A060159 -id:A060159 - cp's OEIS frontend

A060291 Number of twin Harshads, including overlaps, whose sum is prime and where the 2nd Harshad is <= 10^n.

7, 8, 21, 60, 278, 1520, 9583, 61835, 419705, 3023007, 22597508

Offset: 1

Enoch Haga, Mar 24 2001

			a(1)=7 because there are 7 pairs of Harshads whose sum is prime and the 2nd Harshad in the pair is <= 10; these are 1+2=3, 2+3=5, 3+4=7, 5+6=11, 6+7=13, 8+9=17, 9+10=19. (Another sequence does not permit overlapping Harshad numbers.)

Cf. A060288, A060289, A060290, A005349, A060159.

harshadQ[n_] := Divisible[n, Plus @@ IntegerDigits[n]]; c = 0; p = 10; s = {}; n = 0; k = 2; q1 = True; While[n < 6, q2 = harshadQ[k]; If[q1 && q2 && PrimeQ[2*k-1], c++; If[k > p, n++; AppendTo[s, c-1]; p *= 10]]; q1 = q2; k++]; s (* Amiram Eldar, Jan 19 2021 *)

Generate the twin Harshads whose sum is prime. Count how many there are where the 2nd Harshad in the pair is <= to a consecutive power of 10.

a(8)-a(11) from Amiram Eldar, Jan 19 2021

A144378 Initial term of a series of exactly n consecutive non-Niven (or Harshad) numbers.

Original entry on oeis.org

11, 22, 37, 136, 13, 64, 73, 163, 91, 1730, 289, 1639, 379, 1660, 2737, 919, 559, 14878, 7561, 5671, 9753, 2890, 7777, 4888, 5785, 5590, 27973, 47872, 28681, 22681, 3785, 36184, 46281, 71281, 6481, 48952, 48763, 64978, 119773, 69782, 77881, 55973

Offset: 1

Sergio Pimentel, Sep 18 2008

Multiples of 18 seem to be the high water marks, while terms of the form 18n - 1 seem to be the valleys of this sequence.
Many terms end in '81' for some reason.
This sequence is analog to A060159 with non-Niven numbers.
This sequence is infinite, as opposed as A060159.

			a(5) = 13 since 13, 14, 15, 16 and 17 are all non-Niven numbers and this is the first occurrence of exactly 5 non-Niven numbers.

Klaus Brockhaus, Table of n, a(n) for n=1..125
Eric Weisstein's World of Mathematics, Harshad Number

Cf. A005349, A082516, A060159, A144261, A144262.

digitsum(n) = {local(s=0); while(n, s+=n%10; n\=10); s}
{m=120000; z=42; w=vector(z); n=1; while(n<=m, while(n%digitsum(n)==0, n++); a=n; c=0; while(n%digitsum(n)>0, n++; c++); if(c<=z&&w[c]==0, w[c]=a)); j=1; while(j<=z&&w[j]>0, print1(w[j], ","); j++)} /* Klaus Brockhaus, Sep 24 2008 */

a(2), a(22), a(42) corrected by Klaus Brockhaus, Sep 24 2008

A359839 Numbers k such that k, k + 1 and k + 2 are 3 consecutive Niven (Harshad) numbers that are also divisible by a square.

Original entry on oeis.org

2023, 4912, 12103, 17575, 23273, 51424, 52675, 60399, 78650, 80800, 87723, 93624, 100303, 112624, 117962, 121224, 122875, 182182, 193075, 200752, 228175, 235024, 245725, 245726, 249500, 263275, 306963, 320704, 333475, 373490, 403675, 416583, 421072, 444624, 448000

Offset: 1

Bernard Schott, Jan 15 2023

Equivalently, smallest of 3 consecutive numbers each divisible by a square and also divisible by the sum of their digits (Niven numbers).

			2023 = 7 * 17^2 = 289 * (2+0+2+3); 2024 = 506 * 2^2 = 253 * (2+0+2+4) and 2025 = 81 * 5^2 = 225 * (2+0+2+5) hence 2023 is a term.

Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 48, p. 18, Ellipses, Paris, 2008.
Jean-Marie De Koninck, Those Fascinating Numbers, Entry 110, p. 36, American Mathematical Society, 2009.

Curtis Cooper and Robert E. Kennedy, On consecutive Niven numbers, Fibonacci Quarterly, Vol. 21, No. 2 (1993), pp. 146-151.
Giovanni Resta, Harshad numbers.
Wikipedia, Harshad number.

Intersection of A070258 and A154701.
Cf. A013929, A068781, A235578.
Cf. A005349, A060159, A141769, A330927, A330928, A330929, A330930.

q[n_] := Divisible[n, Total@IntegerDigits[n]] && ! SquareFreeQ[n]; tri = q /@ Range[3]; seq = {}; Do[tri = Join[Rest[tri], {q[k]}]; If[And @@ tri, AppendTo[seq, k - 2]], {k, 3, 5*10^5}]; seq (* Amiram Eldar, Jan 15 2023 *)

More terms from Amiram Eldar, Jan 15 2023

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A060291 Number of twin Harshads, including overlaps, whose sum is prime and where the 2nd Harshad is <= 10^n.

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A144378 Initial term of a series of exactly n consecutive non-Niven (or Harshad) numbers.

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A359839 Numbers k such that k, k + 1 and k + 2 are 3 consecutive Niven (Harshad) numbers that are also divisible by a square.

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