A142330 -id:A142330 - cp's OEIS frontend

A202112 Numbers n such that 90n + 79 is prime.

0, 3, 4, 6, 7, 11, 13, 15, 17, 18, 19, 20, 24, 29, 33, 35, 36, 38, 41, 45, 46, 52, 56, 57, 60, 61, 62, 63, 64, 68, 70, 71, 75, 81, 82, 83, 84, 89, 90, 91, 94, 95, 96, 103, 104, 106, 111, 112, 115, 119, 122, 123, 124, 125, 129, 130, 132, 133, 137, 139, 146

Offset: 1

J. W. Helkenberg, Dec 11 2011

This sequence was generated by adding 14 Fibonacci-like sequences [See: PROG]. Looking at the format 90n+79 modulo 9 and modulo 10 we see that all entries of A142330 have digital root 7 and last digit 9. (Reverting the process is an application of the Chinese remainder theorem.) The 14 Fibonacci-like sequences are generated (via the p and q values given in the PERL program) from the base p,q pairs 79*91, 19*61, 37*7, 73*43, 11*89, 29*71, 47*17, 83*53, 13*13, 31*49, 67*67, 23*23, 41*59, 77*77.

Cf. A181732, A198382, A195993, A196000, A196007, A201739, A201734, A201804, A201816, A201817, A201818, A201820, A201822, A202101, A202104, A202105, A202110.

Select[Range[0, 200], PrimeQ[90 # + 79] &]

is(n)=n%90==79 && isprime(n) \\ Charles R Greathouse IV, Jun 01 2016

a(n) ~ 24n log n. - Charles R Greathouse IV, Jun 01 2016

A202113 Numbers n such that 90n + 61 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 11, 13, 14, 20, 21, 23, 24, 25, 29, 31, 34, 36, 37, 39, 43, 44, 45, 46, 50, 51, 53, 55, 56, 58, 62, 64, 67, 69, 70, 71, 77, 81, 84, 90, 93, 94, 99, 101, 102, 104, 105, 106, 108, 109, 112, 114, 116, 119, 120, 123, 125, 127, 132, 135, 136

Offset: 1

J. W. Helkenberg, Dec 11 2011

This sequence was generated by adding 14 Fibonacci-like sequences [See: PROG]. Looking at the format 90n+61 modulo 9 and modulo 10 we see that all entries of A142330 have digital root 7 and last digit 1. (Reverting the process is an application of the Chinese remainder theorem.) The 14 Fibonacci-like sequences are generated (via the p and q values given in the Perl program) from the base p,q pairs 61*91, 19*79, 37*43, 73*7, 11*71, 29*89, 47*53, 83*17, 13*67, 31*31, 49*49, 23*77, 41*41, 59*59.

Cf. A181732, A198382, A195993, A196000, A196007, A201739, A201734, A201804, A201816, A201817, A201818, A201820, A201822, A202101, A202104, A202105, A202110, A202112.

Select[Range[0, 200], PrimeQ[90 # + 61] &]

is(n)=isprime(90*n+61) \\ Charles R Greathouse IV, Feb 17 2017

A333040 Even numbers m such that sigma(m) < sigma(m-1).

Original entry on oeis.org

46, 106, 118, 166, 226, 274, 298, 316, 346, 358, 406, 466, 514, 526, 562, 586, 622, 694, 706, 766, 778, 826, 838, 862, 886, 946, 1006, 1114, 1126, 1156, 1186, 1198, 1282, 1306, 1366, 1396, 1426, 1486, 1522, 1546, 1576, 1594, 1618, 1702, 1726, 1756

Offset: 1

Bernard Schott, Mar 22 2020

nonn

The even terms of A333039 represent about only 7% of the data, so they are proposed in this sequence. Some of these integers are semiprimes with for example these two families:
1) m = 2*p with p prime of the form k^2+k+3 is in A027753. The first few terms are: 46, 118, 226, 766, ... but not all the integers of this form are terms; the first 3 counterexamples are 6, 10, 1018 (see examples).
2) m = 2*p with p prime = (r*s*t+1)/2 and 2A234103. The first few terms are: 106, 166, 274, 346, 358, ... but not all the integers of this form are terms; the first 3 counterexamples are 386, 898 and 958 (see examples).
There is also this subsequence of even m = 2^2*p where p prime, congruent to 34 mod 45, is in A142330. The first few terms are: 316, 1396, 1756, 2416, ... but not all the integers of this form are terms; the first counterexample that comes from the 8th term of A142330 is 5716.
Even (and odd) numbers such that sigma(m) = sigma(m-1) are in A231546.

			166 = 2*83 and 165 = 3*5*11, as 252 = sigma(166) < sigma(165) = 288, hence 166 is a term.
386 = 2*193 and 385 = 5*7*11, as 582 = sigma(386) > sigma(385)= 576, hence 386 is not a term.
766 = 2*383 where 383 = 19^2+19+3 and 765 = 3^2*5*13, as 1152 = sigma(766) < sigma(765) = 1404, hence 766 is a term.
1018 = 2*509 where 509 = 22^2+22+3, and 1017 = 3^2*113, as 1530 = sigma(1018) > sigma(1017) = 1482, hence 1018 is not a term.

J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 620 pp. 82, 280, Ellipses Paris 2004.

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

Intersection of A005843 and A333039.
Subsequence of A333038.
Cf. A231546.

filter:= n -> numtheory:-sigma(n) < numtheory:-sigma(n-1):
select(filter, [seq(i,i=2..2000,2)]); # Robert Israel, Mar 29 2020

Select[2 * Range[1000], DivisorSigma[1, #] < DivisorSigma[1, #-1] &] (* Amiram Eldar, Mar 24 2020 *)

isok(m) = !(m%2) && (sigma(m) < sigma(m-1)); \\ Michel Marcus, Mar 22 2020

A046804 a(n) = p mod (p mod 10) where p = prime(n).

Original entry on oeis.org

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

Offset: 1

Enoch Haga

From Robert G. Wilson v, Feb 12 2014: (Start)
a(n)=0 iff p ends in 1 (A030430) or is a single-digit prime, i.e., 2, 3, 5 or 7 (n = 1, 2, 3 or 4),
a(n)=3 iff n is in A142087,
a(n)=6 iff n is in A142094,
a(n)=7 iff n is in A142330,
a(n)=8 iff n is in A142335.
a(n) can never be 9. (End)

			prime(10) = 29, so a(10) = 29 mod 9 = 2.

Idea derived from "The Creation of New Mathematics: An Application of the Lakatos Heuristic," pp. 292-298 of Philip J. Davis and Reuben Hersh, The Mathematical Experience, Houghton Mifflin Co, 1982. ISBN 0-395-32131-X.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for sequences related to final digits of numbers

Mod[#,Last[IntegerDigits[#]]]&/@Prime[Range[110]] (* Harvey P. Dale, Jan 23 2013 *)
Mod[#,Mod[#,10]]&/@Prime[Range[110]] (* Harvey P. Dale, Aug 22 2020 *)

Name edited by Jon E. Schoenfield, Jan 19 2023

cp's OEIS Frontend

A202112 Numbers n such that 90n + 79 is prime.

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A202113 Numbers n such that 90n + 61 is prime.

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A333040 Even numbers m such that sigma(m) < sigma(m-1).

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A046804 a(n) = p mod (p mod 10) where p = prime(n).

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