Lekraj Beedassy - cp's OEIS frontend

A309440 The number of digits of the greatest product from addends that sum up to 10^n.

1, 2, 16, 160, 1591, 15905, 159041, 1590405, 15904042, 159040419, 1590404183, 15904041824, 159040418240, 1590404182399, 15904041823989, 159040418239888, 1590404182398875, 15904041823988748, 159040418239887480, 1590404182398874791, 15904041823988747910, 159040418239887479099

Offset: 0

Lekraj Beedassy, Aug 03 2019

			The greatest product of numbers that sum up to 10 is 2*2*3*3 = 36 which has 2 digits. Thus a(1) = 2.
The greatest product of numbers that sum up to 100 is 2*2*3^(32) ~ 7.4*10^15 which has 16 digits. Hence a(2) = 16.
The greatest product of numbers that sum up to 1000 is 2*2*3^(332) ~ 1.0*10^159 which has 160 digits. Therefore a(3) = 160.

Cf. A000792.

a(n) = 1 + floor(log(4)/log(10) + ((10^n-1)/3-1)*log(3)/log(10)); \\ Jinyuan Wang, Aug 03 2019

a(n) = 1 + floor(log_10(36) + 10*log_10(27)*R_(n-1)), R_k being the k-th repunit, i.e., 111...111 with only digit 1 appearing k times.

A307748 Weak Skolem-Langford numbers that are prime.

Original entry on oeis.org

2412134003, 2742300437, 141753400357, 171450034753, 191453004359, 246200431613, 267240036473, 293251310059, 386732002687, 394530041519, 426724001617, 451714350037, 460054236253, 463724326007, 491514300539, 497514100579, 497524121579, 592632513169, 642732463007, 683002362181, 685200265181

Offset: 1

Lekraj Beedassy, Apr 26 2019

Prime terms of A108116.

Largest element is a(411) = 973006384792642181. - Jinyuan Wang, Jul 23 2019

Cf. A108116, A132291.

Select[Last /@ Cases[ Import[ "https://oeis.org/A108116/b108116.txt", "Table"], {A108116%20b-file,%20_Giovanni%20Resta">Integer, _Integer}], PrimeQ] (* based on A108116 b-file, _Giovanni Resta, Jul 03 2019 *)

A306890 a(n) is the number of prime digits used in writing out all primes up to and including the n-th prime.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 6, 6, 8, 9, 10, 12, 12, 13, 14, 16, 17, 17, 18, 19, 21, 22, 23, 23, 24, 24, 25, 26, 26, 27, 29, 30, 32, 33, 33, 34, 36, 37, 38, 40, 41, 41, 41, 42, 43, 43, 44, 47, 50, 52, 55, 57, 58, 60, 63, 65, 66, 68, 71, 72, 74, 76, 78, 79

Offset: 1

Lekraj Beedassy, Mar 15 2019

			We have a(10) = 9 since all primes up to the 10th (2, 3, 5, 7, 11, 13, 17, 19, 23, 29) use the 9 prime digits 2, 3, 5, 7, 3, 7, 2, 3, 2.

Partial sums of A109066. Cf. A068670.

With[{s = IntegerDigits[Prime@ Range@ 64]}, Array[Count[Flatten[s[[1 ;; #]] ], ?PrimeQ] &, Length@ s]] (* _Michael De Vlieger, Mar 27 2019 *)

a(n) = sum(k=1, prime(n), if (isprime(k), #select(x->isprime(x), digits(k)))); \\ Michel Marcus, Mar 23 2019

A263242 Larger of emirp pairs that are merely reversals of their end digits.

Original entry on oeis.org

31, 71, 73, 97, 311, 701, 733, 743, 751, 761, 907, 937, 941, 953, 967, 971, 983, 991, 3221, 9001, 9221, 9227, 9551, 9661, 9883, 32321, 33931, 34141, 34841, 35051, 36061, 36761, 37571, 39791, 70001, 71711, 72221, 73331, 74143, 74441, 74843, 74941, 75253, 76261, 76463, 76561

Offset: 1

Lekraj Beedassy, Oct 13 2015

The first digit is always larger than the last digit.

Harvey P. Dale, Table of n, a(n) for n = 1..500

Cf. A006567, A109309, A263240, A263241.

epQ[n_]:=Module[{idn=IntegerDigits[n],mid},mid=Rest[Most[idn]];PrimeQ[ IntegerReverse[n]]&&mid==Reverse[mid]&&idn[[1]]>idn[[-1]]]; Select[ Prime[Range[6,8000]],epQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 10 2016 *)

Corrected by Harvey P. Dale, Oct 10 2016

A263241 Lesser of emirp pairs that are merely reversals of their end digits.

Original entry on oeis.org

13, 17, 37, 79, 107, 113, 149, 157, 167, 179, 199, 337, 347, 359, 389, 709, 739, 769, 1009, 1223, 1229, 1559, 1669, 3889, 7229, 10007, 10009, 10909, 11717, 11719, 12227, 12323, 12829, 13337, 13933, 14143, 14447, 14449, 14549, 14843, 14947, 15053, 16063, 16267, 16567, 16763

Offset: 1

Lekraj Beedassy, Oct 13 2015

The first digit is always smaller than last digit.

Cf. A006567, A109308, A263240, A263242.

Select[Range[20000], And @@ PrimeQ[{#, IntegerReverse[#]}] && ! PalindromeQ[#] && First[(d = IntegerDigits[#])] < Last[d] && PalindromeQ[Most@ Rest@ IntegerDigits[#]] &] (* Amiram Eldar, Oct 16 2021 *)

isok(p) = my(dp=digits(p), dr=Vecrev(dp), r=fromdigits(dr)); if (isprime(r) && (r>p) && isprime(p), sum(i=2, #dp-1, dp[i]==dr[i]) == #dp-2); \\ Michel Marcus, Oct 16 2021

Missing 179 added by Zak Seidov, Oct 15 2021

A263240 Emirps whose reversals are merely their end digits swapped.

