A362058 Contributions to OEIS

 

A362058 The location of the first occurrence of n in the decimal expansion of phi (the golden ratio, 1.6180339887...). 0 4, 0, 19, 5, 11, 22, 1, 10, 3, 7, 231, 34, 121, 55, 254, 366, 0, 35, 2, 188, 19, 54, 62, 131, 78, 213, 67, 63, 51, 174, 40, 137, 181, 5, 26, 56, 28, 98, 32, 6, 105, 90, 347, 27, 58, 21, 70, 102, 15, 11, 214, 394, 66, 111, 57, 768, 30, 48, 22, 166, 68, 1, 50 (list; graph; refs; listen; history; edit; text; internal format) OFFSET 0,1 COMMENTS Locations in the expansion of phi are numbered 0 for the digit before the decimal point, 1 for the first digit after the decimal point, and so on. LINKS Table of n, a(n) for n=0..62. FORMULA a(n) = A088577(n) - 1. EXAMPLE The first occurrence of 0 in phi occurs 4 places after the decimal point, so a(0)=4; 5 first occurs 22 places after the decimal point, so a(5)=22; 10 first occurs 231 places after the decimal point so a(10)=231. MATHEMATICA Table[-1 + SequencePosition[#, IntegerDigits@ n][[1, 1]], {n, 0, 50}] &@ First@ RealDigits@ N[GoldenRatio, 10^4] CROSSREFS Cf. A001622 (phi) Cf. A088577 (1-based locations). Cf. A078197 (for e), A176341 (for Pi), A014777 (for Pi but different indexing). Sequence in context: A058348 A184946 A351615 * A085618 A263626 A282579 Adjacent sequences: A362055 A362056 A362057 * A362059 A362060 A362061 KEYWORD nonn,base AUTHOR James C. McMahon, Apr 06 2023 STATUS approved

Contributions to The On-Line Encyclopedia of Integer Sequences A361746

 I'm making this post to have a place to go to quickly reference OEIS sequences that I have authored.

A361746 Number of occurrences of the most frequently occurring letter(s) in US English name of n. 0

1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 2, 2, 2, 4, 3, 3, 2, 2, 3, 3, 2, 2, 2, 3, 3, 3, 2, 2, 3, 3, 2, 2, 2, 2, 3, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 4, 2, 3, 2, 4, 3, 3, 1, 2, 2, 3, 1, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 4, 2 (list; graph; refs; listen; history; edit; text; internal format)

OFFSET 0,4 LINKS Table of n, a(n) for n=0..100. EXAMPLE For n = 3, a(3) = 2 because the letter e occurs twice in the word "three". For n = 4, a(4) = 1 because the letters f, o, u and r each occur once in the word "four". MATHEMATICA a[n_] := Max[Values[LetterCounts[IntegerName[n, "Words"]]]]; Array[a, 100, 0] (* Amiram Eldar, Mar 24 2023 *) PROG (Python) from num2words import num2words from collections import Counter def a(n): return max(Counter(num2words(n).replace(" and", "").replace(" ", "")).values()) print([a(n) for n in range(112)]) # Michael S. Branicky, Mar 24 2023 CROSSREFS Cf. A005589. Sequence in context: A156775 A064693 A325614 * A369917 A319608 A230850 Adjacent sequences: A361743 A361744 A361745 * A361747 A361748 A361749 KEYWORD easy,nonn,word,less AUTHOR James C. McMahon, Mar 22 2023 STATUS approved

Are Prime Numbers Predictable?

A recent paper by Han-Linn Li, Shu-Cherng Fang, and Way Kuo of the Department of Computer Science, City University of Hong Kong, suggests that there are cyclic effects among the composites, spawning primes distributed periodically. A link to their paper is given here:  The Periodic Table of Primes.

I'm still reading it now, so I will post more later. Meanwhile, I wanted to share it immediately.

Update: The idea proposed by the paper is that the authors identify 48 integers as "roots." These integers are between 11 and 211, and are either prime or not divisible by 2, 3, 5 or 7. The periodic cycle claimed is 210 which is the product of 2, 3, 5 and 7. 

I recently came across a criticism of the paper which I'm linking here: A New Study About Prime Numbers Is Stirring Up a Mess of Controversy.