A045572 -id:A045572 - cp's OEIS frontend

A045807 7-ish numbers (end in 53, 71, 79, 97).

53, 71, 79, 97, 153, 171, 179, 197, 253, 271, 279, 297, 353, 371, 379, 397, 453, 471, 479, 497, 553, 571, 579, 597, 653, 671, 679, 697, 753, 771, 779, 797, 853, 871, 879, 897, 953, 971, 979, 997, 1053, 1071, 1079, 1097, 1153, 1171, 1179, 1197, 1253, 1271

Offset: 1

J. H. Conway

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A045800-A045809.

Cf. A045572, A045797, A045798.

import Data.List (findIndices)
a045807 n = a045807_list !! (n-1)
a045807_list = findIndices (`elem` [53,71,79,97]) $ cycle [0..99]
-- Reinhard Zumkeller, Jan 23 2012

Empirical g.f.: x*(53+18*x+8*x^2+18*x^3+3*x^4)/(1-x-x^4+x^5). - Colin Barker, Jan 23 2012

More terms from Erich Friedman

A158942 Nonsquares coprime to 10.

Original entry on oeis.org

3, 7, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 83, 87, 89, 91, 93, 97, 99, 101, 103, 107, 109, 111, 113, 117, 119, 123, 127, 129, 131, 133, 137, 139, 141, 143, 147, 149, 151, 153, 157, 159, 161, 163

Offset: 1

Eric Desbiaux, Mar 31 2009

Odd primes + odd nonprime integers that have an odd numbers of proper divisors A082686, are the result of a suppression of integers satisfying: 2n (A005843); n^2 (A000290); n^2+n (A002378). Of these, we can suppress the multiples of 5 (A008587).
Decimal expansion of 1/10^(n^2+n) + 1/10^(n^2) + 1/10^(5*n) + 1/10^(2*n) gives a 0 for these integers.
2n + n(n+1) + n^2 = 2n^2 + 3n = A014106.
2n^2 + 3n + 5n = 2n^2 + 8n = 2n(n+4) = A067728 8(8+n) is a perfect square.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Intersection of A000037 and A045572.

Cf. A082868, A002378, A000290, A005843, A008587, A014106, A067728.

Select[Range@ 163, ! IntegerQ@ Sqrt@ # && CoprimeQ[#, 10] &] (* Michael De Vlieger, Dec 11 2015 *)

isok(n) = (n % 2) && (n % 5) && (isprime(n) || (numdiv(n) % 2 == 0)); \\ Michel Marcus, Aug 27 2013

is(n)=gcd(n,10)==1 && !issquare(n) \\ Charles R Greathouse IV, Sep 05 2013

New name from Charles R Greathouse IV, Sep 05 2013

A179192 Numbers n, not relatively prime to 10, such that the decimal form of the period of 1/n is prime.

Original entry on oeis.org

12, 18, 30, 36, 45, 48, 75, 120, 180, 192, 198, 270, 288, 300, 330, 360, 450, 480, 495, 750, 768, 1152, 1200, 1584, 1800, 1875, 1920, 1980, 1998, 2304, 2700, 2880, 3000, 3072, 3300, 3330, 3600, 3690, 4500, 4800, 4950, 4995, 5625, 7500, 7680, 9090, 11520, 12000, 12288, 15840, 18000, 18432, 18750, 19200, 19800, 19980, 19998

Offset: 1

Michel Lagneau, Jul 01 2010

The sequence A175545 (numbers n such that the decimal form of the period of 1/n is prime) is only concerned with numbers n such that gcd(n,10)=1. Each number n such that gcd(n,10)<>1 generates a quotient where there exist a sequence of digits which is periodic after a finite sequence of digits, for example 1/36 = .0277777.... and 7 is periodic.
The prime numbers corresponding to this sequence are :
3, 5, 3, 7, 2, 3, 3, 3, 5, 3, 5, 37, 2, 3, 3, 7, 2, 3, 2,...

			1584 is in the sequence because 1/1584 = .0006313131313131313131... and 31 is prime.

H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), ch. 19, 'Die periodischen Dezimalbrueche'.

Cf. A175557, A175555, A178505, A045572, A002329

Reap[Do[p=RealDigits[1/n][[1,-1]]; If[GCD[10,n]>1 && Head[p] === List, While[p[[-1]] == 0, p=Most[p]]; If[PrimeQ[FromDigits[p]], Sow[n]]], {n, 20000}]][[2,1]]

Union of A179192 and A175545 is A061564.