Original entry on oeis.org

13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389, 701, 709, 733, 739, 743, 751, 761, 769, 907, 937, 941, 953, 967, 971, 983, 991, 1009, 1223, 1229, 1559, 1669, 3221, 3889, 7229, 9001, 9221, 9227, 9551, 9661, 10007, 10009, 10909, 11717, 11719, 12227, 12323, 12829, 13337, 13933, 14143, 14447, 14449, 14549, 14843, 14947

Offset: 1

Lekraj Beedassy, Oct 13 2015

Subsequence of emirps A006567 with which it is identical for the first 37 terms.

Entries with 4 or more digits consist of a middle repdigit or palindrome part inserted between the end digits.

Cf. A006567, A263241, A263242.

A262137 a(n) = round(1500/(n + 5)).

Original entry on oeis.org

300, 250, 214, 188, 167, 150, 136, 125, 115, 107, 100, 94, 88, 83, 79, 75, 71, 68, 65, 62, 60, 58, 56, 54, 52, 50, 48, 47, 45, 44, 43, 42, 41, 39, 38, 38, 37, 36, 35, 34, 33, 33, 32, 31, 31, 30, 29, 29, 28, 28, 27, 27, 26, 26, 25, 25, 25, 24, 24, 23

Offset: 0

Lekraj Beedassy, Sep 11 2015

"Fine-tuned" square counting method to directly read the heartbeat rate a(n) on EKG printout whereby in presence of similar occurring QRS wave complexes one such successive pair is considered (with the first QRS complex specifically spotted whose R wave /|\ falls sharply right on a thick marking) so that the ensuing R complex is read n smaller markings (or mm) farther off the first thick marking following the first spotted R thick-marking wave.

Sequence of first twenty entries or so is usually relevant and worth memorizing in clinical practice.

Dale Dubin, Rapid Interpretation Of EKG's, pp. 89, 335. COVER Pub. FL 2000.

J. S. S. G. de Jong, P. G. Postema, ECGpedia, Rate: Fine-tuned square-counting method
D. Dubin, Chapter 4: Rate, page 89, 335

Table[Round [1500/(n + 5)], {n, 0, 200}] (* Vincenzo Librandi, Sep 12 2015 *)

first(m)=vector(m,i,i--;round(1500/(i + 5))) \\ Anders Hellström, Sep 12 2015

A209620 Emirps that become their own reversals when rotated through 180 degrees (including on calculator display).

Original entry on oeis.org

1021, 1151, 1181, 1201, 1511, 1811, 10151, 11551, 15101, 15511, 100511, 101281, 102181, 102551, 105211, 105251, 108881, 110051, 110221, 110281, 110881, 111211, 111821, 112111, 112181, 112501, 115001, 115021, 118081, 120121, 120511, 121021, 121151, 122011

Offset: 1

Lekraj Beedassy, Mar 21 2012

Such emirps have end digits 1 and use only digits 0, 1, 2, 5, 8, and the sequence naturally includes A155512, the emirps with only digits 0 and 1.

			1181 of this sequence, for instance, belongs to the emirp pair (1181, 1811), where each member is a 180-degree rotation of the other; similarly for the term 112501 of this sequence, that belongs to the emirp pair (105211, 112501) and which, displayed on a calculator and turned upside-down, becomes its own reversal.

Cf. A006567.

t1 = {0, 1, 2, 5, 8}; okQ[n_] := Module[{d = IntegerDigits[n], r}, r = Reverse[d];  r != d && Complement[d, t1] == {} && PrimeQ[FromDigits[r]]]; Select[Prime[Range[100000]], okQ] (* T. D. Noe, Apr 24 2012 *)

A210548 Larger of emirp pairs whose members have prime digital products.

Original entry on oeis.org

31, 71, 311, 1511, 112111, 1171111, 11131111, 71111111, 131111111, 1115111111, 31111111111, 131111111111111111, 1151111111111111111111111111, 11111111111711111111111111111, 131111111111111111111111111111111111111

Offset: 1

Lekraj Beedassy, Mar 22 2012

Beyond the first two terms, a(n) is the intersection of A173596 and A046703.

Cf. A006567, A210546, A210547.

More terms from Alois P. Heinz, Mar 22 2012

A210547 Lesser of emirp pairs whose members have prime digital products.

Original entry on oeis.org

13, 17, 113, 1151, 111211, 1111711, 11111117, 11113111, 111111131, 1111115111, 11111111113, 111111111111111131, 1111111111111111111111111511, 11111111111111111711111111111, 111111111111111111111111111111111111131

Offset: 1

Lekraj Beedassy, Mar 22 2012

Beyond the first two terms, a(n) is the intersection of A173595 and A046703.

Cf. A006567, A210546, A210548.

More terms from Alois P. Heinz, Mar 22 2012

cp's OEIS Frontend

User: Lekraj Beedassy

A309440 The number of digits of the greatest product from addends that sum up to 10^n.

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A307748 Weak Skolem-Langford numbers that are prime.

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A306890 a(n) is the number of prime digits used in writing out all primes up to and including the n-th prime.

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A263242 Larger of emirp pairs that are merely reversals of their end digits.

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A263241 Lesser of emirp pairs that are merely reversals of their end digits.

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A263240 Emirps whose reversals are merely their end digits swapped.

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A262137 a(n) = round(1500/(n + 5)).

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A209620 Emirps that become their own reversals when rotated through 180 degrees (including on calculator display).

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A210548 Larger of emirp pairs whose members have prime digital products.

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