, 1598, 1686, 1716, 1836, 1884, 1916, 1940, 2052, 2352, 2424, 2508, 2616, 2688, 2708

Offset: 1

Robert Bilinski, Sep 22 2019

nonn

Robert Bilinski, Table of n, a(n) for n = 1..10000
Joseph Louis de Lagrange, Additions aux éléments d'algèbre d'Euler. Analyse indéterminée, (1774), pp. 143ff.
J. D. Fulton and W. L. Morris, On arithmetical functions related to the Fibonacci numbers, Acta Arithmetica 16 (1969), 105-110.
James Grime and Brady Haran, Fibonacci Mystery, Numberphile video, 2013.

Cf. A001175.

Module[{nn=1000,fibs},fibs=Fibonacci[Range[nn]];Accumulate[Table[Length[ FindTransientRepeat[ Mod[fibs,n],2][[2]]],{n,70}]]] (* Harvey P. Dale, Jul 08 2023 *)

cp's OEIS Frontend

User: Robert Bilinski

A339470 Decimal expansion of log(phi)^2, where phi is the golden ratio (A002390^2).

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A337789 Numbers k such that trajectory of k under repeated calculation of fecundity (x -> A070562(x)) eventually reaches 0.

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A336605 a(n) is the number of partitions of the n-th tetrahedral number (A000292).

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A334631 a(n) = number of letters in the sum of all previous terms, with a(1) = 1 (in US English, excluding spaces and hyphens).

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A328738 Numbers k such that at least one of sopfr(k-1) and sopfr(k+1) is greater than or equal to 4*sopfr(k).

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A328737 Numbers k such that at least one of sopfr(k-1) and sopfr(k+1) is greater than or equal to 3*sopfr(k).

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A328736 Numbers k such that at least one of sopfr(k-1) and sopfr(k+1) is greater than or equal to 2*sopfr(k).

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A328263 a(n) = number of letters in a(n-1) (in Polish), with a(1) = 1.

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A327749 Natural numbers whose sum of prime factors (with repetition) is palindromic in base 10.

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