Sequence corrected by T. D. Noe, Nov 18 2010

A186694 Numbers ending in 1, 3, 7 or 9 such that changing any one decimal digit produces a composite number.

Original entry on oeis.org

212159, 294001, 505447, 584141, 595631, 604171, 872897, 971767, 1062599, 1203623, 1282529, 1293671, 1524181, 1566691, 1702357, 1830661, 2017963, 2474431, 2690201, 3085553, 3326489, 3716213, 3964169, 4103917, 4134953, 4173921, 4310617, 4376703

Offset: 1

Arkadiusz Wesolowski, Feb 25 2011

Union of A050249 and A143641.

This sequence is infinite because Terence Tao proved that sequence A050249 is infinite.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1500
Chris Caldwell, The Prime Glossary, Weakly prime
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 212159
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 17171...58369 (1000-digits)
Terence Tao, A remark on primality testing and decimal expansions, Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.

Cf. A050249, A045572, A143641.

primeProof[n_] := Module[{d, e, isPP, num}, d=IntegerDigits[n]; isPP=True; Do[e=d; e[[i]]=j; num=FromDigits[e]; If[num != n && PrimeQ[num], isPP=False; Break[]], {i, Length[d]}, {j, 0, 9}]; isPP]; Select[Range[1, 1000000, 2], Mod[#, 5] > 0 && primeProof[#] &] (* T. D. Noe, Feb 26 2011 *)

A235589 The periodic part of the decimal expansion of m/(m+1), for those m/(m+1) that have pure periods.

Original entry on oeis.org

6, 857142, 8, 90, 923076, 9411764705882352, 947368421052631578, 952380, 9565217391304347826086, 962, 9655172413793103448275862068, 967741935483870, 96, 972, 974358, 97560, 976744186046511627906, 9787234042553191489361702127659574468085106382, 979591836734693877551020408163265306122448

Offset: 1

Bill McEachen, Jan 12 2014

A companion sequence stemming from the some of the elements excluded by A156703. The sequence is highly volatile and infinite...as with A156703 the subset elements are encountered in numerical order. a(n) will start with the digit 9 for n>4 I believe. Entries can grow quite large very quickly. Each entry will be encountered once, and they will end in an even digit.

The number of digits of a(n) is given by A002329. - Michel Marcus, Aug 19 2015

			1/2=0.5 non-repeating, so exclude from sequence.
2/3=0.6 repeating, so a(1)=6.
5/6=0.833 (repeating) but has "8" prefix ahead of repeating "3" so exclude from sequence (decimal expansion not purely periodic)
6/7=0.857142 repeating so a(2)=857142.

Subsequence of A259299.

Cf. A156703, A036275, A060283, A114205.

FromDigits@ Flatten@ First@ RealDigits[(# - 1)/#] & /@ Select[Range@ 120, CoprimeQ[#, 10] &] //Rest (* Michael De Vlieger, Aug 18 2015 *)

a(n) = the periodic part of the decimal expansion of (A045572(n+1)-1) / A045572(n+1). - Doug Bell, Aug 17 2015

Missing terms added by Ralf Stephan, Jan 19 2014

Incorrect terms 916, 94 removed and two more terms added by Michael De Vlieger, Aug 18 2015

A243765 Numbers that have all their divisors in A002191 (possible values for sigma(n), A000203).

Original entry on oeis.org

1, 3, 7, 13, 31, 39, 91, 93, 127, 217, 307, 381, 403, 921, 961, 1093, 1209, 1651, 1723, 2149, 2801, 2821, 3279, 3541, 3937, 3991, 4953, 5113, 5169, 7651, 8011, 8191, 8403, 9517, 10303, 10623, 11811, 11973, 12061, 12493, 15339, 17293, 19531, 19607, 22399

Offset: 1

Michel Marcus, Jun 10 2014

nonn

Since 2 does not belong to A002191, all terms are odd.
All primes p that are in A023195 (Prime numbers that are the sum of the divisors of some n), are also in this sequence; and the prime factors of all terms can only belong to A023195.
Up to 10^7, only one term is a prime power: 961=31^2 (being a square, see A038688, A228061 and A243810).

			The divisors of 3 are 1 and 3 that both belong to A002191, 1 as sigma(1) and 3 as sigma(2).
The divisors of 39 are 1, 3, 13 and 39 all of which belong to A002191, 13 as sigma(9) 39 as sigma(18).

Amiram Eldar, Table of n, a(n) for n = 1..2000

Cf. A000203, A002191, A023195.

Cf. A045572 (analog sequence with the sum of proper divisors instead).

N:= 10^6: # to get all terms up to N
A002191:= select(`<=`,{seq(numtheory[sigma](i),i=1..N)},N):
A243765:= select(t -> numtheory[divisors](t) subset A002191, A002191); # Robert Israel, Jun 16 2014

list(lim) = select(n->n<=lim, Set(vector(lim\=1, n, sigma(n))));
isok(n, lists) = {fordiv (n, d, if (!vecsearch(lists, d), return(0))); return(1);}
lista(nn) = {lists = list(nn); for(n=1, nn, if (isok(n, lists), print1(n, ", ")););}

A268044 The odd numbers congruent to {3, 4} mod 5.

Original entry on oeis.org

3, 9, 13, 19, 23, 29, 33, 39, 43, 49, 53, 59, 63, 69, 73, 79, 83, 89, 93, 99, 103, 109, 113, 119, 123, 129, 133, 139, 143, 149, 153, 159, 163, 169, 173, 179, 183, 189, 193, 199, 203, 209, 213, 219, 223, 229, 233, 239, 243, 249, 253, 259, 263, 269, 273, 279, 283, 289, 293, 299

Offset: 1

Mikk Heidemaa, Jan 25 2016

The odd numbers with terminal digit 3 or 9.

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001622, A005408, A045435, A047204, A131229.

Second bisection of A045572.

[5*n-(3-(-1)^n)/2: n in [1..60]]; // Vincenzo Librandi, Jan 25 2016

Table[5 n - (3 - (-1)^n)/2, {n, 1000}] (* or *) Select[ Range [1000], OddQ[#] && MemberQ[{3, 4}, Mod[#, 5]] &]
LinearRecurrence[{1,1,-1},{3,9,13},60] (* Harvey P. Dale, Feb 12 2023 *)

G.f.: x*(3 + 6*x + x^2)/((1 + x)*(1 - x)^2).
a(n) = a(n-1) + a(n-2) - a(n-3).
a(n) = 5*n - (3 - (-1)^n)/2.
a(n) = -A131229(-n+1) with A131229(0) = -3.
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt((5+sqrt(5))/2)*Pi/10 - 3*log(phi)/(2*sqrt(5)), where phi is the golden ratio (A001622). - Amiram Eldar, Apr 15 2023

Edited by Bruno Berselli, Jan 25 2016

A301617 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 1.

Original entry on oeis.org

1, 19, 37, 73, 91, 109, 127, 163, 181, 199, 217, 253, 271, 289, 307, 343, 361, 379, 397, 433, 451, 469, 487, 523, 541, 559, 577, 613, 631, 649, 667, 703, 721, 739, 757, 793, 811, 829, 847, 883, 901, 919, 937, 973, 991, 1009, 1027, 1063, 1081, 1099

Offset: 1

Gary Croft, Mar 24 2018

Numbers == {1, 19, 37, 73} mod 90 with additive sum sequence 1{+18+18+36+18} {repeat ...}. Includes all prime numbers > 7 with digital root 1.

			1+18=19; 19+18=37; 37+36=73; 73+18=91; 91+18=109.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A177704, A045572.

Intersection of A007775 and A017173.

seq(seq(i+90*j,i=[1,19,37,73]),j=0..30); # Robert Israel, Mar 25 2018

LinearRecurrence[{1,0,0,1,-1},{1,19,37,73,91},50] (* Harvey P. Dale, Dec 14 2019 *)

a(n) = 1 + 18 * (n - 1 + n\4) \\ David A. Corneth, Mar 24 2018

Vec(x*(1 + 18*x + 18*x^2 + 36*x^3 + 17*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^60)) \\ Colin Barker, Mar 24 2018

n == {1, 19, 37, 73} mod 90.
a(n + 1) = a(n) + 18 * A177704(n + 1). - David A. Corneth, Mar 24 2018
From Colin Barker, Mar 24 2018: (Start)
G.f.: x*(1 + 18*x + 18*x^2 + 36*x^3 + 17*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
(End)

The missing term 1081 added to the sequence by Colin Barker, Mar 24 2018

A345904 Numbers ending in 1, 3, 7 or 9 that yield no primes if their first digit is changed to another nonzero digit.

Original entry on oeis.org

1071, 1149, 1183, 1401, 1509, 1611, 1771, 1773, 1809, 1891, 1921, 2071, 2149, 2183, 2401, 2509, 2611, 2771, 2773, 2809, 2891, 2921, 3071, 3149, 3183, 3401, 3509, 3611, 3771, 3773, 3809, 3891, 3921, 4071, 4149, 4183, 4401, 4509, 4611, 4771, 4773, 4809, 4891, 4921, 5071

Offset: 1

Tanya Khovanova, Jun 29 2021

If a number is in this sequence, then all the numbers with the first digit changed to another nonzero digit are also in this sequence.

Numbers ending in 0, 2, 4, 5, 6, and 8 are not included by definition, because they are composite independently of the other digits.

			1071, 2071, 3071, 4071, 5071, 6071, 7071, 8071 and 9071 are all composite numbers. Thus, all of them are in this sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Subsequence of A045572.

q:= n-> (l-> l[1] in [1, 3, 7, 9] and andmap(not isprime, [seq(parse
   (cat(j, seq(l[-i], i=2..nops(l)))), j=1..9)]))(convert(n, base, 10)):
select(q, [$1..5080])[];  # Alois P. Heinz, Jun 29 2021

Select[Range[1, 6000, 2], Take[IntegerDigits[#], -1] != {5} && CompositeQ[Table[FromDigits[Prepend[Rest[IntegerDigits[#]], n]], {n, 9}]] == {True, True, True, True, True, True, True, True, True} &]
Select[Range[1,5100,2],NumberDigit[#,0]!=5&&NoneTrue[FromDigits/@Table[ PadRight[ {d},IntegerLength[#],IntegerDigits[#]],{d,9}],PrimeQ]&] (* Harvey P. Dale, Sep 24 2021 *)

from sympy import isprime
def ok(n):
    if n < 10 or n%10 not in {1, 3, 7, 9}: return False
    s = str(n)[1:]
    return all(not isprime(int(d+s)) for d in "123456789")
print(list(filter(ok, range(1, 6000, 2)))) # Michael S. Branicky, Jun 29 2021

A350385 Minimum number of zeros that need to be added to x_n ones such that a combination of these zeros and ones can make a number b with the property gcd(b, rev(b)) = digitsum(b) = x_n where x_n is coprime to 10.

Original entry on oeis.org

0, 1, 36, 1, 12, 66, 3, 3, 6, 4, 2, 3, 4, 10, 75, 16, 7, 3, 3, 7, 2, 5, 4, 3, 3, 6, 2, 2, 2, 10, 10, 5, 2, 3, 2, 2, 2, 4, 3, 10, 304, 4, 3, 3, 1, 3, 12, 6, 124

Offset: 1

Ruediger Jehn, Jan 05 2022

Only for numbers x_n coprime to 10 (A045572, i.e., numbers ending with 1,3,7 or 9) do there exist numbers b such that gcd(b, rev(b)) = x_n and digitsum(b) = x_n (rev(b) is the digit reversal of b, e.g., rev(123) = 321). If b must consist only of zeros and ones, the smallest values of b that satisfy these two constraints are converted to decimal and form sequence A348480. The question arose: How many zeros are needed for each x_n to find a matching number b? In most cases just a few zeros are enough, but some numbers, such as 7, 11, 13 and 37, require more zeros than ones and the corresponding b is called a "long solution". x_n = 101 requires 304 zeros because 101 is a porous number (see A337832).

			a(2) = 1 because x_2 = 3 and if you add 1 zero to 3 ones you can form b = 1011 for which gcd(b,rev(b)) = digitsum(b) = 3.

Rüdiger Jehn, A new 200 Euro math puzzle, Youtube video, Sep 17 2021.
Rüdiger Jehn, Long Solutions of Sequence A348480 of the On-Line Encyclopedia of Integer Sequences, arXiv:2201.00710 [math.GM], 2022.
Rüdiger Jehn, Porous Numbers, arXiv:2104.02482 [math.GM], 2021.

Cf. A045572, A333666, A337832, A348480.

A348480 = [1, 11, 4399137296449, 767, 4543829, 302306413101798081695809]
for m in A348480:
    print(bin(m)[2:].count('0'))

cp's OEIS Frontend

A045807 7-ish numbers (end in 53, 71, 79, 97).

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A158942 Nonsquares coprime to 10.

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A179192 Numbers n, not relatively prime to 10, such that the decimal form of the period of 1/n is prime.

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A186694 Numbers ending in 1, 3, 7 or 9 such that changing any one decimal digit produces a composite number.

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A235589 The periodic part of the decimal expansion of m/(m+1), for those m/(m+1) that have pure periods.

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A243765 Numbers that have all their divisors in A002191 (possible values for sigma(n), A000203).

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A268044 The odd numbers congruent to {3, 4} mod 5.

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A301617 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 1.

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